Herbert M Sauro wrote:
> [...] We've got a
> model at the moment which is full of equilibrium reactions. I'm not sure how
> one should go about this sort of modeling in an automatic way because such
> systems admit two moiety groups, one group which comes from the slow
> reactions and the other from the quasi-steady-state group,
Pardon my ignorance, what exactly is referred to as a 'moiety' here?
Does that mean 'species' or 'reactant'?
---
In any case, with regards to automating a mixed computation that
involves (a) chemical equilibria and (b) reaction dynamics, in DynaFit
[P. Kuzmic (1996) Anal. Biochem. 237, 260-273;
http://www.biokin.com/dynafit/] I have a module that reads the reaction
mechanism and then composes the so-called formula matrix for equilibrium
steps. All _reversible_ steps are considered for participation in rapid
equilibria, except those that are marked with a special symbol:
C <==> D ... (possibly) an equilibrium step
A <=*=> B ... slow step, not included in equilibria
The formula matrix is then fed to an equilibrium solver, and the
composition at equilibrium is computed from it and from the complex
formation constants. Those in turn are automatically compiled from
binary equilibrium constants [simple ratios or elementary rate
constants], using a matrix method based on the theory of chemical
equilibria, see Smith & Missen (1988) for details.
@book {Smit82,
author = "W. R. Smith and R. W. Missen",
title = "Chemical Reaction Equilibrium Analysis",
publisher = "John Wiley",
address = "New York",
year = "1982"}
I rolled my own solver, based on the multi-dimensional Newton Raphson
method, but the EQS solver by Smith & Missen (1988) also works very
well. After the equilibria are computed, depending on the design of the
experiment (say at the end of the pre-incubation phase), the simulated
reaction takes off dynamically.
In the present version of DynaFit, during the dynamic phase of the
simulation the rapid-equilibrium approximation is abandoned, and all
steps are again treated explicitly. However, it is possible that this
kind of approach could be adapted to numerical integration.
Beware of one potentially serious problem though. Any such mixed system
of ODEs and algebraic equations (describing rapid equilibria) is a
"differential-algebraic equation (DAE)" system. The theory of DAEs
seems to suggest that, usually, one is in trouble with global truncation
error. Petzold has written extensively on this, see for example the
first reference on the list below. After I muddled through that article
some year back, I decided not to pursue a numerical machine of the kind
you are talking about. But maybe I was too conservative, so take a look
for yourself.
- Petr
_____________________________________________________________________
P e t r K u z m i c, Ph.D. mailto:pkuzmic at biokin.com
BioKin Ltd. * Software and Consulting http://www.biokin.com
--------
DIFFERENTIAL-ALGEBRAIC EQUATIONS ARE NOT ODES
PETZOLD L
SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
3: (3) 367-384 1982
Cited References: 13
Times Cited: 143
AUTOMATIC SELECTION OF METHODS FOR SOLVING STIFF AND NONSTIFF
SYSTEMS OF ORDINARY DIFFERENTIAL-EQUATIONS
PETZOLD L
SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
4: (1) 136-148 1983
Cited References: 11
Times Cited: 104
NUMERICAL-SOLUTION OF NONLINEAR DIFFERENTIAL-EQUATIONS WITH ALGEBRAIC
CONSTRAINTS .1. CONVERGENCE RESULTS FOR BACKWARD DIFFERENTIATION
FORMULAS
LOTSTEDT P, PETZOLD L
MATHEMATICS OF COMPUTATION
46: (174) 491-516 APR 1986
Cited References: 27
Times Cited: 55
NUMERICAL-SOLUTION OF NONLINEAR DIFFERENTIAL-EQUATIONS WITH ALGEBRAIC
CONSTRAINTS .2. PRACTICAL IMPLICATIONS
PETZOLD L, LOTSTEDT P
SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING
7: (3) 720-733 JUL 1986
Cited References: 21
Times Cited: 38
--------
Gear, C.W. (1988) "Differential algebraic equation index transformation"
SIAM J. Sci. Statist. Comput. 9, 39-47.
Griepentrog, E.; Maerz, R. (1986) "Differential-Algebraic Equations and
Their Numerical Treatment", Teubner, Leipzig.
Hairer, E.; Wanner, G (1991) "Solving Ordinary Differential Equations
II: Stiff and Differential-Algebraic Problems", Springer, Berlin (1991)
Petzold, L. (1984) "ODE methods for the solution of differential
algebraic systems" SIAM J. Numer. Anal. 21, 716-728.
Stoer, J.; Bulirsch, R. (1993) "Introduction to Numerical Analysis", 2nd
Ed., Springer, New York, pp. 494-497.
