This problem utilizes the original experimental data published by Michaelis and Menten in 1913. The Michaelis-Menten paper arguably represents the beginning of enzyme kinetics as a systematic field.

"Die Kinetik der Invertinwirkung" Michaelis, L.; and Menten, M.L. (1913) Biochem. Z. 49, 333-369.

This DynaFit example is based on the analysis of the reaction progress curves, as opposed to the commonly used initial reaction velocities. Indeed, the original paper drew most of their important conclusions from the analysis of the reaction progress.

The DynaFit script file is set up for a model discrimination analysis, which concludes that, based on rigorous statistical analysis, the invertase enzyme is partially inhibited by the substrate (saccharose).

Here we analyze the initial reaction velocities reported by Michaelis and Menten (1913). The fitting model is the simple "Michaelis-Menten" mechanism.

We can see that that the initial velocity data actually does not fit very well. This is not too surprising, because the highest concentration of saccharose was [S] = 333 mM (an extremely high value).

A single reaction progress curve was collected by HPLC measurements of testosterone reduction over time (22 data points). The data were fit to the simplest possible enzyme mechanism (the "Van Slyke – Cullen" mechanism), including only two steps.

The original data are taken from our paper on the mechanism of slow, tight inhibition of 5-alpha-ketosteroid reductase by azasteroids:

   E + S <==> ES     :     k   kd
   ES --> E + P      :     kr

   E + S --> ES      :     k
   ES --> E + P      :     kr

Another look, using confidence intervals for rate constants, at the two mechanisms shown above. For the "Michaelis - Menten" mechanism the outcome of confidence interval estimation is as follows:

1. Only lower limit can be determined for the association rate constant k.
2. Neither the lower limit nor the upper limit can be determined for the dissociation rate constant k_d.
3. Both the lower limit and the upper limit can be determined for the reaction rate constant k_r.

The HIV protease is a dissociative dimer. We have previously measured the rate constants that describe the monomer - dimer equilibrium, and determined that the protease dimer is stabilized by bound substrate:

Here we show that even when the reaction progress curves from the previous example are analyzed simultaneously ("global analysis"), which is a much more stringent requirement for the kinetic model, the proposed denaturation mechanism holds.

Relating again to the HIV denaturation kinetics induced by mechanical stirring, we show here the analysis of initial velocities from the assays analyzed above. The initial velocity data are fit in two different ways:

1. using the "rapid equilibrium approximation", where the binding of enzyme and substrate is assumed to be infinitely rapid;
2. computing the reaction velocity by numerical integration of a system of differential equations corresponding to the full mechanism.

This problem in this directory relate to an enzyme assay in which the substrate and the inhibitor are not simple molecules, but instead are dissociative, noncovalent complexes. In particular, the substrate for hexokinase is a complex of ATP and Mg(2+), whereas other metals (such as europium Eu or lutetium Lu) form ATP complexes that are inhibitory.

Morrison and Cleland [1] were able to derive a rate equation for this system based on the following assumptions:

1. the concentrations of ATP and metals are very much higher than the metal-ATP dissociation constants, so that no free ATP exists in solution;
2. dimerization of ATP at higher concentrations is neglected;
3. only 1:1 metal-ATP complexes are considered;
4. the metal-ATP complex is such a poor inhibitor that the operational concentration is always very much higher than the corresponding inhibition constant.

[mechanism]

   Mg + ATP <===> S
   Eu + ATP <===> I

   E + S <===> E.S
   E.S ----> E + P
   E + I <===> E.I

In this example problem, we analyzed experimental data on europium-ATP inhibition, scanned electronically from Figure 3 of ref. [1].

A possible experiment designed to study the isomerization of enzyme forms during catalysis involves the use of isotopically labeled substrates. The enzyme is equilibrated with a certain amount of labeled S' and/or P', until the equilibrium between S' and P' is completely established. Then we add an excess amount of unlabeled S or P (say, hundred-fold over the amount of radioactive material), and observe that changes in concentration of the labeled reactants. The reaction mechanism can be formulated as follows:

[1] "Isomerization of the free enzyme versus induced fit: effects of steps involving induced fit that bypass enzyme isomerization on flux ratios and countertransport." Britton, H. G. (1997) Biochem. J. 321, 187-199.

[2] "Kinetics of iso mechanisms." Rebholz, K.L. and Northrop, D.B. (1995) Meth. Enzymol. 249, 211-240.

This example problem uses the same experimental data on fumarase isomerization during catalysis that were analyzed above:

"Kinetics of iso mechanisms." Rebholz, K.L. and Northrop, D.B. (1995) Meth. Enzymol. 249, 211-240.

In this particular computation we show that confidence intervals for several important rate constants in the reaction mechanism.

The experimental data in this example problem were provided by Prof. Linda Hsieh-Wilson (Caltech) when she was a graduate student in Peter Schultz’ lab at Berkeley. The data represent a collection of progress curves from the peptidyl-prolyl cis-trans isomerization assay, catalyzed by recombinant human cyclophilin. The salient feature of this assay is that the enzyme reaction is accompanied by a significant uncatalyzed background reaction.

In the early 1990’s we had to apply significant algebraic skills to develop a mathematical model for the kinetics of similar enzymes:

"Mathematical models for the kinetics of peptidyl-prolyl cis-trans isomerases" Kuzmic, P.; et al. (1992) in Peptides - Chemistry and Biology 249, Smith, J.A. (Ed.), ESCOM, Leiden, pp. 470-471.

Indeed, the resulting algebraic expressions are quite complicated and have to be evaluated "recursively". In contrast, in DynaFit we can specify the mathematical model simply by typing:

[mechanism]

   E + S <==> ES   :    k    ks
   ES --> E + P    :    kr
   S --> P         :    kt

This example problem illustrates one of the most common tasks in an enzymology laboratory, the determination of a (competitive) inhibition constant:

1. Determine initial velocities form a collection of reaction progress curves.
2. Assemble from the results a concentration - velocity data file.
3. Fit the initial velocities to the simple competitive inhibition mechanism.

The DynaFit script in this example problem fully automates all three steps.

The experimental data for this example problem were supplied by Kathryn Houseman (Searle Research & Development, currently Monsanto). The data represent initial velocity measurements at multiple substrate concentrations and multiple inhibitor concentrations. The goal of the analysis was to verify that the mechanism of inhibition is competitive.

An important feature of the mechanism is that the HIV protease is treated explicitly as a dissociative dimer:

[mechanism]  

   M + M <==> E      :     Kd      dissoc.
   E + S <==> ES     :     Ks     dissoc.
   ES ---> E + P     :         kr     
   E + I <==> EI     :     Ki      dissoc.

Using the same set of initial velocity data as in the previous example problems, the goal of this analysis is to determine approximate confidence intervals for the kinetic constants. It is important to realize that the formal standard errors ("plus-or-minus" values often reported in the biochemical literature) strictly speaking are always incorrect in the case of nonlinear regression.

The experimental data used in this problem were originally published in the following paper:

"Inhibition of angiotensin converting enzyme: Mechanism and substrate dependence" Shapiro, R.; and Riordan, J.F. (1984) Biochemistry 23 5225-5233.

One difficulty with a particular inhibitor of ACE was that the authors (at Harvard University) were not able to propose, with a reasonable amount of plausibility, a specific inhibition mechanism. Several mechanisms were discussed, including at least hypothetically the partial, mixed-type noncompetitive, two-site inhibition. However, the data were only crudely analyzed by using a parabolic (second order polynomial) regression of double-reciprocal plots.

Here we show that a sufficient mechanism for the "mystery" inhibitor involves partial, mixed-type noncompetitive, single-site binding.

The experimental data used in this example come from Figure 5 of the following paper:

"Inhibition of arginine aminopeptidase by Bestatin and Arphamenine analogues. Evidence for a new mode of binding to aminopeptidases." Harbeson, S.L.; and Rich, D.H. (1988) Biochemistry 27 7301-7310.

The authors found for their particular inhibitor an "unusual noncompetitive double-reciprocal plot", suggesting an unusual mechanism of inhibition. In contrast with their conclusions, which were not supported by a rigorous model discrimination anlaysis, we find that the partial, mixed-type noncompetitive, two-site inhibition tentatively entertained for ACE is in fact operating here.

Once the mechanism of aminopeptidase B inhibition was established in the previous example problem, we then proceeded to determine the full confidence intervals for kinetic constants. The corresponding DynaFit script as well as the results are summarized in the output files below.

In this example problem we determine the 95% confidence intervals for the kinetic constants that appear in the most plausible mechanistic model established in the preceding example.

This example problem is a part of three part sequence related to the study of "slow, tight" inhibition of the HIV protease by the substrate analog inhibitor JG-365

In this particular DynaFit example we prepare for the analysis of the inhibition progress curves by analyzing substrate kinetic data alone. The substrate kinetic constants determined here will be used as a fixed constants in the subsequent investigation of the inhibition mechanism.

Here we analyze a reaction progress curve from a fluorescence assay of the HIV protease, using tyrosine as the fluorophore:

It is interesting that JG-365 has [1] been cited at least 18 times in the biochemical literature, but the fact that it is a slow binding inhibitor of the HIV protease has never been noticed.

After the most plausible mechanism was selected for the "slow, tight" inhibition of HIV protease by JG-365 (see preceding problem), we then determined the confidence intervals for all rate constants in the mechanism. It is important that the confidence intervals are quite different from the formal standard errors (± values usually reported in biochemical literature).

The experimental data for this DynaFit example were provided by Dr. Sergei Gulnik (National Cancer Institute, Frederick, MD). The HIV protease is inhibited by a "slow, tight" inhibitor that is also irreversible. The irreversible step in the reaction mechanism is specified in the DynaFit script as is shown below:

[mechanism]

   M + M <==> E      : ka   kd
   E + S <==> ES     : kon  ks
   ES ---> E + P     : kr
   E + P <==> EP     : kon  kp
   E + I <==> EI     : kon  ki
   EI --> EJ         : kde

Consider an enzyme inhibition mechanism for a dissociative enzyme dimer (e.g., the HIV protease, integrase, or reverse transcriptase) in which the inhibitor binds simultaneously to both the monomer and the dimer. In the DynaFit script, this situation is represented as follows:

[mechanism]  

;  The enzyme dimer dissociates into subunits:

   M + M <==> E      :    kmd    kdm

;  Only the dimer is catalytically active:

   E + S <==> ES     :    kas    kds
   ES ---> E + P     :    kr

;  The inhibitor binds to both enzyme forms:

   M + I <==> MI     :    k1a    k1d
   E + I <==> EI     :    k2a    k2d

In this DynaFit example, we use a heuristic simulation to investigate whether it would be possible to determine all rate constants that characterize this complex enzyme-inhibitor interaction. The answer is found by (a) simulating a set of progress curves and (b) subsequently fitting the theoretical model to these simulated curves.

A textbook problem

The time course of alpha-pinene isomerization is analyzed in view of a postulated molecular mechanism. This particular problem served as a test case in several textbooks on statistics (e.g., [1]). I was also used in original statistical research papers to illustrate particularly difficult issues arising in the analysis of "multi-response" observations [2].

[1] "Nonlinear Regression" Seber, G.A.F. and Wild, C.J. (1989) John Wiley & Sons, Inc., New York, p. 551.

[2] "Some Problems Associated with the Analysis of Multiresponse Data" Box, G.E.P. et al. (1971) Technometrics 15, 33-51.

These "problems" are conveniently avoided in DynaFit simply by typing:

[mechanism]

pinene ---> dipentene        :    k1
pinene ---> allo-ocimene     :    k2
allo-ocimene ---> pyronene   :    k3
allo-ocimene <==> dimer      :    k4   k5

Here we attempt to determine whether the time-dependent binding of conotoxin to nicotinic receptor proceeds with or without the formation of an intermediate complex. The (unpublished) data were provided by Dr. Stewart N. Abramson, Department of Pharmacology, School of Medicine, University of Pittsburgh, PA.

A salient feature of this data set is that two runs at nominally the same concentration of conotoxin are treated as separate. Such segregation of presumably identical experiments greatly improved the goodness of fit and essentially allowed the model discrimination analysis.

Heuristic Simulation Study

Here we simulate data according to the "two-step" binding model for conotoxin / nicotinic receptor binding, and then fit the simulate data to the "one-step" model. Unfortunately, the results show that at the concentrations used in this study, the two binding models cannot be distinguished.

In this example problem, we analyze a gel-shift assay data on nuclease (protein P) binding to DNA (D). The nuclease enzyme is known to exist primarily either as a protein trimer (T) or hexamer (T2). Thus, the binding mechanism is described by three equilibrium constants, K₁, K₂, and K₃:

   P + P + P <=> T    :   K1   dissoc
   T + D <==> TD      :   K2   dissoc
   TD + T <==> T2D    :   K3   dissoc

In this DynaFit example problem we focus on the importance of global analysis of multiple sets of experimental data. In the output files below, please examine the confidence intervals for the equilibrium constants K₁, K₂, and K₃. Only when the three available data sets from gel shift assays are analyzed simultaneously do we obtain a narrow, well-defined confidence interval for all three equilibrium constants.

This script file illustrates the importance of allowing for adjustable base-line signal in the analysis of ligand-receptor binding. The (unpublished) radioligand binding data were provided by Dr. David Raffel, University of Michigan, Department of Nuclear Medicine.

The goal of this example problem is to illustrate the importance of allowing for an adjustable constant background ("baseline") signal in the analysis of equilibrium binding data.

In this example problem we analyze hypothetical data from a ligand - receptor binding experiment where the ligand not only binds at a specific binding site, but also binds nonspecifically. A detailed explanation and a step-by-step tutorial is given in the following online article:

In this DynaFit example problem we analyze experimental data from the following paper:

Time-Resolved NMR Study

A ionophore molecule (a "calix[4]tube") was incubated with a potassium ion salt and the amount of of ionophore-metal ion complex was measured over time.

The (unpublished) experimental data were provided by Dr. Philippe Schmitt, Oxford University, Department of Chemistry, as part of an ongoing investigation of this novel class ionophores:

"Slow, Tight" Binding of an Inhibitor with Multiple Interconverting Forms

We have studied the "slow, tight" binding of the immunosupressant drug Cyclosporin-A to recombinant human cyclophilin:

"Lithium chloride perturbation of cis-trans peptide bond equilibria: effect on conformational equilibria in cyclosporin A and time-dependent inhibition of cyclophilin" Kofron, J.L.; Kuzmic, P.; Kishore, V., Gemmecker, G.; Fesik, S.W. and Rich, D.H. (1992) J. Amer. Chem. Soc. 114, 2670-2675.

In the original paper, the experimental data were analyzed by a specially designed mathematical formalism not applicable to any other biochemical system. Here we subject the same data to a statistical analysis by using the completely general symbolic modelling language of DynaFit.

The Kinetics of Competitive Ligand Binding

The time-course of thrombin binding to hirudin is observed over time. In a separate experiment, the slow dissociation of the thrombin-hirudin complex is also observed over time. Both of these experiments are analyzed simultaneously. Importantly, in both of them, thrombin competes for hirudin binding with a chemical mutant, dehydrothrombin. The purpose of this experiment was to determine all four rate constants.

Wedemeyer, Ashton and Scheraga (Baker Laboratory of Chemistry, Cornell University) analyzed the experiment by using a very elaborate mathematical formalism, involving more than a hundred equations:

IX + TF.VIIa   <==>     IX.TF.VIIa   
IX.TF.VIIa     -->      TF.VIIa + IXa
X + TF.VIIa    <==>     X.TF.VIIa    
X.TF.VIIa      -->      TF.VIIa + Xa
X + VIIIa.IXa  <==>     X.VIIIa.IXa 
X.VIIIa.IXa    -->      VIIIa.IXa + Xa
IX + Xa        -->      Xa + IXa      
V + Xa         -->      Va + Xa       
VIII + Xa      -->      VIIIa + Xa    
V + IIa        -->      IIa + Va      
VIII + IIa     -->      VIIIa + IIa   
II + Va.Xa     <==>     II.Va.Xa      
II.Va.Xa       -->      Va.Xa + mIIa  
mIIa + Va.Xa   -->      Va.Xa + IIa   
VIIIa + IXa    <==>     VIIIa.IXa    
Va + Xa        <==>     Va.Xa        

Proline racemase served as the model enzyme for the development of very complex, esoteric mathematical models for enzyme kinetics:

S1 + E <===> S1.E        :    k      ks1
  S1.E  ---> E + S2      :    kr1
S2 + E <===> S2.E        :    k      ks2

        ---> S1          :    v1
     S2 --->             :    v2

[1] "Oscillations in Biological and Chemical Systems" Sel'kov, E. E. (1967) Nauka (Acad. Sci. USSR), Moscow.

[2] "Symmetry breaking instabilities in biological systems" Prigogine, I., et al. (1969) Nature 223, 913-916.

The mechanism of "slow, tight" inhibition of dihydrofolate reductase is quite complex and yet all twenty eight rate constants have been either directly measured or reasonably inferred.

Substrate rate rate constants for E. Coli enzyme:

"Substrate channeling" relates to two consecutive enzyme reactions, where the product of the first reaction is never released from the enzyme + substrate complex. Instead, the second enzyme binds to form a ternary complex enzyme 1 + substrate + enzyme 2. The final reaction product is formed from this ternary complex.

Note several interesting characteristics of the generated output curves:

• Product is formed with an initial lag phase.
• Channeling cannot be observed at all, unless the corresponding rate constants reach certain critical values.