Overview

A Genome-scale-model (GSM) is a representation of the metabolic network in a form that computers can understand, and make calculations to predict the fluxes (rates of reactions).

We used GSM for two purposes,

1. Explore how Yarrowia Lipolytica (Y.Lip) grows in different environments, and identify optimal conditions.
2. Predict what genes can be amplified to enhance beta-carotene yield.

Theory

Why? to explore how to grow y.lip efficiently. To optimize b-carotene produciton by amplifying or deleting genes.

This is done by using genome-scale modelling,

And we found that...

Genome-scale Modelling

The following explains the mathematical basis for genome-scale modeling (GSM). In short, a metabolic network can be modeled as linear equations and optimized by solving the corresponding linear program. The output is the set of reaction rates for all reactions in the network along with the set of shadow prices associated with the metabolites. Both play important roles in the subsequent phenotype phase plane analysis.

A simple metabolic network

Consider a very simple metabolic network with two metabolites, \(A\) and \(B\), where \(A\) flows into the cell with rate \(r_1\), is converted into \(B\) with rate \(r_2\), which is then excreted from the cell with rate \(r_3\): $$ \begin{split} & \xrightarrow{r_1} A \\ A & \xrightarrow{r_2} B \\ B & \xrightarrow{r_3} \\ \end{split} $$ Under a steady-state assumption, i.e. where all metabolite levels are constant and no metabolite can be accumulated in the system, the formation and degradation rates for each metabolite must cancel each other. For the network above, this is equivalent to the following set of constraints: $$ \begin{split} A:& \quad r_1 - r_2 = 0 \\ B:& \quad r_2 - r_3 = 0 \end{split} $$

The linear program

Now, consider the full genome-scale model of Yarrowia lipolytica with \(M\) = 1683 metabolites and \(R\) = 1985 reactions.

Let \(r = (r_1, ... , r_R)\) be the vector of the \(R\) reaction rates, and similarly let \(m = (m_1, ... ,m_M)\) be the vector of the \(M\) metabolites. The reaction of which the rate is to be maximized, for example the biomass function, is then denoted the objective function and can be expressed as, $$ f(r) = r_{\text{obj}} $$ Now, for \(m_i\), the equality constraint \(c_i\) can be expressed as a sum of all \(r_j\) weighted by the stoichiometric coefficients \(a_{ij}\), where \(a_{ij}\) = 0 for reactions that do not affect the concentration of the considered metabolite. Thus, $$ c_i(r) = \sum\limits_{j=1}^R a_{ij}r_j $$ Which is just a generalization of the constraints of the two-metabolite network presented in the previous section. In addition to the steady-state constraints on the metabolites, for each reaction rate, \(r_j\), there is a lower and an upper bound (\(l_j\) and \(u_j\)). These inequality constraints can be expressed as, $$ \begin{split} c_{j,l} &= r_j \geq l_j \iff r_j - l_j \geq 0 \\ c_{j,u} &= r_j \leq u_j \iff -r_j + u_j \geq 0 \end{split} $$ Having the objective function and the constraints well-stated, the linear program can be formulated in standard form: $$ \begin{split} \max \quad & f(x) \\ s.t. \quad c_i &= 0 \quad \forall i \\ c_j &\geq 0 \quad \forall j \end{split} $$ Solving this program is equal to optimizing the genome-scale model. In order to accomplish that, it is necessary to define the so-called Lagrangian Function.

Solution to FBA and shadow prices

For each constraint \(c_i\) on the metabolite \(m_i\), let \(\lambda_i\) be the associated Lagrange multiplier or shadow price, let \(\lambda = (\lambda_i,...,\lambda_M)\) and let \(c = (c_1,...,c_M, c_{M+1}...,c_{M+R})\). Now, define the Lagrangian function: $$ \mathcal{L}(\lambda,r) = f(r) - \lambda^Tc(r) $$ The solution to the linear program is then found by solving the system, $$ \begin{pmatrix} \bigtriangledown \mathcal{L}(\lambda,c,r) \\ c(r) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ Then, the output will consist of the set \(f(r),r,\lambda)\), i.e. the reaction rates (including the one corresponding to the objective function) and the shadow prices. In our case, the linear program was solved in COBRA, which uses a simplex algorithm by default. The objective function and constraints were based on an SBML model. REFERENCE Further mathematical details are omitted. REFERENCE.

model validation

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phpp

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Adding B_carotene reactions

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what genes to amp? FVA and differential FVA

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what genes to amp FSEOF

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what genes to ko? optgene

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lmoma and room results

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Results

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phpp on y.lip and s.cer on diff. substrates

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what genes to amp and ko

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so whta does this means?

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Section 4

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Section 5

Has ut facer debitis, quo eu agam purto. In eum justo aeterno. Sea ut atqui efficiantur, mandamus deseruisse at est, erat natum cum eu. Quot numquam in vel. Salutatus euripidis moderatius qui ex, eu tempor volumus vituperatoribus has, ius ea ullum facer corrumpit.

Section 6

Has ut facer debitis, quo eu agam purto. In eum justo aeterno. Sea ut atqui efficiantur, mandamus deseruisse at est, erat natum cum eu. Quot numquam in vel. Salutatus euripidis moderatius qui ex, eu tempor volumus vituperatoribus has, ius ea ullum facer corrumpit.

Attributors

Niko, Joao, Mikael, Kristian, Anne Sophie, Patrice, Julian, Anders, Eduard, Martin, Biosustain slack