Difference between revisions of "Team:DTU-Denmark/Model"
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$$ | $$ | ||
| − | c_i( | + | c_i(r) = \sum\limits_{j=1}^R a_{ij}r_j |
$$ | $$ | ||
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$$ | $$ | ||
\begin{split} | \begin{split} | ||
| − | \max & f(x) \\ | + | \max \quad & f(x) \\ |
s.t. \quad c_i &= 0 \quad \forall i \\ | s.t. \quad c_i &= 0 \quad \forall i \\ | ||
c_j &\geq 0 \quad \forall j | c_j &\geq 0 \quad \forall j | ||
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<h4 class="h4">Solution to FBA and shadow prices</h4> | <h4 class="h4">Solution to FBA and shadow prices</h4> | ||
| − | + | ||
| − | For each constraint | + | <p> |
| − | Now, define the Lagrangian function: | + | For each constraint \(c_i\) on the metabolite \(m_i\), let \(\lambda_i\) be the associated Lagrange multiplier or <b>shadow price</b>, let \(\lambda = (\lambda_i,...,\lambda_M)\) and let \(c = (c_i,...,c_M, c_{M+1}...,c_{M+R})\). |
| − | + | </p> | |
| − | + | <p> | |
| − | + | Now, define the Lagrangian function: | |
| − | + | ||
| − | + | $$ | |
| − | + | \mathcal{L}(\bm{\lambda},\bm{c},\bm{r}) = f(\bm{r}) - \bm{\lambda}^T\bm{c} | |
| − | + | $$ | |
| − | + | ||
| − | + | The linear program is then solved, by solving the system, | |
| − | + | ||
| − | + | $$ | |
| − | + | \begin{pmatrix} | |
| − | + | \bigtriangledown \mathcal{L}(\bm{\lambda},\bm{c},\bm{r}) \\ | |
| − | + | \bm{c}(r) | |
| − | + | \end{pmatrix} | |
| + | = | ||
| + | \begin{pmatrix} | ||
| + | 0 \\ 0 | ||
| + | \end{pmatrix} | ||
| + | $$ | ||
| + | |||
| + | 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. Further mathematical details are omitted. REFERENCE. | ||
| + | </p> | ||
<h3 class="h3">model validation</h3> | <h3 class="h3">model validation</h3> | ||
Revision as of 13:59, 18 October 2016
Genome Scale Modelling
What medium should I use to grow my microorganism? Which genes should I mutate to make my organism produce more beta-carotene? Let genome scale modelling help you design your dream organism. A genome scale model (GSM) is a computational representation of the metabolic network, containing information about the metabolites, reactions, genes and more. Using a GSM of Y.Lip, we explored how Y.Lip grows on different substrates. We compared it to S.cer to see the pros and cons of each organism. We also found out what genes need to be manipulated to optimize beta-carotene production.
Section 1
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Someone famous in Source Title
iGEM info Modeling
Mathematical models and computer simulations provide a great way to describe the function and operation of BioBrick Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior in conjunction with experiments in the wetlab.
Inspiration
Here are a few examples from previous teams:
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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 \(c = (c_i,...,c_M, c_{M+1}...,c_{M+R})\).
Now, define the Lagrangian function: $$ \mathcal{L}(\bm{\lambda},\bm{c},\bm{r}) = f(\bm{r}) - \bm{\lambda}^T\bm{c} $$ The linear program is then solved, by solving the system, $$ \begin{pmatrix} \bigtriangledown \mathcal{L}(\bm{\lambda},\bm{c},\bm{r}) \\ \bm{c}(r) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ 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. 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
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 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.
