Transcript - ACA 2009, Applications of Computer Algebra

Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Computer Algebra Approach to
Sensitivity Analysis: Application to TRP
Modeling Use ordinary differential equations to model
mass action kinetics
Sensitivity Analysis Use partial differential equations to model
concentration sensitivities with respect to
parameters
Computer Algebra Use CAS to solve the large system of
Approach equations simultaneously
Conclusion
Tryptophan Application Implementation of the method for E. coli
July 17, 2015
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
Variable Concentrations
Constant Parameters
Modeling Basics
d[X]
 {In Rate}  {Out Rate}
dt

 f1 (X1 ,X 2 ,...,X n ,K1 ,K 2 ,...,K m ) 
dt

d[X 2 ]

 f 2 (X1 ,X 2 ,...,X n ,K1 ,K 2 ,...,K m ) 
  f (X, K)
dt
M
M
M

d[X n ]

 f n (X1 ,X 2 ,...,X n ,K1 ,K 2 ,...,K m ) 
dt

d[X1 ]
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
Parameter Changes Effect System
Dynamics
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
How do we get Sensitivity equations?
d
f
f X d  X 
f (X,K) 

g  

dK
K X K dt  K 
Operon
Application to TRP
[X]
Conclusion
Normalized Unitless Sensitivity Score
K
X
K
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
A Simple Example
Recall,
d[X1 ]
dt
 f1 (X1 ,X 2 ,...,X n ,K1 ,K 2 ,...,K m )
and,
d
f
f X
f (X, K) 

g
dK
K X K
Then,
d[C]
 k1[ A][B]  k2 [C]
14 2 43 {
dt
Gain rate
Loss rate
 d[C] 
d  d[C] 
 k2 
 [ A][B]



dt  dk1 
 dk1 
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
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.
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
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.
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
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.
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
The TRP Operon
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
The TRP Operon
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
CAS Implementation
4 concentrations: Of, Mf, E, T
x 24 parameters = 96 sensitivities
Maple will find these sensitivities quickly with
matrix algebra.
4 concentrations + 96 sensitivities =
100 differential equations
Matlab will solve this system
simultaneously and print sensitivity scores.
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
TRP Sensitivities Revealed
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Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
TRP Sensitivities Revealed
QuickTime™ and a
decompres sor
are needed to see thi s pic ture.
QuickTime™ and a
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QuickTime™ and a
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Casey Henderson and Necmettin Yildirim
NCF LOGO
TRP Sensitivities Revealed
[T]/k-t
Repressor Dissassociation
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
Transcription Termination
[T]/b
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
Correlation to Experimental Results
b = .85
b = .9996
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Future Work
Improve the Model The operon is more complex than the model
presented here. For example, there is a time
delay in transcription.
Parameter Estimation Parameter values directly effect the numeric
solution. Better estimations will give more
accurate results.
Conclusion
Collaborative Work A database of results to check against.
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
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)
Casey Henderson and Necmettin Yildirim
NCF LOGO
Introduction
Math Modeling
Sensitivity Analysis
Computer Algebra
Operon
Application to TRP
Conclusion
Questions? Thank You!