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<html>
<head>
<title>Birth-Death</title>
</head>
<body>
<h1>Birth-Death</h1>
<p>
The models in this section deal with the birth-death process, which is
comprised of the birth reaction <em>X → 2X</em> and the death
reaction <em>X → 0</em>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-01</h2>
<p>
<b>Importing SBML.</b><br>
Use the file open button <img src="fileopen.png"> in the tool
bar to import the file
<tt>examples/sbml/dsmts31/dsmts-001-01.xml</tt>.
One other thing to note is that the SBML model does not specify a size
for the Cell compartment. However, Cain requires that compartments
have a defined size, so it was given the default, unit size.
</p>
<p>
<b>Reference solutions.</b><br>
Now open the Cain file
<tt>examples/cain/dsmts31/dsmts-001-01.xml</tt>. First select
<tt>Reference</tt> in the methods list. Note that the output type
is <tt>Statistics, Transient Behavior</tt>. The methods that store
only statistics (and not trajectories or histograms) are used for
storing solutions that have been generated outside of Cain.
The reference (analytical) solution from the DSMTS has been imported.
In the simulation output panel, click the plot button to bring up the
plot configuration window. At the top of this window select the
"Statistics" tab. In the plot configuration window shown
below we have specified plotting the mean with the standard
deviation shown as error bars.
</p>
<p>
<img src="DsmtsBirthDeath01StatisticsConfiguration.png">
</p>
<p>
The resulting plot of the analytical solution is shown below.
</p>
<p>
<img src="DsmtsBirthDeath01Reference.jpg">
</p>
<p>
The reference solutions from the DSMTS have been imported in all of
the examples in this directory. Now we will show how to import such a
solution. First select <tt>TimeSeries</tt> in the methods list. As we
will import a solution to compare with it, clone the method with the
clone button <img src="editcopy.png">. Then rename the clone method
from "TimeSeries1" to "Ref". In the method editor
change the category (the second pull-down list) from
"Time Series, Uniform" to "Statistics, Transient
Behavior". In the launcher panel note that only the import
button <img src="fileopen.png"> is enabled. Click this button
to import the solution. Select the file <tt>dsmts-001-01.txt</tt>,
which is a text file that contains the means and standard
deviations. Each line in the file contains the data for one time frame.
(You can examine the file with your favorite text editor.) Now the
result appears in the simulation output panel. You can plot the
solution, examine it in a table, or use it for analysis of other
solutions. Since the solution was already imported before, you can
now delete your output and the method that you created.
</p>
<p>
<b>Generating trajectories.</b><br>
Now select the
<tt>TimeSeries</tt> method in the methods list. It uses the direct method
and records the state at the times [0, 1, ..., 50]. Specify that
10,000 trajectories should be generated in the launcher panel.
Then start the simulation by clicking the launch button
<img src="launch.png">.
</p>
<p>
Return to the plot configuration window. (Click the plot button in
the simulation output panel if you closed it.)
At the top of the window select the "Time Series" tab.
Note that the "BithDeath01, TimeSeries" output is selected
in the pull-down menu. In the radio button groups
select <tt>Species</tt> and <tt>Mean</tt>.
Uncheck the <tt>Legend</tt> button since there is only one species. Fill in an
appropriate title and axes labels. The resulting configuration window
is shown below.
</p>
<p>
<img src="DsmtsBirthDeath01PlotConfiguration.png">
</p>
<p>
Below is the plot of the empirical mean and standard deviation of the species
population.
</p>
<p>
<img src="DsmtsBirthDeath01MeanStdDev.jpg">
</p>
<p>
<b>Comparing empirical and reference solutions.</b><br>
Below we plot the empirical
and analytical means and standard deviation of the population. The
empirical solutions are plotted with a dotted line and circles. The
reference solutions are plotted with a solid line. First we plot
the empirical solution using the "Time Series" tab of
the plot configuration window. Then we add the plot of the reference
solution by using the "Plot" button in the
"Statistics" tab. (See the
<a href="VisualizationSignificance.htm">Significance Testing</a>
section for detailed instructions on how to produce the plots.)
</p>
<p>
<img src="DsmtsBirthDeath01MeanEmpiricalReference.jpg">
</p>
<p>
<img src="DsmtsBirthDeath01StdDevEmpiricalReference.jpg">
</p>
<p>
From the plots above we see that the empirical solution appears to
agree with the reference solution. Now we use
<a href="http://en.wikipedia.org/wiki/Student's_t-test">Student's
t-test</a> to test the null hypotheses that the empirical means are
equal to the analytical means. The test yields a
<a href="http://en.wikipedia.org/wiki/P-value">p-value</a> that
is the probability of obtaining a test statistics that is at least
as extreme as the observed result, assuming that the null hypothesis
is true. See the
<a href="VisualizationSignificance.htm">Significance Testing</a>
section for more information.
</p>
<p>
Click the p-value button <img src="pValue.png"> in the
simulation output panel to open the p-value analysis window shown below.
In the left column select the "BirthDeath01, TimeSeries"
output. In the right column select the reference solution. Click the
"Calculate" button to compute the p-value for all of the
species and all of the frames. The row headers list the frame times.
The column headers list the species.
</p>
<p>
<img src="DsmtsBirthDeath01PValueAnalysis.png">
</p>
<p>
Click the "Plot" button to show a plot of p-value versus
frame number. We see that the p-values are consistent with a
correct stochastic simulation method.
</p>
<p>
<img src="DsmtsBirthDeath01PValue.jpg">
</p>
<p>
<b>Histogram of species population.</b><br>
Next we will study the distribution of species populations at t = 50.
Select the <tt>Histogram</tt> method in the methods list panel. Note
that we record a single frame at the end of the simulation. Launch
a simulation to generate 10,000 trajectories. Select the
<tt>Histograms</tt> tab in the plot configuration window. Select the
<tt>Filled</tt> field in the table. Uncheck the <tt>Legend</tt> button.
Fill in an appropriate title and axes labels. The resulting plot
configuration window is shown below.
</p>
<p>
<img src="DsmtsBirthDeath01HistogramConfiguration.png">
</p>
<p>
Below is the distribution of species populations at time t = 50.
</p>
<p>
<img src="DsmtsBirthDeath01Histogram.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-02</h2>
<p>
This model is the same as the one before, except that the rate parameters
Lambda and Mu are declared local to the reactions. Cain does not have
local parameters, so they are left undefined. After importing
<tt>examples/sbml/dsmts31/dsmts-001-02.xml</tt>, one needs to define
Lambda and Mu in the parameters editor panel.
Because of the local parameters, the reactions are not recognized as having
mass-action kinetics.
</p>
<p>
<img src="DsmtsBirthDeath02CustomReactions.png">
</p>
<p>
In order to use the mass-action solvers, we need to edit the reactions
to reflect this.
</p>
<p>
<img src="DsmtsBirthDeath02MassActionReactions.png">
</p>
<p>
Open <tt>examples/cain/dsmts31/dsmts-001-02.xml</tt> to see the result.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-03</h2>
<p>
This model is the same as dsmts-001-01, except that the rate parameters
Lambda and Mu have values 1 and 1.1, respectively. The population
decays more rapidly than before.
</p>
<p>
<img src="DsmtsBirthDeath03Mean.jpg">
</p>
<p>
<img src="DsmtsBirthDeath03StdDev.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-04</h2>
<p>
This model is the same as dsmts-001-01, except that the initial population
of <em>X</em> is 10. Below are plots of the mean and standard deviation
of the population. As before, we compare the empirical solution
using 10,000 trajectories with the analytical solution.
</p>
<p>
<img src="DsmtsBirthDeath04Mean.jpg">
</p>
<p>
<img src="DsmtsBirthDeath04StdDev.jpg">
</p>
<p>
Note that because the population is small, stochastic effects are very
noticable. Open <tt>examples/cain/dsmts31/dsmts-001-04.xml</tt>
and select <tt>AllReactions</tt> in the methods list panel.
This method uses a solver that records every reaction event.
(In general, it is only feasible to use such solvers when the total number of
reaction events is small. This is typically the case when the species
populations are small.) Generate
five trajectories and plot the result. Below we see that the trajectories
have quite different behaviors.
</p>
<p>
<img src="DsmtsBirthDeath04AllReactions.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-05</h2>
<p>
This model is the same as dsmts-001-01, except that the initial population
of <em>X</em> is 10,000. Below is a plot of the mean and standard deviation
of the population. Note that the standard deviation is relatively smaller
than before. Since the initial population is higher, there are more
reaction events in each trajectory. Thus we generated only 1,000 trajectories
in analyzing the mean population.
</p>
<p>
<img src="DsmtsBirthDeath05Mean.jpg">
</p>
<p>
<img src="DsmtsBirthDeath05StdDev.jpg">
</p>
<p>
Now because the population is larger, stochastic effects are less
important. Open <tt>examples/cain/dsmts31/dsmts-001-05.xml</tt>
and select <tt>TimeSeries</tt> in the methods list panel.
Generate five trajectories and plot the result. Below we see that
the trajectories are distinct, but have similar behavior.
</p>
<p>
<img src="DsmtsBirthDeath05Trajectories.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-06</h2>
<p>
In this model death reaction <em>X → 0</em> is replaced by
<em>X →</em> Sink. Sink is declared to be boundary condition, i.e.
the <tt>boundaryCondition</tt> attribute in the <tt>species</tt> element
is true. Thus, despite the reaction <em>X →</em> Sink, the value
of Sink remains 0. Cain does not support boundary conditions. So one
must either make Sink a parameter, or change the reaction so that
the amount of Sink is constant.
</p>
<p>
Open <tt>examples/cain/dsmts31/dsmts-001-06.xml</tt>,
and select <tt>TimeSeries</tt> in the methods list panel.
Generate a single trajectory and plot the result. Below appears that the
Sink species has not been plotted. It has been in fact been plotted, but
the line coincides with the bottom frame of the plotting window. One may use
the toolbar in the plotting window to pan or zoom in order to see the Sink
line.
</p>
<p>
<img src="DsmtsBirthDeath06Default.jpg">
</p>
<p>
One may also change the axes limits in the plot configuration window.
Enter "-5" for the lower bound in the
<tt>Y axis limits</tt> field. The resulting plot is shown below.
</p>
<p>
<img src="DsmtsBirthDeath06Limits.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-07</h2>
<p>
This model is the same as dsmts-001-06, except that Sink is not a boundary
condition. Now there are two reacting species. We generate 10,000 trajectories
and plot the mean and standard deviation. The empirical solutions are
shown with dotted lines and markers. The reference solutions are shown
with solid lines.
</p>
<p>
<img src="DsmtsBirthDeath07Mean.jpg">
</p>
<p>
<img src="DsmtsBirthDeath07StdDev.jpg">
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-08</h2>
<p>
This model is the same as dsmts-001-01, except that the Cell compartment
has size 1. This has no effect on the simulation results.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-09</h2>
<p>
This model is the same as dsmts-001-01, except that the Cell compartment
has size 2. Again, this has no effect on the simulation results because
the species amounts are measured in substance units.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-10</h2>
<p>
This model is the same as dsmts-001-01, except that the Cell compartment
has size 1. Also, the species are measured in concentration units, not substance
units. Ordinarily Cain would not correctly interpret the initial amounts
because it assumes that they are given in substance units. However,
in this case the result is correct because the compartment has unit size.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-11</h2>
<p>
This model is the same as dsmts-001-10, except that the Cell compartment
has size 2. Since the species are measured in concentration units,
Cain does not correctly calculate the initial amounts. One must enter
the correct initial amount of 200 for <em>X</em> in the species editor.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-12</h2>
<p>
This model is the same as dsmts-001-01, except that the rate law for
the birth reaction is written <tt>Lambda*X*0.5*2</tt>. Cain recognizes
this as a mass-action rate law with propensity factor <tt>Lambda</tt>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-13</h2>
<p>
This model is the same as dsmts-001-01, except that Lambda has the
value 0.2 and the rate law for
the birth reaction is written <tt>Lambda*X*0.5</tt>.
This is recognized as a mass-action rate law with propensity factor
<tt>0.5*Lambda</tt>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-14</h2>
<p>
This model is the same as dsmts-001-01, except that the rate law for
the birth reaction is written <tt>Lambda*X/2/0.5</tt>.
This is recognized as a mass-action rate law with propensity factor
<tt>Lambda</tt>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-15</h2>
<p>
This model is the same as dsmts-001-01, except that the rate law for
the birth reaction is written <tt>Lambda*(X/2)/0.5</tt>.
This is recognized as a mass-action rate law with propensity factor
<tt>Lambda</tt>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-16</h2>
<p>
This model is the same as dsmts-001-01, except that the rate law for
the birth reaction is written <tt>Lambda*X/(2/2)</tt>.
This is recognized as a mass-action rate law with propensity factor
<tt>Lambda</tt>.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-17</h2>
<p>
This model is the same as dsmts-001-08, except that the unit compartment
volume is explicitly included in the rate laws. One may use
compartment sizes in the same way that one uses parameter values.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-18</h2>
<p>
This model is the same as dsmts-001-17, except that the compartment
volume is 0.5.
</p>
<!------------------------------------------------------------------------>
<h2>dsmts-001-19</h2>
<p>
This model is the same as dsmts-001-01, except that there is an assignment
rule that defines a new species <em>y = 2 X</em>. Cain does not support
assignment rules.
</p>
</body>
</html>
|
