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<chapter id="model" xreflabel="Model">
<!-- Begin CVS Header -->
<!--   $Source: /Volumes/Home/Users/shoops/cvs/copasi_dev/userdocs/userguide/copasi_model.xml,v $ -->
<!--   $Revision: 1.5 $ -->
<!--   $Name:  $ -->
<!--   $Author: shoops $ -->
<!--   $Date: 2008/03/12 00:35:05 $ -->
<!-- End CVS Header -->

<!-- Copyright (C) 2008 by Pedro Mendes, Virginia Tech Intellectual -->
<!-- Properties, Inc., EML Research, gGmbH, University of Heidelberg, -->
<!-- and The University of Manchester. -->
<!-- All rights reserved. -->

<!-- Copyright (C) 2001 - 2007 by Pedro Mendes, Virginia Tech Intellectual -->
<!-- Properties, Inc. and EML Research, gGmbH. -->
<!-- All rights reserved. -->

<title>The Model in COPASI</title>


<para>
This section discusses the general properties of the model in COPASI and its
mathematical interpretation. Generally a model consists of species which are
placed in a compartment. Species are produced or consumed by reactions which
happen with a speed given by kinetic functions. This model is interpreted
mathematically in different ways in COPASI. 
</para>

<sect1 id="compartments" xreflabel="Compartments">
<title>Compartments</title>
<para>
Every species in COPASI must be placed in a compartment. The compartment has
to have a well defined volume because COPASI relies on being able to calculate
particle numbers from concentrations and vice versa. 
COPASI does not really support models without concentrations (i.e. dealing only
with amounts of substance or particle numbers) but you can work around this
limitation by setting the volume to be 1.0. But if you do this keep in mind that
the dimensions of the different parameters as displayed in the COPASI GUI are
not correct.
The volumes of compartments are constant in this version of COPASI. 
</para>
<para>
If the (chemically) same species is present in several compartments it is
treated as several species in COPASI. E.g. if you have a model that contains
glucose inside a cell and outside a cell these are two species in the model.
They can however have the same name.
</para>
</sect1>

<sect1 id="Species" xreflabel="Species">
<title>Species</title>
<para>
A species is characterized by how much of it is present at a given time. This
can be expressed as a particle number or as a concentration. COPASI makes sure
that those two are consistent at any time. An initial value can be provided for
concentration or particle number of each species. This initial value will be
used as a starting point for simulations or other calculations. 
A species can be defined to be "fixed" which means it current value will
always be the same as the initial value, even if the species participates in
a reaction.
</para>
<para>
Associated to the species is also a value called transition time. It is
calculated as the current particle number of the species devided by either
the sum of all reaction fluxes going into the species or the sum of all
reaction fluxes going out of the species, whichever is smaller (the fluxes
also epxressed in number of particles per second). The transition time gives a
rough heuristic measure for how long a particle of this species will exist on
average before it is consumed. A short transition time in a stable steady state
may indicate that the equilibrium is a "fast" equilibrium which could be used
for a simplification of the model. Note however that this calculation is inexact
if any reversible reactions are involved.
</para>
</sect1>

<sect1 id="GlobalQuantities" xreflabel="Global Quantities">
<title>Global Quantities</title>
<para>
The model also contains a list of global quantities. Roughly speaking the global quantities can be used to describe values that are not necessarily a concentration or a volume. But to be more specific: The global quantities can have three different modes (called "types" in the GUI). 
</para>
<sect2 id="fixedquantities" xreflabel="fixed Quantities">
<title>Fixed Quantities</title>
<para>
"Fixed" quantities always take a constant value. While the GUI still displays an initial value and a transient value those are always the same. This can be used if the same numerical value is used in several parts of the model (e.g. several reactions use a common kinetic parameter) but you want to be able to change this value in a single place.
</para>
</sect2>
<sect2 id="assignedquantities" xreflabel="assigned Quantities">
<title>Quantities with assignment</title>
<para>
Global quantities can also have an assignment rule. This means a mathematical expression is specified (which can involve other variable or constant values from the model) and the value of the global quantity is alway the value of this expression. Circular definitions are not allowed.
Global quantities with assignment do not have an initial value (since their value is always, also at the beginning of a simulation, determined by the mathematical expression).
These assigned quantities could be used for output only (e.g. if you want to plot the sum of some concentrations) or can be used in the model itself.
</para>
</sect2>
<sect2 id="odequantities" xreflabel="ode Quantities">
<title>Quantities with differential equation</title>
<para>
If global quantities are set to the "ode" mode they become true variables of the model. A mathematical expression has to be specified and this expression is interpreted as the right hand side of an ordinary differential equation for this variable. An initial value has to be provided for these quantities.
</para>
<para>
Since there is no obvious probabilistic interpretation of a general ode stochastic simulation is disabled for models containing global quantities of type "ode".
</para>
</sect2>
</sect1>

<sect1 id="Reactions" xreflabel="Reactions">
<title>Reactions</title>
<para>
A reaction is a process by which species will be consumed or produced. It is
characterized by a description of how fast this happens. It contains a list of
substrates, i.e. the species that are consumed if the reaction takes place,
along with the information about how many molecules of each substrate are
consumed when the reaction event happens once (the stoichiometry).
Correspondingly there is a list of products with the respective stoichiometries.
Reactions without substrates are possible, as well as reactions without
products. However reactions without substrates and without products are not
allowed. In addition so called modifiers can be specified which are neither
produced nor consumed in the reaction but which have influence on the speed of
the reaction.
</para>
<para>
The speed of the reaction is always specified by a reference to a kinetic
function. The kinetics can depend on the concentrations of the substrates,
products, and modifiers, on the volume of a compartment, on local or global
parameters, and on the simulation time. The difference between local and global
parameters is that local parameters only specify a numerical value of a kinetic
parameter for one specific reaction. Global parameters can be used in several
reactions.
</para>
<para>
Reactions can be reversible or irreversible. Kinetic functions for irreversible
reactions should always be positive. Also they should not depend on the
concentration of the products (only on the concentrations of the substrates and
modifiers). While all built in kinetic functions satisfy these conditions, they
are not enforced for user defined functions. 
</para>
</sect1>

<sect1 id="Functions" xreflabel="Functions">
<title>Functions</title>
<para>
  A function in COPASI is a mathematical expression that calculates a value from
a given list of other values. Generally functions can be used in two ways in
COPASI: As a kinetic function specified for a reaction or as a function that is
called from another function or expression. 
  The function contains a detailed list of parameters that specifies how many
values (and what kind of values) need to be passed to the function. If a
function is called from another function or expression only the number of values
is checked, not the kind of value. 
  In the case that a function is used as a kinetic function for a reaction
COPASI ensures consistency about the roles of the parameters in the model. Each
parameter has one of six different roles: "Substrate", "Product", "Modifier",
"Volume", "Time", "Parameter". The first three must be connected with the
concentration of a species that has the respective role in the reaction.
"Volume" must be connected with a compartment, "Time" is always the simulation
time of the model. "Parameter" must be connected with a global parameter or a
local value in the reaction.
  A function also specifies if it is appropriate for a reversible or
irreversible reaction. This information is not automatically infered from the
mathematical description of the function, and COPASI does not enforce
restrictions like irreversible kinetics having to be strictly positive. 
  
</para>
</sect1>

<sect1 id="deterministicModel" xreflabel="Deterministic Model">
<title>Deterministic Interpretation of the Model</title>
<para>
  One possible mathematical interpretation of the model is to convert it into a
set of ordinary differential equations. The variables of the equation are the
particle numbers of the species in the model. The right hand side of the
differential equation are constructed as follows: The particle numbers are
converted to concentrations taking into acount the unit for amounts of substance
and dividing by compartment volume. These concentrations are used to calculate
the reaction fluxes. The kinetic functions, as they are defined in COPASI give
as result a value that is a concentration rate (for single compartment
reactions) or an amount of substance rate (for multicompartment reactions),
respectively. So for single compartment reactions the value of the kinetic
function is multiplied by the compartment volume; kinetics for multi compartment
reactions are assumed to already be expressed in units of amount of substance
(e.g. moles) per time. The resulting value is then multiplied by a factor to
convert amount of substance per time to particle numbers per time. Linear
combinations of these values, using the stoichiometries as coefficients, result
in particel number rates for all species. These form the right hand side of
the differential equations.
</para>
<para>
COPASI automatically performs an analysis of the model by which conserved values
are found. The conserved values COPASI is looking for are linear combinations of
particle numbers that do not vary during the time evolution of the system. Each
conservation relation can be used to eliminate one variable of the system,
leading to a reduced system with a smaller number of variables. These variables
are called the independent variables of the system; the dependent variables are
defined as linear combinations of independent variables. COPASI handles this
model reduction transparently, but it is displayed in the GUI which species
are treated as independent or dependent variables.
Technically finding the conservation relation means finding rows in the
stoichiometry matrix that can be expressed as linear combinations of other rows.
COPASI uses Householder QR factorization to do
this<citation>Vallabhajosyula06</citation>.
</para>
 
</sect1>
  
<sect1 id="stochasticModel" xreflabel="Stochastic Model">
  <title>Stochastic Interpretation of the Model</title>
  <para>
    An alternative interpretation is to consider the model as a stochastic
process. In this case the reaction kinetics are not considered to describe the
rates of change for the concentrations of involved species, but rather as a
specification about the probability that a reaction event happens. If a reaction
event happens the particle numbers of the involved species are updated
according to their stoichiometries. That means particle numbers are always
integer numbers and change discretely. 
</para>
<para>
Specifically the value of the kinetic function is interpreted as a so called
propensity, that is a differential probability density that a reaction event will 
happen in the next infinitesimal time interval.
However there are subtle differences between reaction rates and reaction propensities. One of those differences that only matters for rather small particle numbers is that e.g. the rate of a second order mass action reaction is described as  
<inlineequation><mml:math>
<mml:mrow>
  <mml:mi>k</mml:mi>
  <mml:mo>*</mml:mo>
  <mml:msup>
    <mml:mi>S</mml:mi>
    <mml:mn>2</mml:mn>
  </mml:msup>
</mml:mrow>
</mml:math></inlineequation>
, while the propensity of the same reaction is 
<inlineequation><mml:math>
  <mml:mrow>
    <mml:mi>k</mml:mi>
    <mml:mo>*</mml:mo>
    <mml:mi>S</mml:mi>
    <mml:mo>*</mml:mo>
    <mml:mfenced>
      <mml:mrow>
      <mml:mi>S</mml:mi>
      <mml:mo>-</mml:mo>
      <mml:mi>1</mml:mi>
    </mml:mrow>
    </mml:mfenced>
  </mml:mrow>
</mml:math></inlineequation>
. COPASI will apply this kind of corrections automatically. In cases where these corrections have already been done by the modeler explicitly COPASI needs to be told not to apply this correction. This is described in the section
<xref linkend="generalSettings"/>
</para>
<para>
Another issue modelers should be aware of is that the rate laws for enzymatic reactions that are derived using the steady state approximation are not necessarily valid for stochastical simulation. In many cases they are, but the underlying assumptions for using them are not exactly the same as for deterministic simulations. 
</para>
</sect1>


</chapter>

	<chapter id="model" xreflabel="Model">
	<!-- Begin CVS Header -->
	<!-- $Source: /Volumes/Home/Users/shoops/cvs/copasi_dev/userdocs/userguide/copasi_model.xml,v $ -->
	<!-- $Revision: 1.5 $ -->
	<!-- $Name: $ -->
	<!-- $Author: shoops $ -->
	<!-- $Date: 2008/03/12 00:35:05 $ -->
	<!-- End CVS Header -->

	<!-- Copyright (C) 2008 by Pedro Mendes, Virginia Tech Intellectual -->
	<!-- Properties, Inc., EML Research, gGmbH, University of Heidelberg, -->
	<!-- and The University of Manchester. -->
	<!-- All rights reserved. -->

	<!-- Copyright (C) 2001 - 2007 by Pedro Mendes, Virginia Tech Intellectual -->
	<!-- Properties, Inc. and EML Research, gGmbH. -->
	<!-- All rights reserved. -->

	<title>The Model in COPASI</title>


	<para>
	This section discusses the general properties of the model in COPASI and its
	mathematical interpretation. Generally a model consists of species which are
	placed in a compartment. Species are produced or consumed by reactions which
	happen with a speed given by kinetic functions. This model is interpreted
	mathematically in different ways in COPASI.
	</para>

	<sect1 id="compartments" xreflabel="Compartments">
	<title>Compartments</title>
	<para>
	Every species in COPASI must be placed in a compartment. The compartment has
	to have a well defined volume because COPASI relies on being able to calculate
	particle numbers from concentrations and vice versa.
	COPASI does not really support models without concentrations (i.e. dealing only
	with amounts of substance or particle numbers) but you can work around this
	limitation by setting the volume to be 1.0. But if you do this keep in mind that
	the dimensions of the different parameters as displayed in the COPASI GUI are
	not correct.
	The volumes of compartments are constant in this version of COPASI.
	</para>
	<para>
	If the (chemically) same species is present in several compartments it is
	treated as several species in COPASI. E.g. if you have a model that contains
	glucose inside a cell and outside a cell these are two species in the model.
	They can however have the same name.
	</para>
	</sect1>

	<sect1 id="Species" xreflabel="Species">
	<title>Species</title>
	<para>
	A species is characterized by how much of it is present at a given time. This
	can be expressed as a particle number or as a concentration. COPASI makes sure
	that those two are consistent at any time. An initial value can be provided for
	concentration or particle number of each species. This initial value will be
	used as a starting point for simulations or other calculations.
	A species can be defined to be "fixed" which means it current value will
	always be the same as the initial value, even if the species participates in
	a reaction.
	</para>
	<para>
	Associated to the species is also a value called transition time. It is
	calculated as the current particle number of the species devided by either
	the sum of all reaction fluxes going into the species or the sum of all
	reaction fluxes going out of the species, whichever is smaller (the fluxes
	also epxressed in number of particles per second). The transition time gives a
	rough heuristic measure for how long a particle of this species will exist on
	average before it is consumed. A short transition time in a stable steady state
	may indicate that the equilibrium is a "fast" equilibrium which could be used
	for a simplification of the model. Note however that this calculation is inexact
	if any reversible reactions are involved.
	</para>
	</sect1>

	<sect1 id="GlobalQuantities" xreflabel="Global Quantities">
	<title>Global Quantities</title>
	<para>
	The model also contains a list of global quantities. Roughly speaking the global quantities can be used to describe values that are not necessarily a concentration or a volume. But to be more specific: The global quantities can have three different modes (called "types" in the GUI).
	</para>
	<sect2 id="fixedquantities" xreflabel="fixed Quantities">
	<title>Fixed Quantities</title>
	<para>
	"Fixed" quantities always take a constant value. While the GUI still displays an initial value and a transient value those are always the same. This can be used if the same numerical value is used in several parts of the model (e.g. several reactions use a common kinetic parameter) but you want to be able to change this value in a single place.
	</para>
	</sect2>
	<sect2 id="assignedquantities" xreflabel="assigned Quantities">
	<title>Quantities with assignment</title>
	<para>
	Global quantities can also have an assignment rule. This means a mathematical expression is specified (which can involve other variable or constant values from the model) and the value of the global quantity is alway the value of this expression. Circular definitions are not allowed.
	Global quantities with assignment do not have an initial value (since their value is always, also at the beginning of a simulation, determined by the mathematical expression).
	These assigned quantities could be used for output only (e.g. if you want to plot the sum of some concentrations) or can be used in the model itself.
	</para>
	</sect2>
	<sect2 id="odequantities" xreflabel="ode Quantities">
	<title>Quantities with differential equation</title>
	<para>
	If global quantities are set to the "ode" mode they become true variables of the model. A mathematical expression has to be specified and this expression is interpreted as the right hand side of an ordinary differential equation for this variable. An initial value has to be provided for these quantities.
	</para>
	<para>
	Since there is no obvious probabilistic interpretation of a general ode stochastic simulation is disabled for models containing global quantities of type "ode".
	</para>
	</sect2>
	</sect1>

	<sect1 id="Reactions" xreflabel="Reactions">
	<title>Reactions</title>
	<para>
	A reaction is a process by which species will be consumed or produced. It is
	characterized by a description of how fast this happens. It contains a list of
	substrates, i.e. the species that are consumed if the reaction takes place,
	along with the information about how many molecules of each substrate are
	consumed when the reaction event happens once (the stoichiometry).
	Correspondingly there is a list of products with the respective stoichiometries.
	Reactions without substrates are possible, as well as reactions without
	products. However reactions without substrates and without products are not
	allowed. In addition so called modifiers can be specified which are neither
	produced nor consumed in the reaction but which have influence on the speed of
	the reaction.
	</para>
	<para>
	The speed of the reaction is always specified by a reference to a kinetic
	function. The kinetics can depend on the concentrations of the substrates,
	products, and modifiers, on the volume of a compartment, on local or global
	parameters, and on the simulation time. The difference between local and global
	parameters is that local parameters only specify a numerical value of a kinetic
	parameter for one specific reaction. Global parameters can be used in several
	reactions.
	</para>
	<para>
	Reactions can be reversible or irreversible. Kinetic functions for irreversible
	reactions should always be positive. Also they should not depend on the
	concentration of the products (only on the concentrations of the substrates and
	modifiers). While all built in kinetic functions satisfy these conditions, they
	are not enforced for user defined functions.
	</para>
	</sect1>

	<sect1 id="Functions" xreflabel="Functions">
	<title>Functions</title>
	<para>
	A function in COPASI is a mathematical expression that calculates a value from
	a given list of other values. Generally functions can be used in two ways in
	COPASI: As a kinetic function specified for a reaction or as a function that is
	called from another function or expression.
	The function contains a detailed list of parameters that specifies how many
	values (and what kind of values) need to be passed to the function. If a
	function is called from another function or expression only the number of values
	is checked, not the kind of value.
	In the case that a function is used as a kinetic function for a reaction
	COPASI ensures consistency about the roles of the parameters in the model. Each
	parameter has one of six different roles: "Substrate", "Product", "Modifier",
	"Volume", "Time", "Parameter". The first three must be connected with the
	concentration of a species that has the respective role in the reaction.
	"Volume" must be connected with a compartment, "Time" is always the simulation
	time of the model. "Parameter" must be connected with a global parameter or a
	local value in the reaction.
	A function also specifies if it is appropriate for a reversible or
	irreversible reaction. This information is not automatically infered from the
	mathematical description of the function, and COPASI does not enforce
	restrictions like irreversible kinetics having to be strictly positive.

	</para>
	</sect1>

	<sect1 id="deterministicModel" xreflabel="Deterministic Model">
	<title>Deterministic Interpretation of the Model</title>
	<para>
	One possible mathematical interpretation of the model is to convert it into a
	set of ordinary differential equations. The variables of the equation are the
	particle numbers of the species in the model. The right hand side of the
	differential equation are constructed as follows: The particle numbers are
	converted to concentrations taking into acount the unit for amounts of substance
	and dividing by compartment volume. These concentrations are used to calculate
	the reaction fluxes. The kinetic functions, as they are defined in COPASI give
	as result a value that is a concentration rate (for single compartment
	reactions) or an amount of substance rate (for multicompartment reactions),
	respectively. So for single compartment reactions the value of the kinetic
	function is multiplied by the compartment volume; kinetics for multi compartment
	reactions are assumed to already be expressed in units of amount of substance
	(e.g. moles) per time. The resulting value is then multiplied by a factor to
	convert amount of substance per time to particle numbers per time. Linear
	combinations of these values, using the stoichiometries as coefficients, result
	in particel number rates for all species. These form the right hand side of
	the differential equations.
	</para>
	<para>
	COPASI automatically performs an analysis of the model by which conserved values
	are found. The conserved values COPASI is looking for are linear combinations of
	particle numbers that do not vary during the time evolution of the system. Each
	conservation relation can be used to eliminate one variable of the system,
	leading to a reduced system with a smaller number of variables. These variables
	are called the independent variables of the system; the dependent variables are
	defined as linear combinations of independent variables. COPASI handles this
	model reduction transparently, but it is displayed in the GUI which species
	are treated as independent or dependent variables.
	Technically finding the conservation relation means finding rows in the
	stoichiometry matrix that can be expressed as linear combinations of other rows.
	COPASI uses Householder QR factorization to do
	this<citation>Vallabhajosyula06</citation>.
	</para>

	</sect1>

	<sect1 id="stochasticModel" xreflabel="Stochastic Model">
	<title>Stochastic Interpretation of the Model</title>
	<para>
	An alternative interpretation is to consider the model as a stochastic
	process. In this case the reaction kinetics are not considered to describe the
	rates of change for the concentrations of involved species, but rather as a
	specification about the probability that a reaction event happens. If a reaction
	event happens the particle numbers of the involved species are updated
	according to their stoichiometries. That means particle numbers are always
	integer numbers and change discretely.
	</para>
	<para>
	Specifically the value of the kinetic function is interpreted as a so called
	propensity, that is a differential probability density that a reaction event will
	happen in the next infinitesimal time interval.
	However there are subtle differences between reaction rates and reaction propensities. One of those differences that only matters for rather small particle numbers is that e.g. the rate of a second order mass action reaction is described as
	<inlineequation><mml:math>
	<mml:mrow>
	<mml:mi>k</mml:mi>
	<mml:mo>*</mml:mo>
	<mml:msup>
	<mml:mi>S</mml:mi>
	<mml:mn>2</mml:mn>
	</mml:msup>
	</mml:mrow>
	</mml:math></inlineequation>
	, while the propensity of the same reaction is
	<inlineequation><mml:math>
	<mml:mrow>
	<mml:mi>k</mml:mi>
	<mml:mo>*</mml:mo>
	<mml:mi>S</mml:mi>
	<mml:mo>*</mml:mo>
	<mml:mfenced>
	<mml:mrow>
	<mml:mi>S</mml:mi>
	<mml:mo>-</mml:mo>
	<mml:mi>1</mml:mi>
	</mml:mrow>
	</mml:mfenced>
	</mml:mrow>
	</mml:math></inlineequation>
	. COPASI will apply this kind of corrections automatically. In cases where these corrections have already been done by the modeler explicitly COPASI needs to be told not to apply this correction. This is described in the section
	<xref linkend="generalSettings"/>
	</para>
	<para>
	Another issue modelers should be aware of is that the rate laws for enzymatic reactions that are derived using the steady state approximation are not necessarily valid for stochastical simulation. In many cases they are, but the underlying assumptions for using them are not exactly the same as for deterministic simulations.
	</para>
	</sect1>


	</chapter>