1. Numerical Enzyme Kinetics using DynaFit software
Petr Kuzmič, Ph.D. BioKin, Ltd.

2. Statement of the problem
There are no traditional (algebraic) rate equations for many important cases:
Time-dependent inhibition in the general case substrate depletion enzyme deactivation
Tight binding inhibition in the general case impurities in inhibitors dissociative enzymes
Auto-activation inhibition e.g., protein kinases
Many other practically useful situations.
Numerical Enzyme Kinetics

3. Numerical Enzyme Kinetics
Solution
Abandon traditional algebraic formalism of enzyme kinetics
Deploy numerical (iterative) fitting modes instead
This is approach is not new: SOFTWARE
D. Garfinkel (1960’s ’s) BIOSYM
C. Frieden (1980’s ’s) KINSIM P. Kuzmic (2000’s - present) DynaFit K. Johnson (2010’s - present) Kinetic Explorer
Numerical Enzyme Kinetics

4. Introduction: A bit of theory
Numerical Enzyme Kinetics

5. Numerical vs. algebraic mathematical models
FROM A VARIETY OF ALGEBRAIC EQUATIONS TO A UNIFORM SYSTEM OF DIFFERENTIAL EQUATIONS k1
EXAMPLE: Determine the rate constant k1 and k-1 for A + B AB
k-1
ALGEBRAIC EQUATIONS
DIFFERENTIAL EQUATIONS
d[A]/dt = -k1[A][B] + k-1[AB]
d[B]/dt = -k1[A][B] + k-1[AB]
d[AB]/dt = +k1[A][B] – k-1[AB]
Applies only when [B] >> [A]
Applies under all conditions
Numerical Enzyme Kinetics

6. Advantages and disadvantages of numerical models
THERE IS NO SUCH THING AS A FREE LUNCH
ADVANTAGE
ALGEBRAIC MODEL
DIFFERENTIAL MODEL + -
can be derived for any molecular mechanism
can be derived automatically by computer
can be applied under any experimental conditions
can be evaluated without specialized software
requires very little computation time does not always require an initial estimate is resistant to truncation and round-off errors
has a long tradition: many papers published - +
Numerical Enzyme Kinetics

7. Numerical Enzyme Kinetics
A "Kinetic Compiler"
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
E + S ---> ES : k1
ES ---> E + S : k2
ES ---> E + P : k3
Input (plain text file):
Rate terms:
Rate equations:
d[E ] / dt =
- k1  [E]  [S]
+ k2  [ES]
k1  [E]  [S]
+ k3  [ES]
k2  [ES]
d[ES ] / dt =
+ k1  [E]  [S]
- k2  [ES]
- k3  [ES]
k3  [ES]
Similarly for other species...
Numerical Enzyme Kinetics

8. System of Simple, Simultaneous Equations
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
"The LEGO method"
of deriving rate equations
E + S ---> ES : k1
ES ---> E + S : k2
ES ---> E + P : k3
Input (plain text file):
Rate terms:
Rate equations:
k1  [E]  [S]
k2  [ES]
k3  [ES]
Numerical Enzyme Kinetics

9. DynaFit can analyze many types of experiments
MASS ACTION LAW AND MASS CONSERVATION LAW IS APPLIED IN THE SAME WAY
EXPERIMENT
DYNAFIT DERIVES A SYSTEM OF ...
chemistry biophysics
Kinetics (time-course)
Equilibrium binding
Initial reaction rates
Ordinary differential equations (ODE) Nonlinear algebraic equations
Nonlinear algebraic equations
enzymology
Numerical Enzyme Kinetics

10. Numerical Enzyme Kinetics
Example 1: Inhibition of HIV protease “Tight binding” inhibition constant from initial rates Use Kiapp values, not IC50’s
Wha’t wrong with IC50’s ?
Numerical Enzyme Kinetics

11. Measures of inhibitory potency
INTRINSIC MEASURE OF POTENCY:
DG = -RT log K i
Depends on
[S] [E]
Example:
Competitive inhibitor
DEPENDENCE ON
EXPERIMENTAL CONDITIONS
1. Inhibition constant
2. Apparent K i
3. IC50 NO
YES NO
YES
K i
K i* = K i (1 + [S]/KM)
IC50 = K i (1 + [S]/KM) + [E]/2
Points to make:
Ki is the best, but laborious to get (need to vary both [S] and [I]).
Kiapp is the next best. It "costs" as much as getting IC50 (vary [I] only), but is more informative than IC50 (does not depend on [E]).
Of these three, IC50 is the least appropriate measure of potency, because...
... for tight binding inhibitors, IC50 shifts with [E]! That means, with a new protocol, a new staff person who changes [E] for some reason, in a different lab, etc., everyone will be getting a different IC50 for the same tight binding inhibitor (e.g., staurosporine) under otherwise identical conditions!
Remember, "tight binding" has nothing to do with any particular value of Ki (picomolar, nanomolar, etc.). "Tight binding" is not a property of an inhibitor, but rather a characteristic of experimental conditions: whenever [E] is numerically close to Ki.
For example, an inhibitor with Ki = 1 M is "tight binding" when [E] = 500 nM (this could happen, if your substrate is really slow).
[E] « K i: IC50  K i*
"CLASSICAL" INHIBITORS:
[E]  K i: IC50  K i*
"TIGHT BINDING" INHIBITORS:
Numerical Enzyme Kinetics

12. Tight binding inhibitors : [E]  K i
HOW PREVALENT IS "TIGHT BINDING"?
A typical data set:
Completely inactive:
Tight binding:
~ 10,000 compounds
~ 1,100
~ 400
... NOT SHOWN
Points to make:
How prevalent is tight binding? For how many compounds is IC50 really meaningless?
This is a histogram of Kiapp for 10,000+ compounds.
Two groups of compounds, centered at about 200 M and 6 M, respectively.
About 400 (or 4%, or one in 25) are "tight binding", so for one compound out of 25, the IC50 has no meaning.
Data courtesy of Celera Genomics
Numerical Enzyme Kinetics

13. Problem: Negative Ki from IC50
FIT TO FOUR-PARAMETER LOGISTIC:
K i* = IC50 - [E] / 2
Points to make:
Can't fix the problem by the "naïve" method: Measure IC50 as usual (four-parameter logistic) Subtract [E]/2 from resulting IC PROBLEM! The Kiapp is negative, which is a nonsensical result.
We never know exactly what the enzyme concentration really is, so that's why the above "naïve" method fails. __________________________ If there is time:
Physical meaning of parameters in the four-parameter logistic equation:
The Hill exponent (nHill = 1.4 in this case) has not physical meaning at all for a tight binding inhibitor. We do know independently that this compound is very "clean" mechanistically, with exactly 1:1 stoichiometry, a pure competitive inhibitor.
The value nHill = 1.4 simply means that the original Hill equation (transformed into the "four parameters logistic") was derived under the assumption that no tight binding occurs! In other words, the basic assumptions of the four- parameter logistic equation are violated. It should not be used for tight binding.
Data courtesy of Celera Genomics
Numerical Enzyme Kinetics

14. Solution: Do not use four-parameter logistic
FIT TO MODIFIED MORRISON EQUATION:
P. Kuzmic et al. (2000) Anal. Biochem. 281,
P. Kuzmic et al. (2000) Anal. Biochem. 286,
Points to make:
Here is how to fix the problem: forget about IC50's completely - forget about the four-parameter logistic equation (4PLE)
Instead of 4PLE, use a modification of the Morrison equation (paper #1 above).
This will get you the Kiapp directly (it is, of course, positive).
As a bonus, you may learn what your enzyme concentration really was.
The conditions under which this simultaneous determination of [E] and Kiapp is possible are described in paper #2 above.
In this case, the enzyme appears about 65% active (4.5 nM actual concentration, as opposed to 7.0 nM nominal concentration).
Data courtesy of Celera Genomics
Numerical Enzyme Kinetics

15. live demo DynaFit in Kiapp determination
fitting model is very simple to understand: E + I <==> EI
Numerical Enzyme Kinetics

16. Numerical Enzyme Kinetics
Comparison of results
Fitting model
Result
IC50 = (1.3 ± 0.13) nM
10 x !
Ki* = (0.10 ± 0.05) nM
Points to make:
Summarize:
Let's stop talking about IC50's completely, and move to apparent inhibition constants (Kiapp) as the most basic measure of inhibitory potency.
For (in this case) 95% of compounds, IC50 and Kiapp are the same number, but for the 5% of "tight binders", they are not.
I call this the "modified Morrison" equation, because the original equation (Williams & Morrison, 1979) does not have background rate term (Vb).
Not knowing [E] exactly is not a problem, because of an interesting "circular logic" situation: If you can't determine [E] from the data, it does not matter, because under those circumstances [E] does not affect Kiapp ("classical" or weak binding) If [E] is in fact important in the sense that it could affect your Kiapp ("tight binding") then you can determine it from the data.
Ki* = (0.10 ± 0.05) nM
E + I <==> EI : Ki*
Numerical Enzyme Kinetics

17. Fitting models for enzyme inhibition: Summary
MEASURE OF INHIBITORY POTENCY
MATHEMATICAL MODELS
Apparent inhibition constant K i* is preferred over IC50
Modified Morrison equation is preferred over four-parameter logistic equation:
A symbolic model (DynaFit) is equivalent and more convenient:
Points to make:
Summarize:
Let's stop talking about IC50's completely, and move to apparent inhibition constants (Kiapp) as the most basic measure of inhibitory potency.
For (in this case) 95% of compounds, IC50 and Kiapp are the same number, but for the 5% of "tight binders", they are not.
I call this the "modified Morrison" equation, because the original equation (Williams & Morrison, 1979) does not have background rate term (Vb).
Not knowing [E] exactly is not a problem, because of an interesting "circular logic" situation: If you can't determine [E] from the data, it does not matter, because under those circumstances [E] does not affect Kiapp ("classical" or weak binding) If [E] is in fact important in the sense that it could affect your Kiapp ("tight binding") then you can determine it from the data.
E + I <==> EI : Ki*
Numerical Enzyme Kinetics

18. Recent IC50 work from Novartis, Basel
Patrick Chène et al. (2009) "Catalytic inhibition of topoisomerase
II by a novel rationally designed ATP-competitive purine analogue"
BMC Chemical Biology 9:1
IC50 = [E]/2 + Ki*
Points to make:
Summarize:
Let's stop talking about IC50's completely, and move to apparent inhibition constants (Kiapp) as the most basic measure of inhibitory potency.
For (in this case) 95% of compounds, IC50 and Kiapp are the same number, but for the 5% of "tight binders", they are not.
I call this the "modified Morrison" equation, because the original equation (Williams & Morrison, 1979) does not have background rate term (Vb).
Not knowing [E] exactly is not a problem, because of an interesting "circular logic" situation: If you can't determine [E] from the data, it does not matter, because under those circumstances [E] does not affect Kiapp ("classical" or weak binding) If [E] is in fact important in the sense that it could affect your Kiapp ("tight binding") then you can determine it from the data.
Numerical Enzyme Kinetics

19. Challenges of moving from IC50 to Kiapp
Legitimate need for continuity structure of existing corporate databases
Simple inertia (“fear of the unknown”)
Lack of awareness
POSSIBLE SOLUTIONS:
Gradual transition (report both IC50 and Ki* for a period of time)
Re-compute historical data: Ki* = IC50 – [E]/2
Points to make:
Summarize:
Let's stop talking about IC50's completely, and move to apparent inhibition constants (Kiapp) as the most basic measure of inhibitory potency.
For (in this case) 95% of compounds, IC50 and Kiapp are the same number, but for the 5% of "tight binders", they are not.
I call this the "modified Morrison" equation, because the original equation (Williams & Morrison, 1979) does not have background rate term (Vb).
Not knowing [E] exactly is not a problem, because of an interesting "circular logic" situation: If you can't determine [E] from the data, it does not matter, because under those circumstances [E] does not affect Kiapp ("classical" or weak binding) If [E] is in fact important in the sense that it could affect your Kiapp ("tight binding") then you can determine it from the data.
Numerical Enzyme Kinetics

20. Finer points of Kiapp determination
Sometimes [E] must be optimized, but sometimes it must not be:
Kuzmic, P., et al. (2000) “High-throughput screening of enzyme inhibitors:
Simultaneous determination of tight-binding inhibition constants and
enzyme concentration” Anal. Biochem. 286, 45-50
“Robust regression” analysis (exclusion of outliers):
Kuzmic, P. (2004) “Practical robust fit of enzyme inhibition data”
Meth. Enzymol. 383,
Serial dilution is not always the best:
Points to make:
Summarize:
Let's stop talking about IC50's completely, and move to apparent inhibition constants (Kiapp) as the most basic measure of inhibitory potency.
For (in this case) 95% of compounds, IC50 and Kiapp are the same number, but for the 5% of "tight binders", they are not.
I call this the "modified Morrison" equation, because the original equation (Williams & Morrison, 1979) does not have background rate term (Vb).
Not knowing [E] exactly is not a problem, because of an interesting "circular logic" situation: If you can't determine [E] from the data, it does not matter, because under those circumstances [E] does not affect Kiapp ("classical" or weak binding) If [E] is in fact important in the sense that it could affect your Kiapp ("tight binding") then you can determine it from the data.
Kuzmic, P. (2011) “Optimal design for the dose–response screening of
tight-binding enzyme inhibitors” Anal. Biochem. 419,
Numerical Enzyme Kinetics

21. Numerical Enzyme Kinetics
Example 1: TPH1 inhibition Determination of initial reaction rates from nonlinear progress curves
Numerical Enzyme Kinetics

22. First look at raw experimental data
NOTE THE EXCELLENT REPRODUCIBILITY: THESE ARE TWO REPLICATES ON TOP OF EACH OTHER
Numerical Enzyme Kinetics

23. The text-book recipe: Linear fit
STANDARD APPROACH: FIT A STRAIGHT LINE TO THE “INITIAL PORTION” OF EACH CURVE
Stein, R. “Kinetics of Enzyme Action” (2011), sect
Standard curve
1 µM of product (5-HT)
corresponds to an increase
in fluorescence of RFU
Fluorescence change expected at 10% conversion: ~ 1000 RFU
Numerical Enzyme Kinetics

24. Looking for linearity at less than 10% conversion
IS THIS A “STRAIGHT LINE”?
slope cannot
change abruptly
1000 RFU
10% conversion
slope must be
changing continously
Numerical Enzyme Kinetics

25. Numerical analysis: Fit to a full system of differential equations
A NONLINEAR MODEL OF COMPLETE REACTION PROGRESS
[task]
data = progress
task = fit
[mechanism]
E + S <==> ES : kas kds
ES ---> E + P : kdp
[constants]
kas =
kds = ?
kdp = ?
[concentrations]
E = 0.01
[responses]
P = 100 ?
[data]
sheet B01.txt
column | offset auto ? | concentration S = 60
[output]
directory .../output/fit-B01
rate-file .../data/rates/B01v.txt
DYNAFIT OUTPUT
DYNAFIT INPUT
Numerical Enzyme Kinetics

26. DynaFit auto-generated fitting model
A NONLINEAR MODEL OF COMPLETE REACTION PROGRESS
[mechanism]
E + S <==> ES : kas kds
ES ---> E + P : kdp
Numerical Enzyme Kinetics

27. Instantaneous rate plot
THE SLOPE OF THE PROGRESS CURVE DOES CHANGE ALL THE TIME
fluorescence:
rate of change in fluorescence:
t = 0: rate ~ 1.0
t = 600: rate ~ 0.8
20 % drop in reaction rate over the first 10 minutes
(less than 10% substrate conversion)
Numerical Enzyme Kinetics

28. live demo DynaFit in “full automation” mode
suitable for routine processing of many compounds
mechanism is assumed to be known independently
Numerical Enzyme Kinetics

29. Numerical Enzyme Kinetics
Example 2: Inhibition of 5a-ketosteroid reductase “Slow, tight” binding Model discrimination analysis
Numerical Enzyme Kinetics

30. Possible molecular mechanisms of time-dependent inhibition
Slow binding proper
slow
E + I
E.I
Rearrangement of initial enzyme-inhibitor complex
fast
slow
E + I
E.I
E.I*
etc. (several other possibilities)
Numerical Enzyme Kinetics

31. Model discrimination analysis in DynaFit
ANY NUMBER OF ALTERNATE KINETIC MODELS CAN BE COMPARED IN A SINGLE RUN
DYNAFIT INPUT SCRIPT FILE:
[task]
data = progress | task = fit | model = one-step ?
[mechanism]
E + S ---> ES : kaS
ES ---> E + P : kdP
E + I <==> EI : kaI kdI
...
data = progress | task = fit | model = two-step ?
EI <==> EJ : kIJ kJI
Numerical Enzyme Kinetics

32. live demo DynaFit in “model selection” mode
model selection criteria (AIC, BIC)
residual analysis
use common sense to check results
Numerical Enzyme Kinetics

33. Summary and conclusions
Benefits of using DynaFit in the study of enzyme inhibition:
No mathematical models, only symbolic models (E + I <==> EI) Everybody can understand this. This prevents making mistakes and facilitates “transfer of knowledge”.
Automation Can be used to process 1000’s of compounds in a single run. Automatic model selection (Bayesian Information Criterion).
No restrictions on experimental design Not necessary to have large excess of inhibitor (“tight binding”)
No restrictions on reaction mechanism Any number of interactions and molecular species
Points to make:
Ki is the best, but laborious to get (need to vary both [S] and [I]).
Kiapp is the next best. It "costs" as much as getting IC50 (vary [I] only), but is more informative than IC50 (does not depend on [E]).
Of these three, IC50 is the least appropriate measure of potency, because...
... for tight binding inhibitors, IC50 shifts with [E]! That means, with a new protocol, a new staff person who changes [E] for some reason, in a different lab, etc., everyone will be getting a different IC50 for the same tight binding inhibitor (e.g., staurosporine) under otherwise identical conditions!
Remember, "tight binding" has nothing to do with any particular value of Ki (picomolar, nanomolar, etc.). "Tight binding" is not a property of an inhibitor, but rather a characteristic of experimental conditions: whenever [E] is numerically close to Ki.
For example, an inhibitor with Ki = 1 M is "tight binding" when [E] = 500 nM (this could happen, if your substrate is really slow).
Numerical Enzyme Kinetics

34. Possible deployment at Novartis / Horsham
Free support from BioKin Ltd included with site license:
Periodic data review via Send your raw data, get results back in 72 hours (in most cases).
Phone support
Call any time during US (EST) business hours.
Periodic on-site “DynaFit course” Once a year (either in the Fall or Spring); one-day workshop format.
Free upgrades DynaFit continues to evolve (e.g., “Optimal Experimental Design”)
Points to make:
Ki is the best, but laborious to get (need to vary both [S] and [I]).
Kiapp is the next best. It "costs" as much as getting IC50 (vary [I] only), but is more informative than IC50 (does not depend on [E]).
Of these three, IC50 is the least appropriate measure of potency, because...
... for tight binding inhibitors, IC50 shifts with [E]! That means, with a new protocol, a new staff person who changes [E] for some reason, in a different lab, etc., everyone will be getting a different IC50 for the same tight binding inhibitor (e.g., staurosporine) under otherwise identical conditions!
Remember, "tight binding" has nothing to do with any particular value of Ki (picomolar, nanomolar, etc.). "Tight binding" is not a property of an inhibitor, but rather a characteristic of experimental conditions: whenever [E] is numerically close to Ki.
For example, an inhibitor with Ki = 1 M is "tight binding" when [E] = 500 nM (this could happen, if your substrate is really slow).
Numerical Enzyme Kinetics