Computer Algebra Approach to Sensitivity Analysis: Application to TRP

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Computer Algebra Approach to Sensitivity
Analysis Application to TRP
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
November 18, 2009
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Modeling Basics
Variable Concentrations Constant Parameters
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Parameter Changes Effect System Dynamics
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How do we get Sensitivity equations?
Normalized Unitless Sensitivity Score
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A Simple Example
Recall,
and,
Then,
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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(m1)
equations. Maple can do symbolic calculus to
find the required PDEs, building the sensitivity
matrix. Matlab can take this matrix, along with
the modeling ODEs, and solve the resulting
system numerically.
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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.
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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.
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The TRP Operon
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The TRP Operon
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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.
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TRP Sensitivities Revealed
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TRP Sensitivities Revealed
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TRP Sensitivities Revealed
T/k-t
Repressor Dissassociation
Transcription Termination
T/b
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Correlation to Experimental Results
b .85
b .9996
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Future Work
Improve 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.
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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)
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Questions? Thank You!