DynaFit Scripting Manual

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Initial estimates

Nonlinear regression analysis requires an intelligent guess of initial estimates, thus data analysis should not (and cannot) be approached without prior knowledge. One must have at least some ideas about the possible values of rate and equilibrium constants that are relevant to the biochemical system at hand.

Association rate constants

In the case of bimolecular association rate constants, we must keep in mind that their values for the association of enzymes with small molecules (e.g., drugs) usually are between $10^{5}$ M $\phantom{.}^{-1} {\rm sec}^{-1}$ and $10^{9}$ M $\phantom{.}^{-1} {\rm sec}^{-1}$. The bimolecular association rate constants for protein-protein interactions are usually somewhat smaller. This background information is applied when we approach the point in writing down the script file below:

	
[mechanism]
   E + S <=> ES  :  k   ks
   ES -> E + P   :  kr
   E + I <=> EI  :  k   kis
  ES + I <=> EIS :  k   kii
[constants]
   k = ...

It is recommended to decide on the values for bimolecular rate constants first, keeping in mind that in many experimental situations their exact numerical values cannot be determined. Often one can use estimates for the bimolecular rate constants that are based on the theory of molecular diffusion. For many biochemical mechanisms we may start with the value $10^6$ M $\phantom{.}^{-1} {\rm sec}^{-1}$ for all bimolecular rate constants. The fact that all three association rate constants in the above mechanism are supposed to have equal value is represented by the fact that all of them are assigned the same symbol.

Let us assume that in a set of experiments pertaining the mixed-type inhibition mechanism above, all concentrations are on the micromolar scale. In that case all bimolecular association rate constants have to have the scale $\mu{\rm M}^{-1} \times {\rm time}^{-1}$. If the units of time used for the description of the experimental data are seconds, then the approximate nominal value of all bimolecular rate constants is

	
[constants]
   k = 1.0    ;  uM(-1)sec(-1)

because $k \approx 10^6$ M $\phantom{.}^{-1} {\rm sec}^{-1}$ = 1.0 $\mu$M $\phantom{.}^{-1} {\rm sec}^{-1}$. If however the units of time used for the description of the experimental data were minutes, than the same value of the bimolecular rate constant would be expressed as

	
[constants]
   k = 60.0   ;  uM(-1)min(-1)

because $k \approx 10^6$ M $\phantom{.}^{-1} {\rm sec}^{-1}$ = 60.0 $\mu$M $\phantom{.}^{-1} {\rm min}^{-1}$.

Dissociation rate constants

Initial values for dissociation rate constants are much more difficult to estimate. Usually we have some notion about the equilibrium constants, though, so from the equilibrium constants and from the association rate constants (set to their diffusion limit) we can deduce the initial estimate for the dissociation rate constant.

Example: A substrate for an enzyme reaction following the simple Michaelis-Menten mechanism is expected to have the half-saturation point (Michaelis constant) in the millimolar range. The association rate constant is supposed to be diffusion limited ($10^6$ M $\phantom{.}^{-1} {\rm sec}^{-1}$). From the reaction velocity observed at saturation, it seems that one mole of the enzyme-substrate complex would produce approximately 0.1 moles of the reaction product per second (turnover number $k_{cat} \approx 0.1$ sec$^{-1}$). What is the order of magnitude for the dissociation rate constant? First we need to realize that for the Michaelis-Menten mechanism, $K_m = (k_s + k_r)/k$ and $k_{cat} = k_r$. From this we can estimate $k_s \approx K_m \times k - k_{cat} \approx 1 \times 10^{-3} \times 10^{6} - 0.1 \approx 1000$ sec$^{-1}$.

Very often it is sufficient to come up with crude estimates of rate constants, within several orders of magnitude. Even without the arithmetic shown above we can estimate the dissociation rate constants after several trial simulations. The goal is to have the initial estimate of rate constants produce an qualitative agreement of the simulated data with the experimental data. An agreement at least as good as is shown in Figure 4.1 will probably be sufficient.

Figure 4.1: Example of an initial estimate suitable for starting the regression analysis.

Equilibrium constants

Initial values for equilibrium binding constants are somewhat easier to obtain, in comparison with rate constants. In the equilibrium binding experiment we usually monitor a physical property such as fluorescence, or count of radioactivity per unit of time, in dependence on the total concentration of certain biochemical species.

Let us assume that within the range of concentrations that were chosen by the experimenter, the observed physical quantity (absorbance, radioactivity) has changed to a significant degree. Therefore, for the very initial estimate of simple dissociation equilibrium constants we may take the median value of the experimental concentrations.

Example The equilibrium composition of six different biochemical mixtures containing 50 nM of DNA was measured at different amounts of protein P ( $c_{\scriptscriptstyle \rm P} =$ 20, 40, 80, 160, 320, and 640 nM). The experimenter necessarily had to make a conscious choice of these concentrations, based on some previous knowledge, or simply by increasing the concentrations until a desired effect was in fact observed (e.g., partial or complete saturation). Assuming that the choice of concentrations was sensible, the dissociation constant(s) probably fall within the same range. Therefore we many first try $K_{\rm D} \approx$ 300 nM, which approximately the median value of the experimental range.

For more complex binding mechanisms including several simultaneous equilibria we usually already have an idea whether or not these different equilibria are described by widely different equilibrium constants. It is however quite reasonable to start the analysis by setting all equilibrium constants to the same value, because DynaFit can often successfully optimize these values within three to six orders of magnitude.

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Petr Kuzmic | Jul 12 2005