Simulate models
===============

**cameo** uses the model data structures defined by
`cobrapy <https://opencobra.github.io/cobrapy/>`__, our favorite
COnstraints-Based Reconstruction and Analysis tool for Python. **cameo**
is thus 100% compatible with cobrapy. For efficiency reasons, however,
cameo implements its own simulation methods that take advantage of a
more advanced solver interface.

Primer: Constraint-Based Modeling
---------------------------------

Constraint-based modeling is a powerful modeling framework for analyzing
metabolism on the genome scale (`McCloskey et al.,
2013 <http://www.ncbi.nlm.nih.gov/pubmed/23632383>`__). For a model that
encompasses :math:`n` reactions that involve :math:`m` metabolites,
:math:`\mathbf{S}` is a matrix of dimension :math:`m \times n` that
encodes the stoichiometry of the metabolic reaction system; it is
usually referred to as stoichiometric matrix. Assuming that the system
is in a steady state—the concentration of metabolites are constant—the
system of flux-balances can be formulated as:

.. math::


   \begin{align}
   \mathbf{S} \mathbf{v} = 0\,,
   \end{align}

where :math:`\mathbf{v}` is the vector of flux rates. With the addition
of a biologically meaningful objective, flux capacity constraints,
information about the reversibility of reactions under physiological
conditions, an optimization problem can be formulated that can easily be
solved using `linear
programming <https://en.wikipedia.org/wiki/Linear_programming>`__.

For example, given the maximization of growth rate as one potential
biological objective :math:`v_{biomass}` (i.e., the flux of an
artificial reaction that consumes biomass components in empirically
determined proportions), assuming that the cell is evolutionary
optimized to achieve that objective, incorporating knowledge about
reaction reversibility, uptake and secretion rates, and maximum flux
capacities in the form of lower and uppers bounds
(:math:`\mathbf{v}_{lb}` and :math:`\mathbf{v}_{ub}`) on the flux
variables :math:`\mathbf{v}`, one can formulate and solve an
optimization problem to identify an optimal set of flux rates using Flux
Balance Analysis (FBA):

.. math::


   \begin{align}
    Max ~ & ~ Z_{obj} = \mathbf{c}^{T} \mathbf{v}\\
    \text{s.t.}~ & ~ \mathbf{S} \mathbf{v} = 0 \\
    ~ & ~ \mathbf{v}_{lb} \leq \mathbf{v} \leq \mathbf{v}_{ub} \,.
   \end{align}

Flux Balance Analysis
---------------------

Load a model.

.. code:: ipython3

    from cameo import load_model
    model = load_model('iJO1366')

In cameo, flux balance analysis can be performed with the function
`fba`.

.. code:: ipython3

    from cameo import fba
    %time fba_result = fba(model)


.. parsed-literal::

    CPU times: user 256 ms, sys: 6.4 ms, total: 262 ms
    Wall time: 283 ms


Basically, `fba` calls `model.optimize()` and wraps the optimization
solution in a `FluxDistributionResult` object. The maximum objective
values (corresponding to a maximum growth rate) can be obtained from
`result.objective_value`.

.. code:: ipython3

    fba_result.data_frame




.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
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        .dataframe tbody tr th {
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        <tr style="text-align: right;">
          <th></th>
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          <th>12DGR180tipp</th>
          <td>0.000000</td>
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        <tr>
          <th>12DGR181tipp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>12PPDRtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>12PPDRtpp</th>
          <td>0.000000</td>
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        <tr>
          <th>12PPDStex</th>
          <td>0.000000</td>
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        <tr>
          <th>12PPDStpp</th>
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        <tr>
          <th>14GLUCANabcpp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>14GLUCANtexi</th>
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        <tr>
          <th>23CAMPtex</th>
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        <tr>
          <th>23CCMPtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>23CGMPtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>23CUMPtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>23DAPPAt2pp</th>
          <td>0.000000</td>
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        <tr>
          <th>23DAPPAtex</th>
          <td>0.000000</td>
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        <tr>
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        <tr>
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        <tr>
          <th>23PDE7pp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>23PDE9pp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>26DAHtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>2AGPA120tipp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>2AGPA140tipp</th>
          <td>0.000000</td>
        </tr>
        <tr>
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        <tr>
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        <tr>
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        <tr>
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          <td>0.000000</td>
        </tr>
        <tr>
          <th>...</th>
          <td>...</td>
        </tr>
        <tr>
          <th>VALTRS</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>VALabcpp</th>
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        <tr>
          <th>VALt2rpp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>VALtex</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>VPAMTr</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>WCOS</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>X5PL3E</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>XAND</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>XANt2pp</th>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>XANtex</th>
          <td>0.000000</td>
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          <td>0.000000</td>
        </tr>
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          <th>XTSNt2rpp</th>
          <td>0.000000</td>
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          <th>XTSNtex</th>
          <td>0.000000</td>
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        <tr>
          <th>XYLI1</th>
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          <td>0.000000</td>
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        <tr>
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        <tr>
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        <tr>
          <th>ZN2abcpp</th>
          <td>0.000000</td>
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        <tr>
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          <td>0.000000</td>
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        <tr>
          <th>ZN2tpp</th>
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          <th>Zn2tex</th>
          <td>0.000335</td>
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      </tbody>
    </table>
    <p>2583 rows × 1 columns</p>
    </div>



Flux distributions can be visualized using
`escher <https://escher.github.io>`__ :

.. code:: ipython3

    fba_result.display_on_map("iJO1366.Central metabolism")



.. raw:: html

    
    
    
    
    
    
    
    
      <link rel="stylesheet" href="//unpkg.com/escher-vis@1.6.0/css/dist/builder.min.css"/>
    
    
    
    
    <div style="height: 500px; width: 100%;" id="yrqoxvdtwi"></div>
    
    <script>
     /* Closure to make sure data only applies to this map. */
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         /* data from python */
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             // for the static site, use the url attributes for the map and model
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             if (data.static_site_index !== null) {
                 escher.static.load_map_model_from_url(data.map_download_url, data.model_download_url,
                                                       data.static_site_index, data.options, load);
             } else {
                 load(data.map_data, data.model_data, data.options);
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         // Load dependencies and draw
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               // messes up unpkg links. Honestly, I'm not sure if this will lead
               // to downstream bugs in the notebook, but they really shouldn't be
               // doing this cache-busting anyway.
               urlArgs: null,
             })
             require([ 'escher' ], function(escher) {
                 draw_escher(escher);
             })
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         // Put this at the end so Chrome doesn't cut off the code above.
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Parsimonious Flux Balance Analysis
----------------------------------

Parsimonious flux balance analysis (`Lewis et al.,
2010 <http://www.ncbi.nlm.nih.gov/pubmed/20664636>`__), a variant of
FBA, performs FBA in in a first step to determine the maximum objective
value :math:`Z_{obj}`, fixes it in form of an additional model
constraint (:math:`\mathbf{c}^{T} \mathbf{v} \ge Z_{obj}`), and then
minimizes the :math:`L_1` norm of :math:`\mathbf{v}`. The assumption
behind pFBA is that cells try to minimize flux magnitude as well in
order to keep protein costs low.

.. math::


   \begin{align}
    Max ~ & ~ \lvert \mathbf{v} \rvert\\
    \text{s.t.}~ & ~ \mathbf{S} \mathbf{v} = 0 \\
    & ~ \mathbf{c}^{T} \mathbf{v} \ge Z_{obj} \\
    ~ & ~ \mathbf{v}_{lb} \leq \mathbf{v} \leq \mathbf{v}_{ub} \,.
   \end{align}

In cameo, pFBA can be performed with the function `pfba`.

.. code:: ipython3

    from cameo import pfba
    %time pfba_result = pfba(model)


.. parsed-literal::

    CPU times: user 6.46 s, sys: 77 ms, total: 6.53 s
    Wall time: 10.3 s


The `objective_function` value is :math:`\lvert \mathbf{v} \rvert` …

.. code:: ipython3

    pfba_result.objective_value




.. parsed-literal::

    699.0222751839516



… which is smaller than flux vector of the original FBA solution.

.. code:: ipython3

    abs(fba_result.data_frame.flux).sum()




.. parsed-literal::

    765.0977334751339



Setp 2: Simulate knockouts phenotypes
-------------------------------------

Although PFBA and FBA can be used to simulate the effect of knockouts,
other methods have been proven more valuable for that task: MOMA and
ROOM. In *cameo* we implemented a linear version of MOMA.

--------------

Simulating knockouts:

-  Manipulate the bounds of the reaction (or use the shorthand method
   knock_out)

.. code:: ipython3

    model.reactions.PGI




.. raw:: html

    
    <table>
        <tr>
            <td><strong>Reaction identifier</strong></td><td>PGI</td>
        </tr><tr>
            <td><strong>Name</strong></td><td>Glucose-6-phosphate isomerase</td>
        </tr><tr>
            <td><strong>Memory address</strong></td>
            <td>0x011313e908</td>
        </tr><tr>
            <td><strong>Stoichiometry</strong></td>
            <td>
                <p style='text-align:right'>g6p_c <=> f6p_c</p>
                <p style='text-align:right'>D-Glucose 6-phosphate <=> D-Fructose 6-phosphate</p>
            </td>
        </tr><tr>
            <td><strong>GPR</strong></td><td>b4025</td>
        </tr><tr>
            <td><strong>Lower bound</strong></td><td>-1000.0</td>
        </tr><tr>
            <td><strong>Upper bound</strong></td><td>1000.0</td>
        </tr>
    </table>




.. code:: ipython3

    model.reactions.PGI.knock_out()
    model.reactions.PGI




.. raw:: html

    
    <table>
        <tr>
            <td><strong>Reaction identifier</strong></td><td>PGI</td>
        </tr><tr>
            <td><strong>Name</strong></td><td>Glucose-6-phosphate isomerase</td>
        </tr><tr>
            <td><strong>Memory address</strong></td>
            <td>0x011313e908</td>
        </tr><tr>
            <td><strong>Stoichiometry</strong></td>
            <td>
                <p style='text-align:right'>g6p_c --> f6p_c</p>
                <p style='text-align:right'>D-Glucose 6-phosphate --> D-Fructose 6-phosphate</p>
            </td>
        </tr><tr>
            <td><strong>GPR</strong></td><td>b4025</td>
        </tr><tr>
            <td><strong>Lower bound</strong></td><td>0</td>
        </tr><tr>
            <td><strong>Upper bound</strong></td><td>0</td>
        </tr>
    </table>




-  Simulate using different methods:

.. code:: ipython3

    %time fba_knockout_result = fba(model)
    fba_knockout_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]


.. parsed-literal::

    CPU times: user 104 ms, sys: 4.13 ms, total: 108 ms
    Wall time: 124 ms




.. parsed-literal::

    0.9761293262947268



.. code:: ipython3

    %time pfba_knockout_result = pfba(model)
    pfba_knockout_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]


.. parsed-literal::

    CPU times: user 5.78 s, sys: 38.7 ms, total: 5.82 s
    Wall time: 7.48 s




.. parsed-literal::

    0.9761293262947268



MOMA and ROOM rely on a reference (wild-type) flux distribution. We can
use the one previously computed.

**Parsimonious FBA references seem to produce better results using this
methods**

.. code:: ipython3

    from cameo.flux_analysis.simulation import room, lmoma

.. code:: ipython3

    %time lmoma_result = lmoma(model, reference=pfba_result.fluxes)
    lmoma_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]


.. parsed-literal::

    CPU times: user 27.4 s, sys: 201 ms, total: 27.6 s
    Wall time: 35.6 s




.. parsed-literal::

    0.8724092397035721



ROOM is a difficult computational problem. If the bounds of the system
are not large enough, it can take many hours to simulate. To improve the
speed of the simulation and the chances of finding a solution, we
increase the bounds.

.. code:: ipython3

    for reaction in model.reactions:
        if reaction.upper_bound == 1000:
            reaction.upper_bound = 99999999
        if reaction.lower_bound == -1000:
            reaction.lower_bound = -99999999

.. code:: ipython3

    %time room_result = room(model, reference=pfba_result.fluxes)
    room_result[model.reactions.BIOMASS_Ec_iJO1366_core_53p95M]


.. parsed-literal::

    CPU times: user 35 s, sys: 257 ms, total: 35.3 s
    Wall time: 44.4 s




.. parsed-literal::

    0.9519006583451706