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

[Rainer M. <ra...@tb...>]

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.

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?


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