ncf logo casey henderson and necmettin yildirim sensitivity analysis computer algebra operon...

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

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

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!