Re: [SOSlib-devel] spatial dimension zero[Scanned]

[Andrew F. <af...@ph...>]

Rainer

you wrote:

> Is the stochastic test suite with such models available yet?

yes.  I'm pretty sure Darren Wilkinson announced it on sbml-discuss

yours Andrew=20

> -----Original Message-----
> From: Rainer Machne [mailto:ra...@tb...]=20
> Sent: 26 June 2006 11:56
> To: Andrew Finney
> Cc: SOSlib Development
> Subject: Re: [SOSlib-devel] spatial dimension zero[Scanned]
>=20
>=20
>=20
> >> can a compartment of dimension zero have a size?
> >
> > No
> >
> >>
> >> would we currently handle such structures correctly, at least for=20
> >> fixed compartment sizes?
> >
> > I don't see why not
>=20
> Mhm, I guess we might get an error for Compartment_getValue,=20
> because at the moment I think we treat all compartments as a=20
> constant having a size.=20
> E.g. all ODEs are automatically divided by the compartment size.
>=20
> Is the stochastic test suite with such models available yet?
>=20
> Rainer
>=20
>=20
> On Tue, 20 Jun 2006, Andrew Finney wrote:
>=20
> > Rainer
> >
> >> do you know of any examples of models with a compartment with=20
> >> spatialDimension zero or hasOnlySubstanceUnits true?
> >>
> >
> > Darren Wilkinson created a stoctastic test suite which consists of=20
> > models of this form.  They should operate correctly in a=20
> deterministic=20
> > simulator as well.
> >
> >> can a compartment of dimension zero have a size?
> >
> > No
> >
> >>
> >> would we currently handle such structures correctly, at least for=20
> >> fixed compartment sizes?
> >
> > I don't see why not
> >
> > yours Andrew
> >
> >> -----Original Message-----
> >> From: Rainer Machne [mailto:ra...@tb...]
> >> Sent: 20 June 2006 15:48
> >> To: Andrew Finney
> >> Cc: SOSlib Development
> >> Subject: Re: [SOSlib-devel] variable compartment bug:=20
> >> example[Scanned]
> >>
> >>
> >>>>> The units of the species are of the form substance/size units
> >>
> >>> "The units of the species are of the form substance/size
> >> units (i.e.,
> >>> concentration units, using a broad definition of
> >> concentration) if the
> >>> compartment's spatialDimensions is non-zero and
> >> hasOnlySubstanceUnits
> >>> has the value ``false''. The units of the species are of the form=20
> >>> substance if spatialDimensions is zero or=20
> hasOnlySubstanceUnits has=20
> >>> the value ``true''. The units of substance are those=20
> defined in the=20
> >>> substanceUnits, and the size units are those given in the=20
> >>> spatialSizeUnits field."
> >>> ...
> >>
> >> oops, i guess it was some freudian process that blinded me to the=20
> >> continuation of this sentence :)
> >>
> >> do you know of any examples of models with a compartment with=20
> >> spatialDimension zero or hasOnlySubstanceUnits true?
> >>
> >> can a compartment of dimension zero have a size?
> >>
> >> would we currently handle such structures correctly, at least for=20
> >> fixed compartment sizes?
> >>
> >> BTW, there are several models with variable compartments=20
> in the test=20
> >> suite, all in the group rulesForParametersAndCompartments.
> >>
> >> Could it be that SOSlib only succeeded because the program used to=20
> >> generate tested results also does it wrong?
> >>
> >> Rainer
> >>
> >>
> >> On Tue, 20 Jun 2006, Andrew Finney wrote:
> >>
> >>> Rainer
> >>>
> >>>>>> of course you have to check if the species in your kinetic
> >>>> law have
> >>>>>> dimension conc or substance!
> >>>>>> sbml allows the usage of both, doesn't it?
> >>>>>
> >>>>> See section 4.6.4 here
> >>>>>
> >>>>
> >>=20
> http://sbml.org/specifications/sbml-level-2/version-1/html/sbml-level
> >>>> -
> >>>>> 2.html#SECTION00046000000000000000
> >>>>>
> >>>>> "
> >>>>> The units of the species are of the form substance/size units
> >>>>>
> >>>>> ...
> >>>>>
> >>>>> The units of the species are used in the following ways:
> >>>>>
> >>>>> ...
> >>>>>    * The species identifier has these units when it=20
> appears as a=20
> >>>>> numerical quantity in a mathematical formula expressed=20
> in MathML=20
> >>>>> (discussed in Section 3.6.3).
> >>>>> "
> >>>>>
> >>>>> So i guess, the units of species in math expressions are
> >> always in
> >>>>> concentrations!
> >>>>>
> >>>
> >>> WRONG
> >>>
> >>> They are either substance or concentration units depending on the=20
> >>> attributes of the species element.
> >>>
> >>> "The units of the species are of the form substance/size
> >> units (i.e.,
> >>> concentration units, using a broad definition of
> >> concentration) if the
> >>> compartment's spatialDimensions is non-zero and
> >> hasOnlySubstanceUnits
> >>> has the value ``false''. The units of the species are of the form=20
> >>> substance if spatialDimensions is zero or=20
> hasOnlySubstanceUnits has=20
> >>> the value ``true''. The units of substance are those=20
> defined in the=20
> >>> substanceUnits, and the size units are those given in the=20
> >>> spatialSizeUnits field."
> >>> ...
> >>>
> >>> "The units of the species are used in the following ways:
> >>>
> >>>
> >>> The species initialConcentration field has these units=20
> (see Section=20
> >>> 4.6.3).
> >>>
> >>> The species identifier has these units when it appears as a
> >> numerical
> >>> quantity in a mathematical formula expressed in MathML
> >> (discussed in
> >>> Section 3.6.3).
> >>>
> >>> The math field of an AssignmentRule structure determining
> >> the species'
> >>> quantity (see Section 4.8.2) has these units.
> >>>
> >>> In RateRule structures that set the rate of change of the species'
> >>> quantity (Section 4.8.3), the units on the rule's math
> >> field are the
> >>> units of the species divided by the built-in time units. "
> >>>
> >>> yours Andrew
> >>>
> >>>
> >>>
> >>>
> >>>> -----Original Message-----
> >>>> From: sbm...@li...
> >>>> [mailto:sbm...@li...] On
> >> Behalf Of
> >>>> Rainer Machne
> >>>> Sent: 20 June 2006 15:26
> >>>> To: Stefan Mueller
> >>>> Cc: sbm...@li...
> >>>> Subject: Re: [SOSlib-devel] variable compartment bug:
> >>>> example[Scanned]
> >>>>
> >>>>
> >>>>
> >>>> This
> >>>>
> >>>>> current ODE: d[S1]/dt =3D f(S1, S2, ...) / V planned ODE: d
> >> S1 /dt =3D
> >>>>> f(S1, S2, ...)
> >>>>
> >>>> should of course be
> >>>>
> >>>> current ODE: d[S1]/dt =3D f([S1], [S2], ...) / V planned ODE: d
> >>>> S1 /dt =3D f([S1], [S2], ...)
> >>>>
> >>>>
> >>>> On Tue, 20 Jun 2006, Rainer Machne wrote:
> >>>>
> >>>>> stefan
> >>>>>
> >>>>>
> >>>>>>> divide k by V(0) and the whole kinetic law again by=20
> V^2 for 2nd=20
> >>>>>>> order reactions
> >>>>>>
> >>>>>> yes.
> >>>>>> and that's exactly the same what you are planning, if you
> >>>> want to use
> >>>>>> conc ans subst in parallel.
> >>>>>> only the point of conversion differs.
> >>>>>
> >>>>> not really. i wasn't planning any conversion of the math,
> >>>> except for:
> >>>>>
> >>>>> current ODE: d[S1]/dt =3D f(S1, S2, ...) / V planned ODE: d
> >> S1 /dt =3D
> >>>>> f(S1, S2, ...)
> >>>>>
> >>>>> then we have two data->value arrays, let's say
> >>>>>
> >>>>> data->odeValues =3D S1, S2, S3, etc.
> >>>>> and
> >>>>> data->concValues =3D [S1], [S2], [S3] etc.
> >>>>>
> >>>>>
> >>>>> The first array `odeValues' corresponds to the SUNDIALS
> >>>> values in NVector y.
> >>>>> The second array `concValues' will be used in evaluateAST, for=20
> >>>>> evaluation the math.
> >>>>>
> >>>>>
> >>>>> This would be equivalent of course, with just replacing
> >> all [S1] in
> >>>>> all math expressions with S1/volume, and only using one array
> >>>>> data->odeValues
> >>>>>
> >>>>> While the latter approach might be easier to implement it has a=20
> >>>>> probably the not so small drawback, that for each species
> >>>> in each math
> >>>>> expression the evaluation needs one more operation, i.e.
> >>>> the division,
> >>>>> while this division could be done only once for each
> >>>> species whenever
> >>>>> data->odeValues is updated (i.e. e.g. in the
> >>>> SOSlib/SUNDIALS funciton f).
> >>>>>
> >>>>> However, this might still only be correct, if compartments are=20
> >>>>> determined by assignment rules.
> >>>>> If they are determined by rate rules, then the replacement
> >>>> S1 -> S1/V
> >>>>> would also be required for constructing the jacobi matrix.
> >>>>>
> >>>>>> of course you have to check if the species in your kinetic
> >>>> law have
> >>>>>> dimension conc or substance!
> >>>>>> sbml allows the usage of both, doesn't it?
> >>>>>
> >>>>> See section 4.6.4 here
> >>>>>
> >>>>
> >>=20
> http://sbml.org/specifications/sbml-level-2/version-1/html/sbml-level
> >>>> -
> >>>>> 2.html#SECTION00046000000000000000
> >>>>>
> >>>>> "
> >>>>> The units of the species are of the form substance/size units
> >>>>>
> >>>>> ...
> >>>>>
> >>>>> The units of the species are used in the following ways:
> >>>>>
> >>>>> ...
> >>>>>    * The species identifier has these units when it=20
> appears as a=20
> >>>>> numerical quantity in a mathematical formula expressed=20
> in MathML=20
> >>>>> (discussed in Section 3.6.3).
> >>>>> "
> >>>>>
> >>>>> So i guess, the units of species in math expressions are
> >> always in
> >>>>> concentrations!
> >>>>>
> >>>>> Rainer
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>>
> >>>>>> i mean, every species is in a well-defined=20
> compartment, isn't it?
> >>>>>> so you can use the formula [S]=3DS/V whenever you want...
> >>>>>>
> >>>>>> i would eliminate conc as early as possile, but that's a
> >> matter of
> >>>>> taste.
> >>>>>
> >>>>>
> >>>>>
> >>>>> On Tue, 20 Jun 2006, Stefan Mueller wrote:
> >>>>>
> >>>>>> hi rainer!
> >>>>>>
> >>>>>>> stefan
> >>>>>>>
> >>>>>>>>> Currently the ODE would be:
> >>>>>>>>>
> >>>>>>>>> d[S1]/dt =3D k1 * [S1] / V1
> >>>>>>>>
> >>>>>>>> this is not valid for variable compartments.
> >>>>>>>
> >>>>>>> i said that it's invalid for variable compartments some
> >>>> lines below:
> >>>>>>>>> which is only valid for constant compartments V1 and V2.
> >>>>>>>>
> >>>>>>>> please don't use it for comparison!
> >>>>>>>
> >>>>>>> it's not used for comparison, but it is the current
> >>>> implementation
> >>>>>>> in SOSlib!
> >>>>>>
> >>>>>> i said this just to avoid any additional misunderstanding.
> >>>>>> the whole topic seems to be quite confusing...
> >>>>>>
> >>>>>>>>> At least for the second ODE, depending in two different
> >>>> volumes,
> >>>>>>>>> the conversion to ODEs depending on substance instead of=20
> >>>>>>>>> concentration is not as simple as
> >>>>>>>>>
> >>>>>>>>>> dS1/dt =3D k * [S1] * [S2] / V(0) * V =3D k/V(0) * S1 * S2 =
/ V
> >>>>>>>>
> >>>>>>>> why dou you refer to this law for a reaction of order 2?
> >>>>>>>
> >>>>>>> sorry, at first i wanted to make the second reaction in
> >> V2 of 2nd
> >>>>>>> order, e.g.,
> >>>>>>>
> >>>>>>> in V2:
> >>>>>>> S2 + S3 -> P: k2*[S2]*[S3]
> >>>>>>>
> >>>>>>>>> as your example before,
> >>>>>>>>> but for S2 I get:
> >>>>>>>>>
> >>>>>>>>> dS2/dt =3D k1/V1(0) * S1/V1 * V2  -  k2/V2(0) * S2
> >>>>>>>>>
> >>>>>>>>> Right?
> >>>>>>>>
> >>>>>>>> no!
> >>>>>>>> dS2/dt =3D k1/V1(0) * S1 - k2/V2(0) * S2 it's as=20
> simple as that!
> >>>>>>>> (since dS2/dt =3D -dS1/dt as far as reaction S1->S2 is
> >> concerned.)
> >>>>>>>> that's the advantage of kinetic laws (and consequently
> >> odes) for
> >>>>>>>> substances!
> >>>>>>>
> >>>>>>> of course! sorry, i got quite confused obviously.
> >>>>>>>
> >>>>>>> and for the above second order reaction in V2 it would be:
> >>>>>>>
> >>>>>>>   dS2/dt =3D k1/V1(0) * S1 - k2/V2(0) * S1 * S3 / V2
> >>>>>>>
> >>>>>>> Right this time?
> >>>>>>
> >>>>>> yes :)
> >>>>>>
> >>>>>>>>> Thus I would really have to check the type of reaction
> >>>> and edit a
> >>>>>>>>> lot of kinetic laws before constructing the ODEs.
> >>>>>>>>
> >>>>>>>> the above example shows that this is not necessary.
> >>>>>>>> maybe there are other examples...
> >>>>>>>
> >>>>>>> so you say, we could convert kinetic laws that don't=20
> explicitly=20
> >>>>>>> contain the volume by simply differentiating these
> >> three cases and
> >>>>>>>
> >>>>>>>>>> order 1:
> >>>>>>>>>> S1->junk: k*[S1]
> >>>>>>>>>> dS1/dt =3D k * [S1] / V(0) * V =3D k/V(0) * S1
> >>>>>>>
> >>>>>>> divide k by V(0) for 1st order reactions
> >>>>>>>
> >>>>>>>>>> order 2:
> >>>>>>>>>> S1+S2->bla: k*[S1]*[S2]
> >>>>>>>>>> dS1/dt =3D k * [S1] * [S2] / V(0) * V =3D k/V(0) * S1 * S2 =
/ V
> >>>>>>>
> >>>>>>> divide k by V(0) and the whole kinetic law again by=20
> V^2 for 2nd=20
> >>>>>>> order reactions
> >>>>>>>
> >>>>>>>>>> order 3:
> >>>>>>>>>> S1+S2+S3->wow: k*[S1]*[S2]*[S3]
> >>>>>>>>>> dS1/dt =3D k * [S1] * [S2] * [S3] / V(0) * V =3D k/V(0) *
> >>>> S1 * S2 *
> >>>>>>>>>> S3 /
> >>>>>>>
> >>>>>>> V^2
> >>>>>>>
> >>>>>>> divide k by V(0) and the whole kinetic law again by=20
> V^2 for 2nd=20
> >>>>>>> order reactions
> >>>>>>
> >>>>>> yes.
> >>>>>> and that's exactly the same what you are planning, if you
> >>>> want to use
> >>>>>> conc ans subst in parallel.
> >>>>>> only the point of conversion differs.
> >>>>>>
> >>>>>> i mean, every species is in a well-defined=20
> compartment, isn't it?
> >>>>>> so you can use the formula [S]=3DS/V whenever you want...
> >>>>>>
> >>>>>> i would eliminate conc as early as possile, but that's a
> >>>> matter of taste.
> >>>>>>
> >>>>>>> but there are also enzymatic reactions, with in often
> >>>> quite complex
> >>>>>>> kinetic laws. we would really have to do unit checking,
> >>>> recognizing
> >>>>>>> the implemented reaction mechanisms etc.
> >>>>>>
> >>>>>> of course you have to check if the species in your kinetic
> >>>> law have
> >>>>>> dimension conc or substance!
> >>>>>> sbml allows the usage of both, doesn't it?
> >>>>>>
> >>>>>>> also, we don't really know in which compartment
> >> reactions happen,
> >>>>>>> but could just try to get this information from the
> >>>> reactants' compartments.
> >>>>>>
> >>>>>> this does not matter, for 2 reasons:
> >>>>>>
> >>>>>> 1. if the kinetic law is specified correctly (including the=20
> >>>>>> compartment size), e.g. A+B->C: k*V*[A]*[B], the the
> >>>> compartment size
> >>>>>> is considered automatically.
> >>>>>>
> >>>>>> 2. the conversion [S]=3DS/V does not depend on reactions,
> >>>> but only on
> >>>>>> species (and their compartments).
> >>>>>>
> >>>>>>> so i guess, it would be much easier to
> >>>>>>>
> >>>>>>> a) use ODEs for substances but dependent on
> >>>> concentrations, here the
> >>>>>>> only thing we need to do is to leave out the compartment
> >>>> division at
> >>>>>>> the end of Species_odeFromReactions and have a double
> >>>> bookkeeping of
> >>>>>>> amounts and concentrations
> >>>>>>
> >>>>>> to my taste, double book keeping is dangerous and produces
> >>>> overhead.
> >>>>>>
> >>>>>>> and
> >>>>>>>
> >>>>>>> b) just ignore WRONG models that have a variable
> >> compartment but
> >>>>>>> don't have the compartment explicitly in their=20
> kinetic laws (or=20
> >>>>>>> check for such cases and issue an error);
> >>>>>>>
> >>>>>>> isn't it an issue of correct model writing rather then
> >>>> correct model
> >>>>>>> solving?
> >>>>>>
> >>>>>> i agree.
> >>>>>>
> >>>>>>> However, another problem might be that to calculate=20
> in amounts=20
> >>>>>>> rather then in concentrations also changes the use of=20
> absolute=20
> >>>>>>> errors and maybe also the general the performance of the
> >>>> integrator, i guess. ??
> >>>>>>
> >>>>>> abs errors might change, but that's not an issue.
> >>>>>> rel errors should stay the same.
> >>>>>> (in one-compartment models everything is just multiplied by the
> >>>>>> volume.)
> >>>>>>
> >>>>>>> maybe we will have to really implement both in parallel,
> >>>>>>
> >>>>>> i hope not :)
> >>>>>>
> >>>>>>> * ODEs for concentrations/time if no variable
> >> compartments are in
> >>>>>>> the model
> >>>>>>> * ODEs for ammounts/time if any compartment is variable
> >>>>>>>
> >>>>>>> ?
> >>>>>>>
> >>>>>>> Rainer
> >>>>>>
> >>>>>> maybe we discuss the next round via telephone :)
> >>>>>>
> >>>>>> cheers, s
> >>>>>>
> >>>>>>
> >>>>>>>>> Thus I think it just doesn't work with kinetic laws
> >> that don't
> >>>>>>>>> include the compartment explicitly!
> >>>>>>>>
> >>>>>>>> i don't know exactly what you mean.
> >>>>>>>> of course kinetic laws with explicit compartment are
> >>>> "nicer", but
> >>>>>>>> you can always transform any kinetic law to a "nice" one...
> >>>>>>>>
> >>>>>>>>> But if they include the compartment we can write ODEs for=20
> >>>>>>>>> substances but have concentrations in the right hand
> >>>> side of the
> >>>>>>>>> ODEs. We then just need to a second array for the
> >>>> concentrations, as discussed before.
> >>>>>>>>
> >>>>>>>> if we still use concentrations is a separate problem,
> >>>> independent
> >>>>>>>> of which we discussed until now.
> >>>>>>>> it's a matter of taste, if we eliminate the
> >> concentrations  from
> >>>>>>>> the kinetic laws and get a (possibly nonlinear)
> >>>> dependence on the
> >>>>>>>> compartments OR if we keep concentrations and always
> >> have linear
> >>>>>>>> dependence on the comp.
> >>>>>>>>
> >>>>>>>>> Rainer
> >>>>>>>>
> >>>>>>>> cheers,
> >>>>>>>> stefan.
> >>>>>>>>
> >>>>>>>>> On Mon, 19 Jun 2006, Stefan Mueller wrote:
> >>>>>>>>>> let me summarize:
> >>>>>>>>>> kinetic laws have units substance/time, and they depend on=20
> >>>>>>>>>> concentrations or substances.
> >>>>>>>>>>
> >>>>>>>>>> we SHOULD write down odes for substances, and we COULD
> >>>> eliminate
> >>>>>>>>>> concs completely.
> >>>>>>>>>> (of course, we would have to convert initial concs to
> >>>>>>>>>> substances.)
> >>>>>>>>>>
> >>>>>>>>>> in any case, odes for substances depend on the
> >>>> compartment size
> >>>>>>>>>> (linearly or according to the order of the reaction):
> >>>>>>>>>>
> >>>>>>>>>> order 1:
> >>>>>>>>>> S1->junk: k*[S1]
> >>>>>>>>>> dS1/dt =3D k * [S1] / V(0) * V =3D k/V(0) * S1
> >>>>>>>>>>
> >>>>>>>>>> order 2:
> >>>>>>>>>> S1+S2->bla: k*[S1]*[S2]
> >>>>>>>>>> dS1/dt =3D k * [S1] * [S2] / V(0) * V =3D k/V(0) * S1 * S2 =
/ V
> >>>>>>>>>>
> >>>>>>>>>> order 3:
> >>>>>>>>>> S1+S2+S3->wow: k*[S1]*[S2]*[S3]
> >>>>>>>>>> dS1/dt =3D k * [S1] * [S2] * [S3] / V(0) * V =3D k/V(0) *
> >>>> S1 * S2 *
> >>>>>>>>>> S3 /
> >>>>>>>>>> V^2
> >>>>>>>>>>
> >>>>>>>>>> ...
> >>>>>>>>>>
> >>>>>>>>>> if the kinetic laws included the compartment
> >>>> explicitly (which is
> >>>>>>>>>> desirable), we would have the following situation:
> >>>>>>>>>>
> >>>>>>>>>> order 1:
> >>>>>>>>>> S1->junk: r*[S1]*V
> >>>>>>>>>> dS1/dt =3D  r * [S1] * V=3D r * S1
> >>>>>>>>>>
> >>>>>>>>>> order 2:
> >>>>>>>>>> S1+S2->bla: r*[S1]*[S2]*V
> >>>>>>>>>> dS1/dt =3D r * [S1] * [S2] * V=3D r * S1 * S2 / V
> >>>>>>>>>>
> >>>>>>>>>> order 3:
> >>>>>>>>>> S1+S2+S3->wow: r*[S1]*[S2]*[S3]*V
> >>>>>>>>>> dS1/dt =3D r * [S1] * [S2] * [S3] * V =3D r * S1 * S2=20
> * S3 / V^2
> >>>>>>>>>>
> >>>>>>>>>> as stated earlier, odes for substances have got another=20
> >>>>>>>>>> advantage. for the compartment size we could use
> >> either a rate
> >>>>>>>>>> rule or an assignment
> >>>>>>>>>> rule: dV/dt =3D f(t, V, ...)
> >>>>>>>>>> or
> >>>>>>>>>> V=3D f(t, V, ...)
> >>>>>>>>>>
> >>>>>>>>>> cheers,
> >>>>>>>>>> stefan.
> >>>>>>>>>>
> >>>>>>>>>> Am Montag 19 Juni 2006 11:17 schrieb Stefan Mueller:
> >>>>>>>>>>> andrew:
> >>>>>>>>>>>
> >>>>>>>>>>> for the SBML system
> >>>>>>>>>>> 	S1 in C
> >>>>>>>>>>> 	S1->junk : k * [S1]
> >>>>>>>>>>> 	dC/dt =3D 1
> >>>>>>>>>>>
> >>>>>>>>>>> the right math is:
> >>>>>>>>>>> 	dS1/dt =3D k * [S1] / C(0) * C
> >>>>>>>>>>> 	dC/dt =3D 1
> >>>>>>>>>>> with initial cond:
> >>>>>>>>>>> 	S1(0) =3D [S1](0) * C(0)
> >>>>>>>>>>> 	C(0)
> >>>>>>>>>>>
> >>>>>>>>>>> notation:
> >>>>>>>>>>> 	S1 ... substance
> >>>>>>>>>>> 	[S1] ... conc
> >>>>>>>>>>>
> >>>>>>>>>>> the "proposed solution" (who proposed this?) has=20
> the right=20
> >>>>>>>>>>> units, but is wrong otherwise... :)
> >>>>>>>>>>>
> >>>>>>>>>>> as rainer and me already discussed, everything
> >> would be much
> >>>>>>>>>>> clearer, if the kinetic law included the compartment
> >>>> explicitly:
> >>>>>>>>>>> 	S1 in C
> >>>>>>>>>>> 	S1->junk : C * r * [S1]
> >>>>>>>>>>> 	dC/dt =3D 1
> >>>>>>>>>>>
> >>>>>>>>>>> with:
> >>>>>>>>>>> 	r ... the "real" chemical rate constant.
> >>>>>>>>>>>
> >>>>>>>>>>> rainer:
> >>>>>>>>>>>
> >>>>>>>>>>> chemists always knew that rate constants are not
> >>>> constant with
> >>>>>>>>>>> respect to pH, T, ... as a physicist i have to be fair! :)
> >>>>>>>>>>>
> >>>>>>>>>>> to conclude: the math is not really tricky.
> >>>>>>>>>>> odes for substances are better suited to handle
> >> both multiple
> >>>>>>>>>>> and variable compartments. of course, we have to
> >> take care of
> >>>>>>>>>>> many implementation details...
> >>>>>>>>>>>
> >>>>>>>>>>> cheers,
> >>>>>>>>>>> stefan.
> >>>>>>>>>>
> >>>>>>>>>> _______________________________________________
> >>>>>>>>>> sbmlsolver-devel mailing list=20
> >>>>>>>>>> sbm...@li...
> >>>>>>>>>>=20
> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>>>>>>
> >>>>>>>>> _______________________________________________
> >>>>>>>>> sbmlsolver-devel mailing list
> >>>>>>>>> sbm...@li...
> >>>>>>>>>=20
> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>>>>>
> >>>>>>>> _______________________________________________
> >>>>>>>> sbmlsolver-devel mailing list
> >>>>>>>> sbm...@li...
> >>>>>>>> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>>>>
> >>>>>>> _______________________________________________
> >>>>>>> sbmlsolver-devel mailing list
> >>>>>>> sbm...@li...
> >>>>>>> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>>>
> >>>>>>
> >>>>>> _______________________________________________
> >>>>>> sbmlsolver-devel mailing list
> >>>>>> sbm...@li...
> >>>>>> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>>>
> >>>>>
> >>>>
> >>>>
> >>>> _______________________________________________
> >>>> sbmlsolver-devel mailing list
> >>>> sbm...@li...
> >>>> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>>
> >>>
> >>>
> >>> _______________________________________________
> >>> sbmlsolver-devel mailing list
> >>> sbm...@li...
> >>> https://lists.sourceforge.net/lists/listinfo/sbmlsolver-devel
> >>>
> >>
> >
> >
> > _______________________________________________
> > sbmlsolver-devel mailing list
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> >
>=20