Challenges in the Optimization of Bio-Systems Gerhard-W ilhelm Weber

Challenges in the Optimization

of Bio-Systems

Gerhard-Wilhelm Weber Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

http://www.iam.metu.edu.tr/research/groups/compbio/index.html

IAM, METU Ankara CUBIC, Uni Cologne


Content

Motivation: Bio-Systems

Computational Biology

Modeling and Prediction of Gene Expression Patterns

Population Dynamics, Gene Dynamics and Variation

Optimization

Computational Metabolism, Medicine and Tomography Earth Warming and Sustainable Development

Conclusion and Perspectives

Bio-Systems


Bad Berleburg,Wemlighausen

Bio-Systems



Bio-Systems



Bio-Systems



Bio-Systems



Bio-Systems



Bio-Systems



Bio-Systems



Bio-Systems



Ankara

Bio-Systems


DNA microarray chip experiments

Comp. Bio. & Med.

anticipation of gene patterns based on

M.U. Akhmet, H. Öktem

S.W. Pickl, E. Quek Ming Poh

T. Ergenç, B. Karasözen

J. Gebert, N. Radde, W.

Ö. Uğur, R. Wünschiers

M. Taştan, F.B. Yılmaz


least squares – ML

statistical learning

time-contin.

EEME )(

Expression data

time-discr.

kkk EE M1

)(Μ jik em M

Gene Patterns Modeling & Prediction

0)0( EE

Ex.: Euler, Runge-Kutta


For which parameters, i.e., for which set M (or: dynamics), is stability guaranteed ?

Def.: M is stable : B : (compl.) bounded neighbourhood of ,n

M : 110 M,...,M,M, mΙΝm

.)M...MM( 021 mm

development of processits feasibilitygoodness-of-fit (model) test

Gene Patterns Model. & Pred.


Analysis with Polytopes

Theorem (Brayton, Tong 1979) :

Given a set M : of m distinct complex matrices.

Then,

M is stable is bounded .

0B n0

:kB H

,mod)1(: mkk H : convex hull .

kkBB

0

* U

,M 1'0

kiki

BU

0 1M ,...,Mm

Here, is a bounded neighbourhood of ,

and for k > 0

where


The ‘‘discrete” power of the algorithm is based on using

polyhedra and focussing on the extremal points

of the sets .kB

kB

Theorem 1 : If z is an extremal point of , then there

exist and an extremal point u of , such that

z

kB

0j

.M uj

k

1kB

Extremal Points


Construction Principle


Stopping Criterion

Theorem 2 :

Let ( as above,

Then,

}).,...,2,1{M riuz ijki

1 1{ } M { } 1, 2,..., .r k k i rz ,...,z B z z ,...,z i r H H


Construction Principle

stabilityunboundedB

BB k 00

dynamicsmatricesof

boundedBB

kstepatstopping

kkBB iikk

/

,

,)(""

)(0

0

00U

stability

in Gebert, Laetsch, Pickl, W., Wünschiers 2005

Ergenç, W. 2004


Numerical Example

Ex. (Pickl 1999) :

M = }M,M{ 10 ,01

M0

ba

010

M1 b

algorithm

instability

region of stability


A

)()()()()()()()()()()()()()()()(

34333231

24232221

14131211

04030201

tEtEtEtEtEtEtEtEtEtEtEtEtEtEtEtE

08017025525570180255050200255

2550250255

2001039.02.000061.04.00000

M

Gene Network

Ex. : 1

h kEME

Ė 4Ė 2

Ė 0

Ė 5

Ė 1

Ė 3

0123456789

0 2 4 6 8

Time, t

Expr

essi

on le

vel,

Ė


Gene Network

gene2

gene3

gene1

gene4


0.4 x1

0.2 x2 1 x1

Comp. Bio. & Med.

F.B. Yilmaz 2004

: exper. data

: approx. increase/decrease

Ex.:

nonlinearities


Inference of Gene Regulatory Networks

genes mRNA proteins

disintegration disintegration

protein complexes

cellularprocesses

Gene transcription also regulates the transcription factors defining a dynamic regulatory network involving highly nonlinear feedback mechanisms.

externalfactors


Proposed Model Class

)()( )(M)1( ksks bkEkE

else0)(if1

))((

))1(()(

iii

B

kEkEQ

kEQFks

ipi

p

qEEEQ

tEtEtEQFts

,...,1,0),...,,(

)))(),...,(),((()(121

1

piecewise linear:


Akhmet, Gebert, Pickl, Öktem, W. 2004

Gebert, Radde, W. 2004

Dynamics and Variation

population dynamics

M.U. Akhmet , V. Tkachenko H. Öktem, S.W. Pickl, W., et al.

gene dyn. & variation

I. Togan A. Kence C. Berkman et al. et al.

with delay and impulse


time

( ) ( , ( ), ([ ]))E t f t E t E t

anticipation

... Tomography

inverse problems

Ö. Yaşar, O. Özgür, W.

C. Diner, A. DoğanF. Özbudak, A. Tiefenbach

M. Eyüboğlu, N. Gençer Y. Serinağoğlu, A. KurtS. Sarıkaya, W.

VLSI chip design

medicial imaging

inverse

Discrete ...

Brain / Heart ...


http://www.mun.ca/

Metabolic Engineering

object oriented

modules

DAE ODE

training program

math. modeling

parameter estimation

sports (and general) medicine


A. Schulte Thomas 2004

S. Özöğür 2005

http://www-rt.upb.de/~anna/sicem/sicem_main.html

http://www.uni-paderborn.de/home/

Generalized Semi-Infinite Opt.

I, K, L finite

2C

W. 2003


Reverse Chebychev Approx.

Ex.: approx. of a thermo-couple characteristic Hoffmann, Reinhard

thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]

to be approx. by :

bounds on error

some interpol.

(= y)

Bernhard



Ex.: approx. of a thermo-couple characteristic


to be approx. by :

bounds on error

some interpol.

(= y)



Ex.: approx. of a thermo-couple characteristic


to be approx. by :

bounds on error

some interpol.

(= y)


time

Time-Optimal Control

r R

B

Ex.: time-minimal cooling (or heating) of

!0 T


Anticipation

Ex.’s.: thermo-regulation

control of earth warming

Pickl, W.

),,( txr local-global

anticipation

maximization of time-horizon longest term approximation / description


TEM Model

Z. Alparslan

B. K., W.

S.W. Pickl


JET

EU project application

“Joint International Emissions Trading, Analysis, Forecast, and Optimal Strategies in Sustainable Low Carbon Energy Management”

U. Leopold-W., B. K., S.W. P., M. Türkay, W.

Risk Management H. Körezlioğlu et al.

ValidiertesEmissionsreduktions-

projekt

common learning


improvement of living conditions learning

modeling

optimization

data mining

Sustainable Living

D. DeTombe, A., I., H. Gökmen

S. Kayalığil, M., Y. Ecevit

T. Bali, W. et al.


Balaban Valley

METU


Turkey

http://www.kultur.gov.tr/portal/default_EN.asp?BELGENO=1918

http://www.turkiye-online.com/travel/anatol/ankara/

http://www.hitit.co.uk/regions/Ankara/About.html

http://goturkey.turizm.gov.tr/

http://goturkey.turizm.gov.tr/

http://www.about-turkey.com/tourism

http://www.about-turkey.com/cuisine

http://www.metu.edu.tr/

Conclusions and Perspectives


Association of European

Operational Research Societies

Operations Research Society

of Germany

EURO Working Group

on Continuous Optimization


of Israel

Operations Research

Society of Turkey


of Hungary

EURO XXI in Iceland July 2-5, 2006

21st European Conference on Operational Research

OR for Better Management of Sustainable Development

5th EUROPT Workshop

“Advances in

Continuous Optimization”

Reykjavik, Iceland

June 29 – July 1, 2006

Conclusions and Perspectives


EURO Summer Institute

“Optimization Challenges in Engineering: Methods, Software, and Applications”

Wittenberg, Germany August 18 – September 2, 2006

Earth Institute

Young People for OR in Developing Countries

Balaban Valley Project

to

EURO Working Group

in Continuous Optimizationhttp://www.iam.metu.edu.tr/EUROPT/

and Computational Biology and Medicine Group

Sincere invitation ...


[email protected]


two new EURO WGs planned:

Computational Biology & Bioinformatics

OR for Development

http://www.iam.metu.edu.tr/EUROPT/





mailto:[email protected]

http://www.cs.elte.hu/opres/europt/

Documents

Challenges in the Optimization of Bio-Systems Gerhard-W ilhelm Weber