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NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Modeling
Sensitivity Analysis
Computer Algebra Approach
Tryptophan Application
Use ordinary differential equations to model mass action kinetics
Use partial differential equations to model concentration sensitivities with respect to parameters
Use CAS to solve the large system of equations simultaneously
Implementation of the method for E. coli
Computer Algebra Approach to Sensitivity Analysis: Application to TRP
April 18, 2023
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Modeling BasicsVariable Concentrations
Constant Parameters
d[X1]
dt= f1(X1,X 2 ,...,Xn,K 1,K 2 ,...,K m)
d[X 2 ]dt
= f2(X1,X 2 ,...,Xn,K 1,K 2 ,...,K m)
M M Md[Xn]
dt= fn(X1,X 2 ,...,Xn,K 1,K 2 ,...,K m)
⎫
⎬
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
= f (X,K )
d[X]
dt={ In Rate} −{Out Rate}
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Parameter Changes Effect System Dynamics
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
How do we get Sensitivity equations?
d
dKf (X,K) =
∂f∂K
+∂f∂X
g∂X∂K
=ddt
∂X∂K
⎛⎝⎜
⎞⎠⎟
∂[X]X
∂KK
Normalized Unitless Sensitivity Score
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
A Simple Example
d[C]
dt=k1[A][B]
Gain rate1 24 34
−k2[C]Loss rate{
d
dt
d[C]
dk1
⎛
⎝⎜⎞
⎠⎟=−k2
d[C]dk1
⎛
⎝⎜⎞
⎠⎟+[A][B]
Recall,
d[X1]
dt= f1(X1,X 2 ,...,Xn,K 1,K 2 ,...,K m)
and,
d
dKf (X,K) =
∂f∂K
+∂f∂X
g∂X∂K
Then,
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Computer Algebra Software
Sensitivity Analysis requires a PDE for each variable with respect to each parameter. For m variables and n parameters, this is n(m+1) equations.
Maple can do symbolic calculus to find the required PDE’s, building the sensitivity matrix.
Matlab can take this matrix, along with the modeling ODE’s, and solve the resulting system numerically.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
What is an Operon?
A operon is a genetic regulatory network. It is defined by a set of common genes with one operator. The operator is a binding site for a regulatory protein.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
What is the TRP Operon?
The tryptophan operon in E. Coli is a repressive operon, that shuts down tryptophan production when tryptophan is present in the environment. The presence of tryptophan enables a repressor to bind to the operator, disabling the operon.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
The TRP Operon
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
The TRP Operon
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
CAS Implementation4 concentrations: Of, Mf, E, Tx 24 parameters = 96 sensitivities
4 concentrations + 96 sensitivities = 100 differential equations
Maple will find these sensitivities quickly with matrix algebra.
Matlab will solve this system simultaneously and print sensitivity scores.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
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are needed to see this picture.
TRP Sensitivities Revealed
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QuickTime™ and a decompressorare needed to see this picture.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
QuickTime™ and a decompressor
are needed to see this picture.
TRP Sensitivities Revealed
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a decompressorare needed to see this picture.
QuickTime™ and a decompressorare needed to see this picture.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
TRP Sensitivities Revealed
[T]/b
[T]/k-t
Transcription Termination
Repressor Dissassociation
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Correlation to Experimental Results
b = .85
b = .9996
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Future WorkImprove the Model
Parameter Estimation
Collaborative Work
The operon is more complex than the model presented here. For example, there is a time delay in transcription.
Parameter values directly effect the numeric solution. Better estimations will give more accurate results.
A database of results to check against.
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
References
Dynamic regulation of the tryptophan operon: A modeling study and comparison with experimental data
Moises Santillan and Michael C. Mackey (2001)
Modeling operon dynamics: the tryptophan and lactose operons as paradigms
Michael C. Mackey, Moises Santillan, Necmettin Yildirim (2004)
NCF LOGO
Casey Henderson and Necmettin Yildirim
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Math Modeling
Introduction
Conclusion
Questions? Thank You!