Flux-Variability Analysis for Optimization

Background

What is Metabolic Flux Analysis?

Metabolic Flux balance analysis (FBA) is an application of linear programming to metabolic network that uses the stoichiometric coefficients for each reaction in the system as the set of constraints for the optimization. Simply, it is a mathematical method for analyzing metabolism. This analysis requires the Steady State Assumption, which is, in chemistry, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. In addition, Flux Variability Analysis (FVA) is an extension of Flux balance analysis. FVA determines the ranges of fluxes that correspond to an optimal solution determined through FBA. Hence, FBA is able to calculate the full range of numerical values for each reaction flux within the network.

What are the constraints in model?

Networks can be encoded as stoichiometric matrices (S), in which each row represents a unique metabolite and each column represents a biochemical reaction. The entries in each column of this matrix are the stoichiometric coefficients of the metabolites in the reaction. Metabolites are consumed have a negative coefficient and metabolites that are produced have a positive coefficient. Since most reactions involve only a few metabolites, S is a sparse matrix. The size of S is m*n for a network with m metabolites and n reactions.

Why use Flux Variability Analysis?

Biological systems often contain redundancies that contribute to their robustness. However, flux balance analysis only returns a single flux distribution that corresponds to maximal growth under given growth conditions regardless alternate optimal solutions may exist. FVA is capable to exanimate these redundancies by calculating the full range of numerical values for each reaction flux in a network. Consequently, FVA can be employed to study the entire range of achievable cellular functions as well as the redundancy in optimal phenotypes.

Introduction

What is it modeling?

It models flux rate of metabolic pathways responding to different growth media conditions. The model is expected to generate an optimal set of metabolites that should be added to growth media in order to improve production rate.

Why needs this kind of model?

Same as chemical reactions need optimal environmental conditions to achieve maximum production rate, microbes also require optimized growth conditions to accomplish their tasks in maximum speed. In industrial scale, the optimal conditions for chemical reactions could increase the production rate in hundreds times which means millions of money. This principle also works even if taking biology pathways over chemical reactions. The difference is that in chemical way, the conditions of enzyme, pH, temperature and pressure are more important; however, in microbiological method, the conditions of growth media is more crucial. Further, the selection of media compounds is one of the most significant conditions for growth media. If a model can predict an optimal set of metabolites that need to be added into media, there will be tons of time and lots of money got saved.

How dose the program work?

This program is built upon constraint-based reconstruction analysis and flux variability analysis. It uses published E.coli (iAF1260) and E.coli core models as base chassis. Upon those models, we reconstruct new models of interests. Specifically, new reactions corresponding to the genes we try to engineer into E.coli from other organisms will be added into base chassis. By running flux variability analysis, program will give different sets of flux rates based on distinct constraints. Finally, the program will analysis those data with designed algorithm to generate a set of media compounds that is expected to accelerate production rate.

Algorithm

Conceptual

How to improve the products flux rates through data from FVA? The answer was remained unknown. There were only two things being noticed. One was that FVA could determine full range of numerical values for each reaction flux within the network and its output were able to use for analyze, and the other one was the biomass rate normally had trade-off relation with production rate. Since biomass rate reflects the growth condition, cell must have positive value of biomass flux rate in order to producing. On the other hand, the production flux rate should be higher than zero as well. This implied among all possible set of fluxes, the optimal flux set should locate a place where growth rate multiplies production rate is maximum.

Once the optimal flux rate of biomass was obtained, the value would be set as a new constraint of biomass. Then flux variability analysis would find out the full range of numerical values for each reaction flux within the network that was restricted to the new biological objective.

The differences of values for each reaction in a set of flux that maximized production rate and a set of flux that minimized production rate became interesting. By comparing two sets of fluxes based on visual maps, the results showed some reactions had higher flux rates in production maximum set than production minimum set, some were higher in production minimum set than production maximum set and some had opposite flux directions as most of biological reactions were reversible. In chemical, adding the amount of reactants would force the reactions equilibrium to move forwards, and adding the amount of products could drive the reactions equilibrium to go backwards. Consequently, the question becomes how to find out metabolites that need additional amount to improve the production rate.

One of the possible solutions could be comparing two sets of fluxes, determining differences of each reaction between two sets and changing constraints according to reaction needs. For example, if a metabolite needs more in production maximum set than production minimum set, then add more amount of this metabolite by change constraints to improve the production. However, in reality, cell could only uptake limited kinds of metabolites. Some metabolites were able to be produced by cell but not able to be absorbed from growth media. Hence, only the metabolites that had natural transporters in cell would count.

Last but not least, to improve production by adding more metabolites to growth media, the analysis should start from a model that was built upon glucose minimum growth media.

Concrete

Precondition: The original model is built with glucose minimum media.

1. Define relationship between growth rate and production rate.

2. Find out the optimal growth rate that can maximize the production.

3. Get the difference percentage of flux rate for each reaction between production maximum set and production minimum set.

4. Collect all reactions have difference percentage between two sets that exceed threshold.

5. Score each compound in all collected reactions (Initial score is zero for each compound).

5.1 The difference of flux rates of one reaction from production maximum set to production minimum set is added to the score for all reactants of this reaction.

5.2 The difference of flux rates of one reaction from production minimum set to production maximum set is added to the score for all products of this reaction.

5.3 Repeat 3.1 to 3.2 till all collected reactions are analyzed.

6. Determine whether compounds with positive scores have natural transporters in cell. If so, mark the compound as candidate.

7. Add each candidate to growth media, and run FVA under optimal growth rate computed in Step 2. Compare the production rate from novel model to that from raw model, if the rate is improved, mark as effector.

Demo

Drawbacks

Validation

Source Code

Documents

Reference