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 March 25, 2022
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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion Modeling.
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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