Algebraic solution for time course of enzyme assays
DynaFit in the Analysis of Enzyme Progress Curves
ONLY THE SIMPLEST REACTION MECHANISMS CAN BE TREATED IN THIS WAY
EXAMPLE: “slow binding” inhibition
Irreversible enzyme inhibition
Petr Kuzmic, Ph.D.
BioKin, Ltd.
WATERTOWN, MASSACHUSETTS, U.S.A.
TASK: compute [P] over time
TOPICS
1.
Numerical integration vs. algebraic models: DYNAFIT
2.
Case study: Caliper assay of irreversible inhibitors
3.
Devil is in the details: Problems to be aware of
SIMPLIFYING ASSUMPTIONS:
1.
No substrate depletion
2.
No “tight” binding
Kuzmic (2008) Anal. Biochem. 380, 5-12
DynaFit: Enzyme Progress Curves
Enzyme kinetics in the “real world”
2
Progress curvature at low initial [substrate]
SUBSTRATE DEPLETION USUALLY CANNOT BE NEGLECTED
SUBSTRATE DEPLETION IS MOST IMPORTANT AT [S]0 << KM
[S]0 = 10 µM, KM = 90 µM
~40% decrease in rate
final
slope
1.34
8% systematic error
0.82
initial
slope
linear fit:
10% systematic
error
residuals
not random
Sexton, Kuzmic, et al. (2009) Biochem. J. 422, 383-392
DynaFit: Enzyme Progress Curves
Kuzmic, Sexton, Martik (2010) Anal. Biochem., submitted
3
DynaFit: Enzyme Progress Curves
4
Problems with algebraic models in enzyme kinetics
Numerical solution of ODE systems: Euler method
THERE ARE MANY SERIOUS PROBLEMS AND LIMITATIONS
COMPLETE REACTION PROGRESS IS COMPUTED IN TINY LINEAR INCREMENTS
• Can be derived for only a limited number of simplest mechanisms
mechanism:
A
k
B
• Based on many restrictive assumptions:
practically useful methods
are much more complex!
differential
rate
equation
d [A] / d t = - k [A]
- no substrate depletion
- weak inhibition only (no “tight binding”)
• Quite complicated when they do exist
Δ [A] / Δ t = - k [A]
[A]
[A]0
Δ [A] = - k [A] Δ t
[A] t
The solution: numerical models
straight line segment
[A] t+Δt
[A] t + Δ t - [A] t = - k [A] t Δ t
• Can be derived for an arbitrary mechanism
time
• No restrictions on the experiment (e.g., no excess of inhibitor over enzyme)
• No restrictions on the system itself (“tight binding”, “slow binding”, etc.)
[A] t + Δ t = [A] t - k [A] t Δ t
Δt
• Very simple to derive
DynaFit: Enzyme Progress Curves
5
DynaFit: Enzyme Progress Curves
6
1
Automatic derivation of differential equations
Software DYNAFIT (1996 - 2010)
IT IS SO SIMPLE THAT EVEN A “DUMB” MACHINE (THE COMPUTER) CAN DO IT
PRACTICAL IMPLEMENTATION OF “NUMERICAL ENZYME KINETICS”
Example input (plain text file):
Rate terms:
Rate equations:
“E disappears”
“multiply
[E] × [S]”
k1
E + S ---> ES
k1 × [E] × [S]
k2
ES ---> E + S
k2 × [ES]
k3
ES ---> E + P
d[E ]/dt = - k1 × [E] × [S]
“E is
formed”
+ k2 × [ES]
2009
+ k3 × [ES]
k3 × [ES]
DOWNLOAD http://www.biokin.com/dynafit
similarly for other species
(S, ES, and P)
Kuzmic (2009) Meth. Enzymol., 467, 247-280
DynaFit: Enzyme Progress Curves
7
DynaFit: Enzyme Progress Curves
8
DYNAFIT: What can you do with it?
DYNAFIT applications: mostly biochemical kinetics
ANALYZE/SIMULATE MANY TYPES OF EXPERIMENTAL DATA ARISING IN BIOCHEMICAL LABORATORIES
BUT NOT NECESSARILY: ANY SYSTEM THAT CAN BE DESCRIBED BY A FIRST-ORDER ODEs
• Basic tasks:
- simulate artificial data (assay design and optimization)
- fit experimental data (determine inhibition constants)
- design optimal experiments (in preparation)
• Experiment types:
- time course of enzyme assays
- initial rates in enzyme kinetics
- equilibrium binding assays (pharmacology)
• Advanced features:
- confidence intervals for kinetic constants
-
* Monte-Carlo intervals
* profile-t method (Bates & Watts)
~ 650 journal articles total
goodness of fit - residual analysis (Runs-of-Signs Test)
model discrimination analysis (Akaike Information Criterion)
robust initial estimates (Differential Evolution)
robust regression estimates (Huber’s Mini-Max)
DynaFit: Enzyme Progress Curves
9
DynaFit: Enzyme Progress Curves
10
Traditional analysis of irreversible inhibition
DynaFit in the Analysis of Enzyme Progress Curves
Irreversible enzyme inhibition
BEFORE 1981 (IBM-PC) ALL LABORATORY DATA MUST BE CONVERTED TO STRAIGHT LINES
1962
E+I
Ki
E•I
kinact
X
Kitz-Wilson plot
Petr Kuzmic, Ph.D.
BioKin, Ltd.
WATERTOWN, MASSACHUSETTS, U.S.A.
TOPICS
1.
Numerical integration vs. algebraic models: DYNAFIT
2.
Case study: Caliper assay of irreversible inhibitors
3.
Devil is in the details: Problems to be aware of
log (enzyme activity/control)
1 / slope of straight line (kobs)
increasing
[inhibitor]
1/kinact
time
1/Ki
1 / [inhibitor]
Kitz & Wilson (1962) J. Biol. Chem. 237, 3245-3249
DynaFit: Enzyme Progress Curves
12
2
Traditional analysis – “Take 2”: nonlinear
Traditional analysis: Three assumptions (part 1)
AFTER 1981 STRAIGHT LINES ARE NO LONGER NECESSARY (“NONLINEAR REGRESSION”)
“LINEAR” OR “NONLINEAR” ANALYSIS – THE SAME ASSUMPTIONS APPLY
1981
E+I
Ki
E•I
kinact
X
1. Inhibitor binds only weakly to the enzyme
IBM-PC (Intel 8086)
kobs
enzyme activity/control
increasing
[inhibitor]
kinact
0.5 kinact
no “tight binding”
time
[I], Ki must not be comparable with [E]
[inhibitor]
Ki
kobs =kinact / (1 + Ki / [I])
A = A0 exp(-kobs t)
DynaFit: Enzyme Progress Curves
13
DynaFit: Enzyme Progress Curves
14
Traditional analysis: Three assumptions (part 2)
Traditional analysis: Three assumptions (part 3)
“LINEAR” OR “NONLINEAR” ANALYSIS – THE SAME ASSUMPTIONS APPLY
“LINEAR” OR “NONLINEAR” ANALYSIS – THE SAME ASSUMPTIONS APPLY
3. Initial binding/dissociation is much faster than inactivation
2. Enzyme activity over time is measured “directly”
(“rapid equilibrium approximation”)
In a substrate assay, plot of product [P] vs. time must be a straight line at [I] = 0
ASSUMED MECHANISM:
E+I
Ki
E•I
kinact
X
ACTUAL MECHANISM IN MANY CASES:
E+I
E+S
Ki
Km
E•I
E•S
kinact
kcat
X
E+P
DynaFit: Enzyme Progress Curves
15
DynaFit: Enzyme Progress Curves
16
Simplifying assumptions: Requirements for data
Actual experimental data (COURTESY OF Art Wittwer, Pfizer)
HOW MUST OUR DATA LOOK SO THAT WE CAN ANALYZE IT BY THE TRADITIONAL METHOD ?
NEITHER OF THE TWO MAJOR SIMPLIFYING ASSUMPTION ARE SATISFIED !
1. control curve = straight line
[I] = 0
SIMULATED:
[I] = 0
1. control curve
[E] = 1 nM
[I] = 100 nM
[I] = 200 nM
[I] = 2.5 nM
[S] = 10 μM
Km = 1 μM
[E] = 0.3 nM
[I] = 400 nM
[I] = 800 nM
2. inhibitor concentrations
must be much higher than
enzyme concentration
DynaFit: Enzyme Progress Curves
17
2. inhibitor concentrations
within the same order of
magnitude
DynaFit: Enzyme Progress Curves
18
3
Numerical model for Caliper assay data
Caliper assay: Results of fit – optimized [E]0
NO ASSUMPTIONS ARE MADE ABOUT EXPERIMENTAL CONDITIONS
THE ACTUAL ENZYME CONCENTRATION SEEMS HIGHER THAN THE NOMINAL VALUE
Ki ~ [E]0
“tight binding”
units: μM, minutes
DynaFit input:
[mechanism]
E + S ---> E + P
E + I <===> EI
EI ---> X
:
:
:
kdp
kai
kx
koff ~ kinact
not “rapid equilibrium”
kdi
Ki
= koff/kon
= 1 nM
Automatically generated fitting model:
E+I
E+S
DynaFit: Enzyme Progress Curves
kon
koff
kinact
E•I
kcat/Km
X
kon
koff
7.1 × 104
0.00007
kinact
kcat/Km
0.00014
1.5 × 105
[E]0
1.0
E+P
19
M-1.sec-1
sec-1
sec-1
M-1.sec-1
nM
DynaFit: Enzyme Progress Curves
20
Caliper assay violates assumptions of classic analysis
Numerical model more informative than algebraic
ALL THREE ASSUMPTIONS OF THE TRADITIONAL ANALYSIS WOULD BE VIOLATED
MORE INFORMATION EXTRACTED FROM THE SAME DATA
Traditional model
1.
Enzyme concentration is not much lower than [I]0 or Ki
2.
The dissociation of the E•I complex is not much faster than inactivation
3.
E+I
Ki
General numerical model
(Kitz & Wilson, 1962)
E•I
kinact
X
E+I
kon
koff
E•I
kinact
X
The control progress curve ([I] = 0) is not a straight line
(Substrate depletion is significant)
very fast
no assumptions !
very slow
Two model parameters
Three model parameters
If we used the traditional algebraic analysis,
the results (Ki, kinact) would likely be incorrect.
DynaFit: Enzyme Progress Curves
Add another dimension
(à la “residence time”)
to the QSAR ?
21
DynaFit: Enzyme Progress Curves
22
Numerical modeling looks simple, but...
DynaFit in the Analysis of Enzyme Progress Curves
A RANDOM SELECTION OF A FEW TRAPS AND PITFALLS
Irreversible enzyme inhibition
Petr Kuzmic, Ph.D.
BioKin, Ltd.
• Residual plots
we must always look at them
WATERTOWN, MASSACHUSETTS, U.S.A.
• Adjustable concentrations
we must always “float” some concentrations in a global fit
TOPICS
1.
Numerical integration vs. algebraic models: DYNAFIT
2.
Case study: Caliper assay of irreversible inhibitors
3.
Devil is in the details: Problems to be aware of
• Initial estimates: the “false minimum” problem
nonlinear regression requires us to guess the solution beforehand
• Model discrimination: Use your judgment
the theory of model discrimination is far from perfect
DynaFit: Enzyme Progress Curves
24
4
Residual plots
Direct plots of data: Example 1
RESIDUAL PLOTS SHOWS THE DIFFERENCE BETWEEN THE DATA AND THE “BEST-FIT” MODEL
THESE TWO PLOTS LOOK “INDISTINGUISHABLE”, DO THEY NOT ?
Two-step inhibition
One-step inhibition
residual
signal
+
data
model
0
residual
time
time
E+I
DynaFit: Enzyme Progress Curves
kon
koff
E•I
kinact
DynaFit: Enzyme Progress Curves
X
26
WE DON’T HAVE TO RELY ON VISUALS (“LOG” VS. “HORSESHOE”)
Two-step inhibition
One-step inhibition
-10
E+I
Residual plots: Runs-of-signs test
THESE TWO PLOTS LOOK “VERY DIFFERENT”, DO THEY NOT ?
“horseshoe”
X
25
Residual plots: Example 1
+5
kinact
Two-step inhibition
One-step inhibition
“log”
+2.5
-5
probability 31%
probability 0%
E+I
kinact
X
E+I
DynaFit: Enzyme Progress Curves
kon
koff
E•I
kinact
passes p > 0.05 test
X
27
DynaFit: Enzyme Progress Curves
Residual plots: Example 2
Relaxed inhibitor concentrations: Example 1
SOMETIMES IT’S O.K. TO HAVE “OUTLIERS” – USE YOUR JUDGMENT
WE ALWAYS HAVE “TITRATION ERROR” !
Residual plots: fixed [I]
It’s not always easy
to judge “just how good”
the residuals are:
28
Residual plots: relaxed [I]
“something” happened
with the first three
time-points
DynaFit: Enzyme Progress Curves
29
DynaFit: Enzyme Progress Curves
30
5
Relaxed inhibitor concentrations: Example 1 (detail)
Relaxed inhibitor concentrations: not all of them
WE ALWAYS HAVE “TITRATION ERROR” !
ONE (USUALLY ANY ONE) OF THE INHIBITOR CONCENTRATIONS MUST BE KEPT FIXED
Residual plots: fixed [I]
Residual plots: relaxed [I]
DynaFit script:
...
[data]
directory
extension
file
file
file
file
file
file
./users/COM/.../100514/C1/data
txt
0nM
2p5nM
5nM
10nM
20nM
40nM
|
|
|
|
|
|
offset
offset
offset
offset
offset
offset
auto
auto
auto
auto
auto
auto
?
?
?
?
?
?
|
|
|
|
|
conc
conc
conc
conc
conc
I
I
I
I
I
=
=
=
=
=
0.0025
0.0050
0.0100
0.0200
0.0400
?
?
FIXED
?
?
...
nD = 100, nP = 44
nR = 18
p < 0.0000001
nD = 100, nP = 55
nR = 44
p = 0.08
DynaFit: Enzyme Progress Curves
31
DynaFit: Enzyme Progress Curves
32
Initial estimates: the “false minimum” problem
Large effect of slight changes in initial estimates
NONLINEAR REGRESSION REQUIRES US TO GUESS THE SOLUTION BEFOREHAND
IN UNFAVORABLE CASES EVEN ONE ORDER OF MAGNITUDE DIFFERENCE IS IMPORTANT
E + S ---> E + P :
E + I <===> EI
:
EI ---> X
:
kdp
kai
kx
E + S ---> E + P :
E + I <===> EI
:
EI ---> X
:
kdi
iterative
refimenement
residuals
“best fit”
initial estimate
[constants]
kdp =
10 ?
kai =
10 ?
kdi =
1 ?
kx = 0.01 ?
[concentrations]
S
= 0.85 ?
kdp
kai
kdi
kx
[S] =
initial estimate
E + S ---> E + P :
E + I <===> EI
:
kdi
iterative
refimenement
[constants]
kdp =
10 ?
kai = 0.1 ?
kdi = 0.01 ?
kx = 0.1 ?
[concentrations]
S
= 0.85 ?
=
72
=
4.5
=
1.8
= 0.048
0.22
DynaFit: Enzyme Progress Curves
kdp
kai
kx
33
kdp
kai
kdi
kx
[S] =
DYNAFIT IMPLEMENTS TWO MODEL-DISCRIMINATION CRITERIA
{
{
{
{
10,
10,
10,
10,
1,
1,
1,
1,
0.1,
0.1,
0.1,
0.1,
0.01
0.01
0.01
0.01
One-step
model:
1. Fischer’s F-ratio for nested models
kdp
kai
kx
kdi
[constants]
=
=
=
=
0.27
34
DYNAFIT-4 ALLOWS HUNDREDS, OR EVEN THOUSANDS, OF DIFFERENT INITIAL ESTIMATES
kdp
kai
kdi
kx
=
96
= 0.36
=0.0058
=
~ 0
DynaFit: Enzyme Progress Curves
Model discrimination: Use your judgment
:
:
:
kdi
best fit
residuals
Solution to initial estimate problem: systematic scan
[mechanism]
E + S ---> E + P
E + I <===> EI
EI ---> X
kdp
kai
}
}
}
}
relative
sum of
squares
?
?
?
?
Two-step model:
MEANS:
2. Akaike Information Criterion for all models
“Try all possible combinations of initial estimates.”
• 4 rate constants
• 4 estimates for each rate constant
• 4 × 4 × 4 × 4 = 44 = 256 initial estimates
probability (0 .. 1)
DynaFit: Enzyme Progress Curves
35
DynaFit: Enzyme Progress Curves
36
6
Summary and Conclusions
Questions?
NUMERICAL MODELS ENABLE US TO DO MORE USEFUL EXPERIMENTS IN THE LABORATORY
MORE INFORMATION AND CONTACT:
ADVANTAGES of “Numerical Enzyme Kinetics” (the new approach):
• No constraints on experimental conditions
EXAMPLE: large excess of [I] over [E] no longer required
Petr Kuzmic, Ph.D.
• No constraints on the theoretical model
BioKin Ltd.
EXAMPLE: dissociation rate can be comparable with deactivation rate
• Theoretical model is automatically derived by the computer
•
•
•
•
No more algebraic rate equations
• Learn more from the same data
EXAMPLE: Determine kON and kOFF, not just equilibrium constant Ki = kOFF/kON
Software Development
Consulting
Employee Training
Continuing Education
since 1991
DISADVANTAGES:
• Change in standard operating procedures
Is it better stick with invalid but established methods ? (Continuity problem)
http://www.biokin.com
• Training / Education required
Where to find time for continuing education ? (Short-term vs. long-term view)
DynaFit: Enzyme Progress Curves
37
DynaFit: Enzyme Progress Curves
38
7