Re: [SOSlib-devel] mail for SBML list: variable Compartments in SBML

[Stefan M. <ste...@oe...>]

rainer,
there are more mistakes!
(please see below)

> hi,
>
> find below a draft for bringing this issue to the SBML list, as suggested
> by Andrew.
>
> could you review? is the problem stated clearly and correctly?
>
>
> Dear SBML Community,
>
> As we recently wanted to model variable compartments we realized that our
> tool - SOSlib - actually does not solve models with variable compartments
> correctly.
>
> We wonder how other tools approach this topic.
>
> In summary:
>
> SOSlib constructs ODEs as recommended by the SBML specs, e.g.:
>
> species X1 in compartment V1, with concentration x1 = X1/V1
> species X2 and X3 in V2, with conc. as above
>
> Reactions:
> X1->X2; k1*X1*V1 	// a translocation, k1 [1/sec]
> X2->X3; k2*X2*V2    // a unimolecular reaction, k2 [1/sec]
>
> Derived ODEs:
> dx1/dt = - ( k1 * X1* V1 ) / V1
> dx2/dt =  ( k1 * X1* V1 - k2 * X2 * V2 ) / V2
>
> the kinetic laws are the SBML specific conversion of `classic' rate law
> to substance/time units, instead of concentrations/time to allow for
> easy software handling of multiple compartment models.
>
> If however, the volumes in the model are variable with time, i.e. either
> an assignment rule or a rate rule, these equations would not be correct.

the following section is not very clear and/or wrong:
(please don't send this to the sbml list!)

> A numerical integration routine would start with the initial concentration
> x(0) and add the changes during time t:
>
> x(t) = x(0) +  dx/dt * t
>
> i.e.
>
> x(t) = X(0)/V(0) + dx/dt * t
>
> dx/dt is calculated correctly from above ODE because it also accounts for
> changing compartments by explicit inclusion of the volume in the kinetic
> law.
>
> However, the change of the initial concentration x(0) is not accounted for
> anywhere. The solver should rather return:
>
> x(t) = X(0)/V(t) + dx/dt * t
>
> If x is a function of time and of volume V, x(t,V(t)),
> then the correct ODE would actually be
>
> dx/dt = dx/dt + dx/dV * dV/dt
>
> Right?

no!
please use the following math:
(i already wrote it down in one of my previous mails :) )

definitions:
x(t) ... conc
X(t) ... substance
V(t) ... compartment size

given a kinetic law for a reaction,
e.g. X->Y: r*x*V or X+Y->Z: r*x*y*V,
which explicitly includes the compartment size (!),
the corresponding ode for substance X amounts to:
dX/dt = - r*x*V or - r*x*y*V,
or replacing conc by substances:
dX/dt = - r*X or - r*X*Y/V

the ode for conc can be derived by differentiating x=X/V:
dx/dt = (dX/dt)/V - x*(dV/dt)/V
where dX/dt is given by the kinetic law above.
i.e. we would get a correction term for variable compartment size.
this is the most important reason for using odes for substances!

> We are currently discussing different solutions and their dangers
> for SOSlib, i.e. how to correctly convert the ODE system to an ODE system
> in amount/time instead of concentration/time or whether the above partial
> differential equation could or should be implemented instead.
>
> It would be helpful for us to hear how other tools handle variable
> compartments.
>
>
> And BTW, it might be good to indicate it as good practice NOT to convert
> parameters e.g. from [1/sec] to [litre/sec] to obtain correct units for
> SBML kinetic laws, but to explicitly keep the volume in the formula! This
> would allow changing the compartment size or even making the variable, at
> least for tools which do it correctly.
> E.g. the model in the specs, section 5.6 will not be valid anymore if
> someone wanted to change the volume sizes!
>
> Such rather unconvential volume-dependent rate `constants' can be
> quite dangerous!
>
>
> Rainer
>
>
>
> Rainer Machne
> Institute for Theoretical Chemistry
> University of Vienna
> Austria
>
>
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