# Source code for findChemSteadyState

```
#########################################################################
## This program is part of 'MOOSE', the
## Messaging Object Oriented Simulation Environment.
## Copyright (C) 2013 Upinder S. Bhalla. and NCBS
## It is made available under the terms of the
## GNU Lesser General Public License version 2.1
## See the file COPYING.LIB for the full notice.
#########################################################################
"""
This example sets up the kinetic solver and steady-state finder, on
a bistable model of a chemical system. The model is set up within the
script.
The algorithm calls the steady-state finder 50 times with different
(randomized) initial conditions, as follows:
* Set up the random initial condition that fits the conservation laws
* Run for 2 seconds. This should not be mathematically necessary, but
for obscure numerical reasons it is much more likely that the
steady state solver will succeed in finding a state.
* Find the fixed point
* Print out the fixed point vector and various diagnostics.
* Run for 10 seconds. This is completely unnecessary, and is done here
just so that the resultant graph will show what kind of state has
been found.
After it does all this, the program runs for 100 more seconds on the
last found fixed point (which turns out to be a saddle node), then
is hard-switched in the script to the first attractor basin from which
it runs for another 100 seconds till it settles there, and then
is hard-switched yet again to the second attractor and runs for 400
seconds.
Looking at the output you will see many features of note:
* the first attractor (stable point) and the saddle point (unstable
fixed point) are both found quite often. But the second
attractor is found just once.
It has a very small basin of attraction.
* The values found for each of the fixed points match well with the
values found by running the system to steady-state at the end.
* There are a large number of failures to find a fixed point. These are
found and reported in the diagnostics. They show up on the plot
as cases where the 10-second runs are not flat.
If you wanted to find fixed points in a production model, you would
not need to do the 10-second runs, and you would need to eliminate the
cases where the state-finder failed. Then you could identify the good
points and keep track of how many of each were found.
There is no way to guarantee that all fixed points have been found
using this algorithm! If there are points in an obscure corner of state
space (as for the singleton second attractor convergence in this
example) you may have to iterate very many times to find them.
You may wish to sample concentration space logarithmically rather than
linearly.
"""
from __future__ import print_function
import math
import pylab
import numpy
import moose
def main():
compartment = makeModel()
ksolve = moose.Ksolve( '/model/compartment/ksolve' )
stoich = moose.Stoich( '/model/compartment/stoich' )
stoich.compartment = compartment
stoich.ksolve = ksolve
stoich.path = "/model/compartment/##"
state = moose.SteadyState( '/model/compartment/state' )
moose.reinit()
state.stoich = stoich
state.showMatrices()
state.convergenceCriterion = 1e-6
moose.seed( 111 ) # Used when generating the samples in state space
for i in range( 0, 50 ):
getState( ksolve, state )
# Now display the states of the system at more length to compare.
moose.start( 100.0 ) # Run the model for 100 seconds.
a = moose.element( '/model/compartment/a' )
b = moose.element( '/model/compartment/b' )
# move most molecules over to b
b.conc = b.conc + a.conc * 0.9
a.conc = a.conc * 0.1
moose.start( 100.0 ) # Run the model for 100 seconds.
# move most molecules back to a
a.conc = a.conc + b.conc * 0.99
b.conc = b.conc * 0.01
moose.start( 400.0 ) # Run the model for 200 seconds.
# Iterate through all plots, dump their contents to data.plot.
displayPlots()
quit()
[docs]def makeModel():
""" This function creates a bistable reaction system using explicit
MOOSE calls rather than load from a file
"""
# create container for model
model = moose.Neutral( 'model' )
compartment = moose.CubeMesh( '/model/compartment' )
compartment.volume = 1e-15
# the mesh is created automatically by the compartment
mesh = moose.element( '/model/compartment/mesh' )
# create molecules and reactions
a = moose.Pool( '/model/compartment/a' )
b = moose.Pool( '/model/compartment/b' )
c = moose.Pool( '/model/compartment/c' )
enz1 = moose.Enz( '/model/compartment/b/enz1' )
enz2 = moose.Enz( '/model/compartment/c/enz2' )
cplx1 = moose.Pool( '/model/compartment/b/enz1/cplx' )
cplx2 = moose.Pool( '/model/compartment/c/enz2/cplx' )
reac = moose.Reac( '/model/compartment/reac' )
# connect them up for reactions
moose.connect( enz1, 'sub', a, 'reac' )
moose.connect( enz1, 'prd', b, 'reac' )
moose.connect( enz1, 'enz', b, 'reac' )
moose.connect( enz1, 'cplx', cplx1, 'reac' )
moose.connect( enz2, 'sub', b, 'reac' )
moose.connect( enz2, 'prd', a, 'reac' )
moose.connect( enz2, 'enz', c, 'reac' )
moose.connect( enz2, 'cplx', cplx2, 'reac' )
moose.connect( reac, 'sub', a, 'reac' )
moose.connect( reac, 'prd', b, 'reac' )
# Assign parameters
a.concInit = 1
b.concInit = 0
c.concInit = 0.01
enz1.kcat = 0.4
enz1.Km = 4
enz2.kcat = 0.6
enz2.Km = 0.01
reac.Kf = 0.001
reac.Kb = 0.01
# Create the output tables
graphs = moose.Neutral( '/model/graphs' )
outputA = moose.Table ( '/model/graphs/concA' )
outputB = moose.Table ( '/model/graphs/concB' )
outputC = moose.Table ( '/model/graphs/concC' )
outputCplx1 = moose.Table ( '/model/graphs/concCplx1' )
outputCplx2 = moose.Table ( '/model/graphs/concCplx2' )
# connect up the tables
moose.connect( outputA, 'requestOut', a, 'getConc' );
moose.connect( outputB, 'requestOut', b, 'getConc' );
moose.connect( outputC, 'requestOut', c, 'getConc' );
moose.connect( outputCplx1, 'requestOut', cplx1, 'getConc' );
moose.connect( outputCplx2, 'requestOut', cplx2, 'getConc' );
return compartment
def displayPlots():
for x in moose.wildcardFind( '/model/graphs/conc#' ):
t = numpy.arange( 0, x.vector.size, 1 ) #sec
pylab.plot( t, x.vector, label=x.name )
pylab.legend()
pylab.show()
[docs]def getState( ksolve, state ):
""" This function finds a steady state starting from a random
initial condition that is consistent with the stoichiometry rules
and the original model concentrations.
"""
scale = 1.0 / ( 1e-15 * 6.022e23 )
state.randomInit() # Randomize init conditions, subject to stoichiometry
moose.start( 2.0 ) # Run the model for 2 seconds.
state.settle() # This function finds the steady states.
for x in ksolve.nVec[0]:
print(x * scale, end=' ')
print(state.nIter, state.status, state.stateType, state.nNegEigenvalues, state.nPosEigenvalues, state.solutionStatus)
moose.start( 10.0 ) # Run model for 10 seconds, just for display
# Run the 'main' if this script is executed standalone.
if __name__ == '__main__':
main()
```