File: Simulation.htm

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<html>
<head>
<title>Simulation Methods</title>
</head>
<body>
<h1>Simulation Methods</h1>

<p>
A simulation method may be either deterministic or stochastic. One can obtain a
deterministic method by modelling the reactions with ordinary differential
equations. Numerically integrating the equations gives an approximate solution.
<!--CONTINUE.-->
</p>

<p>
The simulations may be performed with exact or approximate methods.
<a href="http://en.wikipedia.org/wiki/Gillespie_algorithm">Gillespie's
direct method</a> and Gibson and Bruck's next reaction method
are both exact methods. Various formulations of both of these methods
are available. For the direct method, there are a variety of ways of
generating a discrete deviate, that determines which reaction fires.
The next reaction method uses a priority queue. Several data structures
can be used to implement a priority queue. The choice of data structure
will influence the performance, but not the output.
</p>

<p>
Tau-leaping is an approximate, discrete, stochastic method. It is used
to generate an ensemble of trajectories, each of which is an
approximate realization of the stochastic process. The tau-leaping
method takes jumps in time and uses Poisson deviates to determine how
many times each reaction fires.  One can choose fixed time steps or
specify a desired accuracy. The latter is the preferred method.  There
is a hybrid method which combines the direct method and
tau-leaping. An adaptation of the direct method is used for reactions
that are slow or involve small populations; the tau-leaping method is
used for the rest.  This offers improved accuracy and performance for
the case that some species have small populations. For this hybrid
method, one specifies the desired accuracy.
</p>

<p>
One can model the reactions with a set of ordinary differential equations.
In this case one assumes that the populations are continuous and
not discrete (integer). One can numerically integrate the differential
equations to obtain an approximate solution. Note that since this is a
deterministic model, it generates a single solution instead of an
ensemble of trajectories.
</p>

<p>
Each of the stochastic simulation methods use discrete, uniform deviates
(random integers). We use the 
<a href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html">Mersenne Twister 19937</a>
algorithm to generate these.
Both of the exact methods also use exponential deviates that determine
the reaction times. For these we use the ziggurat method.
</p>

</body>
</html>