Computation of Kinetics of Chemical Systems with Many Species
85
Discussion
WALTER:
With regard to the definition of the steady state and the assertion that some
Xi = 0: (1) for closed systems Xi (I) = 0 (0 < 1 < 00) only at extrema
(instantaneously). However, for closed systems there exist mass constraints of
the form: 2:Xi -- constant. In this case the conditions for an approximate
steady state derive from ~Xi ~,O. Whenever some I Xi I are small compared
to other I Xi I the small Xi are considered to be approximately zero. Thus
the steady state approximation is never exact (except at extrema or equilibrium)
in closed systems. For this reason one refers to "quasi-steady-states" for
closed systems (see for example: C. WALTER, J. Theoret. BioI. 11, 181 (1966)).
(2) For open systems Xi (t) = 0 at extrema or during the stable state period
of minimum entropy production (the steady state). Thus one can refer to
"steady states" for certain open systems.
BRDIER~IANN :
The equations can describe both an open system and a closed system depending
upon the coefficients. With ai-S non-zero, and appropriate mass balance
equations, there can be a net flow. "Steady state" means Xi = 0; if the
system happens to be closed Xi = 0 would correspond to the equilibrium.
BURNS:
We are interested in a rather similar problem, that of finding a stationary state
for the equations of motions of an enzyme catalyzed network. For this problem we have found that "netflux" expressions, similar to those described by
W. W. CLELAND (Biochim. Biophys. Acta 67, 104 (1963)), are advantageous.
These reduce substantiallv the number of differential equations for the motion
of the system bv eliminating equations concerned with the movement of
enzyme substrate complexes. Furthermore the equations so eliminated are
often those containing the "fast" movements in the system; the remaining
differential equations can then be integrated, using a much larger step length,
until a stationary state is achieved.
BRE~!ER~IANN :
Using the system itself for adjusting the Xi seems like a good idea. We plan,
however, to experiment with a variety of methods to determine the most useful ones. There is a lack of theory about the efficiency of numerical methods
(theoretical error estimates are usually above practically obtainable values).
Different groups of workers have developed their own methods, but there is
a lack of communication and publication. We have just started and plan to
experiment with a variety of methods. We have a few ideas that look promising.
85
Discussion
WALTER:
With regard to the definition of the steady state and the assertion that some
Xi = 0: (1) for closed systems Xi (I) = 0 (0 < 1 < 00) only at extrema
(instantaneously). However, for closed systems there exist mass constraints of
the form: 2:Xi -- constant. In this case the conditions for an approximate
steady state derive from ~Xi ~,O. Whenever some I Xi I are small compared
to other I Xi I the small Xi are considered to be approximately zero. Thus
the steady state approximation is never exact (except at extrema or equilibrium)
in closed systems. For this reason one refers to "quasi-steady-states" for
closed systems (see for example: C. WALTER, J. Theoret. BioI. 11, 181 (1966)).
(2) For open systems Xi (t) = 0 at extrema or during the stable state period
of minimum entropy production (the steady state). Thus one can refer to
"steady states" for certain open systems.
BRDIER~IANN :
The equations can describe both an open system and a closed system depending
upon the coefficients. With ai-S non-zero, and appropriate mass balance
equations, there can be a net flow. "Steady state" means Xi = 0; if the
system happens to be closed Xi = 0 would correspond to the equilibrium.
BURNS:
We are interested in a rather similar problem, that of finding a stationary state
for the equations of motions of an enzyme catalyzed network. For this problem we have found that "netflux" expressions, similar to those described by
W. W. CLELAND (Biochim. Biophys. Acta 67, 104 (1963)), are advantageous.
These reduce substantiallv the number of differential equations for the motion
of the system bv eliminating equations concerned with the movement of
enzyme substrate complexes. Furthermore the equations so eliminated are
often those containing the "fast" movements in the system; the remaining
differential equations can then be integrated, using a much larger step length,
until a stationary state is achieved.
BRE~!ER~IANN :
Using the system itself for adjusting the Xi seems like a good idea. We plan,
however, to experiment with a variety of methods to determine the most useful ones. There is a lack of theory about the efficiency of numerical methods
(theoretical error estimates are usually above practically obtainable values).
Different groups of workers have developed their own methods, but there is
a lack of communication and publication. We have just started and plan to
experiment with a variety of methods. We have a few ideas that look promising.