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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 May 10, 2015

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion Modeling Basics Variable Concentrations Constant Parameters

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion Parameter Changes Effect System Dynamics

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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? Normalized Unitless Sensitivity Score

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion A Simple Example Recall, and, Then,

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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.

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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.

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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.

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion The TRP Operon

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion The TRP Operon

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion CAS Implementation 4 concentrations: Of, Mf, E, T x 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.

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion TRP Sensitivities Revealed

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion TRP Sensitivities Revealed

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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

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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

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion 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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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)

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NCF LOGO Casey Henderson and Necmettin Yildirim Sensitivity Analysis Computer Algebra Operon Application to TRP Math Modeling Introduction Conclusion Questions? Thank You!

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