DynaFit Scripting Manual

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Rapid-equilibrium vs. dynamic methods

The initial reaction rates (typically, of enzyme reactions) can be computed in DynaFit with or without the rapid-equilibrium approximation ([6], p. 18; [7], p. 18). The steady-state approximation, also used frequently in the analysis of initial reaction rates, is not available in DynaFit.

Rapid-equilibrium approximation

Under the rapid equilibrium approximation, it is assumed that the non-covalent binding of substrates to the enzyme and the dissociation of newly formed products from the intermediate complexes is infinitely faster than any chemical steps. If this assumption can be used profitably, the DynaFit script file must include the sequence

[progress]
   rapid equilibrium
[velocity]
   ...
   equilibrate <list of reactants>
   ...

or the sequence

[progress]
   rapid equilibrium
[velocity]
   ...
   equilibrate all
   ...

Slow steps in the reaction mechanism

Using the rapid equilibrium approximation has implications for the [mechanism] section of the script file. If the keyword rapid equilibrium is present in the [progress] section, all reversible steps will be considered effectively infinitely rapid unless the special notation <==*==> is used, instead of the usual double sided arrow (<====>).

Example 1

In the mechanism below, the reaction steps characterized by rate constants $k_{1}$, $k_{-1}$, $k_{3}$, and $k_{-3}$ will be considered as infinitely rapid compared with $k_{2}$ and $k_{-2}$. Transparently to the user, DynaFit will internally represent the first and third reversible steps by using the equilibrium constants $K_{1} = k_{1} / k_{-1}$ and $K_{2} = k_{3} / k_{-3}$, respectively.

[mechanism]
   E + S <=====> ES  :  k1  k-1
   ES <==*==> EP     :  k2  k-2
   EP <=====> E + P  :  k3  k-3
[progress]
   rapid equilibrium
[response]
   P = 1.0

Kinetic and equilibrium steps

When DynaFit computes initial reaction rates under the rapid-equilibrium approximation, the mechanism must contain at least one ``slow'' step, that is, a step which does not participate in rapid equilibria. In the preceding example there are two such steps, characterized by rate constants $k_2$ and $k_{-2}$.

Example 2

Alternate and equivalent representation of the reaction mechanism in Example 1 above.

[mechanism]
   E + S <=====> ES  :  K(s)   dissoc
   ES ---> EP        :  k2  
   EP ---> ES        :  k-2
   EP <=====> E + P  :  K(p)   dissoc
[progress]
   rapid equilibrium
[response]
   P = 1.0

There can be as many kinetic (``slow'') steps in any given mechanism as is necessary. However, it is important that at least one is present. For example, an omission of the asterisk in ``<==*==>'' in Example 1 would imply that all reversible steps are at equilibrium:

Example 3

This mechanism is notated incorrectly for the computation of initial reaction velocities, because it does not contain any kinetic steps.

[mechanism]
   E + S <===> ES  :  K(s)   dissoc
   ES <===> EP     :  Keq    equil   ; ERROR !!
   EP <===> E + P  :  K(p)   dissoc
[progress]
   rapid equilibrium
[response]
   P = 1.0

Molar response coefficients

The present version of DynaFit places an important limitation on the nature of (bio)chemical species that appear in ``rapid-equilibrium'' reaction mechanisms. In particular, a non-zero molar response coefficient can be assigned only to those species that are directly formed in a step not participating in rapid equilibria.

Example 4

The initial reaction velocity is computed as the rate of formation for the species P, $v = \epsilon_{\rm P} \times {\rm d} [P] / d t$, where ${\rm d} [P] / {\rm d} t = k_{\rm cat} \times [ES]_{\rm eq}$.

[task]
   data = velocity
   task = simulate
[mechanism]
   E + S <===> ES  :  Ks  dissoc
   ES --> E + P    :  kcat
[constants]
   Ks = 1
   kcat = 1
[concentrations]
   E = 0.001
[responses]
   P = 1
[progress]
   rapid equilibrium
[velocity]
   mesh from 0 to 10 step 1
   variable = S
   file ./output/rapeq_1.txt
[end]

If a species with non-zero molar response coefficient is not formed directly in a ``slow'' step (not a ``rapid equilibrium'' step), it is still possible to compute the initial reaction velocity using the rapid equilibrium approximation. However, the present version of DynaFit still requires that the molar response coefficient is attached to a species that does not participate in rapid equilibria. Therefore, one should choose a chemical species from which the truly observable species is formed in a 1:1 stoichiometric ratio, as is explained on the example below.

Example 5

[task]
   data = velocity
   task = simulate
[mechanism]
   E + S <===> ES  :  Ks  dissoc
   ES <=*=> EP     :  kf  kr
   EP <===> E + P  :  Kp  dissoc
[constants]
   Ks = 1, Kp = 1
   kf = 1, kr = 1
[concentrations]
   E = 0.001
[responses]
   EP = 1  ; standing in for 'P' !
[progress]
   rapid equilibrium
[velocity]
   mesh from 0 to 10 step 1
   variable = S
   file rapeq_2.txt
[end]

The dynamic method

An alternate method for computing initial reaction velocities in DynaFit is based on the simulation of pre-steady state dynamics of the (bio)chemical system. In this method, the reaction components are presumed to be mixed at time zero. The chemical composition then changes very rapidly as intermediate molecular complexes are formed by (reversible) molecular association. At a suitable point in time (e.g., one second or ten seconds, depending on the experimental setup) the reaction velocity is computed from a system of differential equations. In this method of computing initial reaction velocities, it is then very important to specify a nonzero mixing delay time, by using the keyword delay in the [progress] section of the script file.

Example 6 The initial reaction velocity is computed at time $t$ = 5 seconds as $v = \epsilon_{\rm P} \times {\rm d} [P] / {\rm d} t$ (see also Example 4). In this case the elementary rate ${\rm d} [P] / d t$ is computed by solving numerically the system of differential equations

$$\dot{c}_{\scriptscriptstyle \rm E} = -k_{\rm 1}\,c_{\scriptscriptstyle \rm S}+k_{\rm -1}\,c_{\scriptscriptstyle \rm ES}+k_{\rm 2}\,c_{\scriptscriptstyle \rm ES}$$ (8.1)

$$\dot{c}_{\scriptscriptstyle \rm S} = -k_{\rm 1}\,c_{\scriptscriptstyle \rm E}\,c_{\scriptscriptstyle \rm S}+k_{\rm -1}\,c_{\scriptscriptstyle \rm ES}$$ (8.2)

$$\dot{c}_{\scriptscriptstyle \rm ES} = +k_{\rm 1}\,c_{\scriptscriptstyle \rm E}\,c_{\scriptscriptstyle \... ...\rm -1}\,c_{\scriptscriptstyle \rm ES}-k_{\rm 2}\,c_{\scriptscriptstyle \rm ES}$$ (8.3)

$$\dot{c}_{\scriptscriptstyle \rm P} = +k_{\rm 2}\,c_{\scriptscriptstyle \rm ES}$$ (8.4)

[task]
   data = velocity
   task = simulate
[mechanism]
   E + S <===> ES  :  k1  k-1
   ES --> E + P    :  k2
[constants]
   k1 = 1, k-1 = 1
   kcat = 1
[concentrations]
   E = 0.001
[responses]
   P = 1
[progress]
   delay = 5
[velocity]
   mesh from 0 to 10 step 1
   variable = S
   file ./output/dynamic.txt
[end]

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biokin.com/dynafit/scripting/html/node49.html
Petr Kuzmic | Jul 12 2005