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Table of Contents With the time course simulation, you can calculate the trajectory for the species in your model over a given time interval. There are different methods to calculate such trajectories and depending on your model, one or several of them may be appropriate to do a time course simulation of your model. COPASI supports three different methodologies to calculate a trajectory. The first method is to do a deterministic time course simulation of your model using the LSODA [Petzold83] algorithm. For systems with small particle numbers, it is sometimes better to do a stochastic simulation rather than a deterministic one. COPASI supports a method for the stochastic calculation of time series, which is called stochastic and uses the next reaction method described by Gibson and Bruck. Since the deterministic simulation is inappropriate for some systems but on the other hand, the stochastic simulation is too time consuming, there are some methods that try to combine the advantages of both deterministic and stochastic simulation. Most of those methods are termed hybrid methods. COPASI also includes such a hybrid method which in some systems where deterministic simulation would lead to incorrect results will give the correct time series but is still computationally less demanding than a pure stochastic simulation.
The default method in COPASI to calculate a time course is
LSODA [Petzold83]. LSODA is part of the
ODEPACK library [Hindmarsh83].
LSODA was written by Linda R. Petzold and Alan C. Hindmarsh.
It solves systems
Options for LSODA
This stochastic simulation method utilizes the algorithm developed by Gibson and Bruck (see [Gibson00] for details). For each reaction a putative stochastic reaction time is calculated and the reaction with the shortest reaction time will be realized. The set of reactions is organized in a priority queue to allow for the efficient search for the fastest reaction. In addition, by using a so-called dependency graph only those reaction times are recalculated in each step, that are dependent on the reaction, which has been realized. This simulation method requires all the reactions to be irreversible. However, COPASI provides a tool, that converts all reversible reactions into irreversible ones. Because the algorithm internally works on discrete particle numbers rather than concentrations, the particle numbers in the system must not exceed a value of approximately
CautionThe current implementation of the Next Reaction Method is very inefficient when the model contains assignment rules, which leads to extended calculation times. There is a restriction regarding global quantities: If a differential equation is provided for a global quantity (the quantity is of type "ode") the model cannot be simulated stochastically with the current version of COPASI. If the global quantity is of type "assignment" stochastic simulation is possible but not as efficient as for models without assignments. No restrictions apply for "fixed" global quantities. Options for Stochastic (Gibson + Bruck)
This hybrid simulation method developed by us combines a deterministic numerical integration of ODEs with a stochastic simulation algorithm. The whole biochemical network is partitioned into a deterministic and a stochastic subnet internally. The deterministic subnet contains all reactions, in which only species with high particle numbers take part. All reactions with at least one low-numbered species are in the stochastic subnet, because here stochastic effects are expected. Which particle numbers are considered low or high can be specified by the user with the Lower Limit and the Upper Limit parameters (Species with particle numbers between those limits do not change their status. This leads to a hysteresis-like behavior and avoids many unnecessary swaps, if the particle numbers fluctuate in the middle range). The partitioning of the biochemical network can change dynamically during the simulation. After a certain number of steps, which the user can define using the parameter Partitioning Interval, the partitioning is recalculated using the current particle numbers in the system. During one run the deterministic subnet and the stochastic subnet are simulated in parallel. A 4th-order Runge-Kutta method is used to numerically integrate the deterministic part of the system. For the stochastic part the simulation method by Gibson and Bruck ([Gibson00]) is utilized. The reaction probabilities of the stochastic subnet are approximated as constant during one stochastic step, even though in theory they can change due to the effects of the deterministic subnet. CautionThe current implementation of the Hybrid Runge Kutta Method is very inefficient when the model contains assignment rules, which leads to extended calculation times. Options for Hybrid (Runge-Kutta)
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with a dense or banded Jacobian when the problem is stiff, but it
automatically selects between non-stiff (Adams) and stiff (BDF)
methods. It uses the non-stiff method initially, and dynamically
monitors data in order to decide which method to use.
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Please note that best achievable relative tolerance is approximately
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