Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks

Modelling, Dynamics and Development of Gene-Environment

and Eco-Finance Networks

4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009

and Eco-Finance NetworksGerhard-Wilhelm Weber *, Ba şak Akteke-ÖztürkZeynep Alparslan-Gök, Ömür Uğur, Hakan Öktem

Institute of Applied Mathematics, METU, Ankara, Turkey

Pakize Taylan Dicle University, Diyarbakır, Turkey

Erik Kropat University of Erlangen-Nuremberg, Germany

Özlem Deferli Department of Mathematics, Cankaya University, Ankara, Tu rkey

* Faculty of Economics, Management Science and Law, University of Siegen, GermanyCenter for Research on Optimization and Control, University of Aveiro, Portugal

Bio-Systems

Bio-Systems

Bio-Systems

sustainability

Bio-Systems

sustainability

• Computational Biology, Medicine, Health Care and Environment

• Gene-Environment Networks and Eco-Finance Networks

• Dynamical Systems

• Hybrid and Anticipatory Systems

Outline

• Stability

• Optimization and Control Theory

• Regression and Clustering

• Financial Mathematics and Risk Management

• Regulatory Networks under Uncertainty and Ellipsoidal Calculus

• Conclusion

DNA microarray chip experiments

Comp. Bio. & Med.

prediction of gene patterns based on

with

M.U. Akhmet, H. Öktem M.U. Akhmet, H. Öktem

S.W. Pickl, E. Quek Ming Poh

T. Ergenç, B. Karasözen

J. Gebert, N. Radde

Ö. Uğur, R. Wünschiers

M. Taştan, A. Tezel, P. Taylan

F.B. Yılmaz, B. Akteke-Öztürk

S. Özöğür, Z. Alparslan-Gök

A. Soyler, B. Soyler, M. Çetin

Comp. Bio. & Med.

GENE time 0 9.5 11.5 13.5 15.5 18.5 20.5

'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811

'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275

'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239

'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935

Ex.: yeast data

'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935

'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533

'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449

'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192

'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027

http://genome-www5.stanford.edu/

Comp. Bio. & Med.

Comp. Bio. & Med.

least squares – ML

statistical learning

time-contin.Expression data

Gene Patterns Modeling & Prediction

0)0( EE =( ) ( )E M E E C E•

= +

time-discr.

kkk EE M1 =+

∈= )(Μ jik em M

Ex.: Euler, Runge-Kutta

environmental effects


( ), ΜkM E

=

E

=

Uğur, W. 2006, W., Taylan, Alparsan-Gök, Özöğür, Akteke-Öztürk 2006


( ), ΜkM E

=

E

=

Kropat, W. , Tezel, Özöğür-Akyüz 2008

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

Gene Patterns Model. & Pred.

Def.: M is stable : ⇔ B : (complex) bounded neighbourhood of∃ ,nΟM :

0 1 1, M ,M ,..., Mkk ΙΝ −∀ ∈ ∈

1 2 0(M M ... M ) .k k− − Β ⊆ Β

Stability Analysis

kB

1kB +

Stability Theorems

Akhmet, Gebert, Pickl, Öktem, W. 2005

Öktem 2005, Akçay 2005

Gebert, Radde, W. 2005

Yılmaz 2004 Yılmaz, Öktem, W. 2005

Uğur, Pickl, Taştan, W. 2005

Weber, Tezel 2006 Uğur, W. 2006

W, Tezel, Taylan, Soyler, Çetin 2007

W, Ugur, Taylan, Tezel 2007

Uğur, Pickl, W, Wünschiers 2007

Analysis with Polytopes

Theorem (Brayton, Tong 1979) :

Given a set M : of m distinct complex matrices.

Then,

M is stable is bounded .

B 0

kk

BB∞

==⇔

0

* U

10 M,...,M −= m

Here, is a bounded neighbourhood of ,0B n0

=:kB H

,mod)1(: mkk −=′ H : convex hull .

,M 1'0

−

∞

= kik

iBU

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

Extremal Points

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

there exists a and an extremal point u of ,

in short : E , such that

z

kB

0ΙΝ∈j

.M uj

k ′=

1−kB)( 1−∈ kBu

Construction Principle

Stopping Criterion

Theorem 2:

Let ( as above,

Then,

).,...,2,1M riuz ijki ∈= ′

.,...,2,1M 11 ri,...,zzzB,...,zz rikkr =∀∈⇔= ′ HH

Construction Principle

=⇒≥=•∞

=

∗

dynamicsmatricesof

boundedBB

kstepatstopping

kkBB ii

kk

/

,

,)(""

)(0

0

00U

stability

⇒/=∂∩∂•∗

stability

unboundedBBB k 00

in

Gebert, Laetsch, Pickl, We. Wünschiers 2004

Ergenc, We. 2004

Ex.:

M = 0 1M , M ,0

1M 0

=

b

a

=

0

10M1 b

Stability Analysis

algorithm

instability

region of stability

)()()()(

)()()()(

)()()()(

)()()()(

34333231

24232221

14131211

04030201

tEtEtEtE

tEtEtEtE

tEtEtEtE

tEtEtEtE

=

080170255

25570180255

050200255

2550250255

Genetic Network

Ex. : , 1,E M E hκ

•= = scalar - valued case

−−

−=

2001

039.02.00

0061.04.0

0000

Mö 4ö 2

ö 0

ö 5

ö 1

ö 3

0123456789

0 2 4 6 8

Time, t

Exp

ress

ion

leve

l, ö

gene2

gene3

gene1

gene4

0.4 x1

0.2 x2 1 x1

Genetic Network

Gene-Environment Networks - Hybrid Systems

( ) ( )( 1) M ( )s k s kE k E k C+ = +( ) : ( ( 1))

1 if ( )( ( )) :

0 else

B

i ii

s k F Q E k

E kQ E k

= −> Ω

=

( ( )) :0 elseiQ E k =

Akhmet, Gebert, Pickl, Öktem, W. 2005

Öktem 2005, Akçay 2005

Gebert, Radde, W. 2005

Yılmaz 2004 Yılmaz, Öktem, W. 2005

Uğur, Pickl, Taştan, W. 2005

Weber, Tezel 2006

( ) ( )( 1) M ( )s k s kE k E k C+ = +

( 1) IM ( )kIE k IE k+ =

Gene-Environment Networks

( ) ( )IE t IM IE t•

= locally

)

( ) ( ) ( )( ) ( ) ( )s t s t s tE t M E t C E t D•

= + +

)

( ) ( ) ( )( 1) M ( ) ( )s k s k s kE k E k C E k D+ = + +

( 1) IM ( )kIE k IE k+ =



= locally

)


= + +


( 1) IM ( )kIE k IE k+ =

modules


=


)))(((:)( tEQFts =

1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t=

where

)


= + +

1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t=

1,)( ii tE θ<

2,1, )( iii tE θθ <<

)(, tEidi i<θ

0 for

1 for( ( )) :

...

for

i

i

Q E t

d

=


)))(((:)( tEQFts =

1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t=

where

parameter estimation:

)


= + +

1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t=

1,)( ii tE θ<

2,1, )( iii tE θθ <<

)(, tEidi i<θ

0 for

1 for( ( )) :

...

for

i

i

Q E t

d

=

parameter estimation:

(i) estimation of thresholds

(ii) calculation of matrices and vectorsdescribing the system in between thresholds

)2

1

0

l

M E C E D Eαα ακκ κ

α

∗ −∗ ∗ ∗

= ∞

+ + −∑ &

( ), ( ), ( )ij i im c d∗ ∗ ∗l


min

Chebychev (maximum) norm



if gene j regulates gene i

otherwise

,i iξ ζl

1:

0i jχ =

mixed integer programming

( 1,2,..., )j n=

subject to

min


)2

1

0

l


α

∗ −∗ ∗ ∗

= ∞

+ + −∑ &

( ), ( ), ( ), ( ), ( ), ( )ij i i ij i im c d χ ξ ζ∗ ∗ ∗l l

n

χ α≤∑ ( 1,2,..., )j n=

( 1,2,..., )m=l1

1

1

,min

&

ij ji

n

ii

n

ii

ii im

χ α

ξ β

ζ γ

δ

=

=

=

≤

≤

≤

≥

∑

∑

∑

l l

overall box constraints

( 1,2,..., )i n=

Trehalose UDP-Glucose

Glycogen

Glucose-1-Phosphate

TPS3 GSY2

NTH2

UGP1

GLC3

GPH1


Glucose Glucose-6-Phosphate

Glycolysis pathway

HXK1

PGM1

knockoutglycogen metabolism pathway in yeast Saccharomyces cerevisiae


min

( ), ( ), ( )ij i im c d∗ ∗ ∗l

( 1, . . . , )( , ) ( )n

i j i j j j np m y yα∗ =≤∑

subject to

)2

1

0

l


α

∗ −∗ ∗ ∗

= ∞

+ + −∑ &

GSIP relaxation

set of combined environmental effects combined environmental effects combined environmental effects combined environmental effects :

( , ) :Y C D∗ ∗ =

1,..., 1,...,1,...,

( 0, ) ( 0, )i ii n i n

m

c d∗ ∗

= ==

× ∏ ∏l

l

1

1

1

, m in

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

( , ) ( )

&

i j i j ji

n

i ii

n

i ii

i i i

mq c y y

d y y

m

β

ζ γ

δ

=

∗

=

∗

=

=≤

≤

≥

∑

∑

∑

l l ll

o v e ra ll b o x c o n s t ra in ts

( ( , ))y Y C D∗ ∗∈

( 1, . . . , )i n=

General. Semi-Infinite Programming

2C

I, K, L finite

)(τψτ

:),(),( 0C∈⋅⋅⋅ ϕψ∈∀∃ ∃

ψ

GSIP – Structural Stability

Jongen, W.

: structurally stable

global local global

)(⋅εnIR

asymptotic

effect

homeom.

⇔

ψ

),( τϕ ⋅

Thm. (W. 1999/2003, 2006):

⇔

ξ



Spline Approximation

( )22

1, , 1, ,

1 1 1

2

= ( )U

L

n n n Ei j i jj EE

i= i= =

f E E dEλ α

ααα α α α

α ∞

′′

∑∑ ∑ ∫:penalty term

21

=0

( ,C,D) = ( ) C( ) D( )l

PRSS M M E E E E E E−

κ κ κ κ κ κκ ∞

+ + −

+

∑)

min :

penalty term

.

Tikhonov regularization (ridge regression)

( )

( )

222, , 2, ,

1 1 1

223, 3,

1 1

( )

( )

U

L

U

L

n m n Ei i

E Ei= = =

n n Ei i

EEi= =

f E E dE

f E dE

µ

ς

α

α α

α

αα

α α α αα ∞

α α α αα ∞

′′+

′′+

∑∑ ∑ ∫

∑ ∑ ∫

l l

l

l

)


( )22

1, , 1, ,

1 1 1

2

= ( )U

L

n n n Ei j i jj EE

i= i= =

f E E dEλ α

ααα α α α

α ∞

′′

∑∑ ∑ ∫:penalty term

21

=0

( ,C,D) = ( ) C( ) D( )l



+ + −

+

∑)

min :

penalty term

.

conic quadratic programming

interior points methods

21 2 3 2

21

22

23

( )

( )

( )

( )

0

2

i j i j

2

i i

2

i i

t

U , , t

V M

W N

Z R

t

θ θ θ

θ

θ

θ

∞

α α∞

α α∞

α α∞

≤

≤

≤

≤

≤

l l

m in

s .t.( )

( )

222, , 2, ,

1 1 1

223, 3,

1 1

( )

( )

U

L

U

L

n m n Ei i

E Ei= = =

n n Ei i

EEi= =

f E E dE

f E dE

µ

ς

α

α α

α

αα

α α α αα ∞

α α α αα ∞

′′+

′′+

∑∑ ∑ ∫

∑ ∑ ∫

l l

l

l

)


21

=0

( ,C,D) = ( ) C( ) D( )l



+ + −

+

∑)

min :

penalty term

.

conic quadratic programming

interior points methods

21 2 3 2

21

22

23

( )

( )

( )

( )

0

2

i j i j

2

i i

2

i i

t

U , , t

V M

W N

Z R

t

θ θ θ

θ

θ

θ

∞

α α∞

α α∞

α α∞

≤

≤

≤

≤

≤

l l

m in

s .t.

TEM Model

TEM Model

Example of stabilityArticle 2, Kyoto Protocol

( 1) ( ) ( )

( )

k k k

kM+

=

E E EM M M

Dynamics and Control in CO 2-Emission Reduction

( 1) ( ) ( )0

k k k+ = +

E E E

( +1) ( ) ( ) IE IM IEk k k=

( )( )

0kkM

u

= +

E E EM M M

( 1) ( ) ( )

( )

k k k

kM+

=

E E EM M M

Dynamics and Control in CO 2-Emission Reduction

( 1) ( ) ( )0

k k k+ = +

E E E

( +1) ( ) ( ) IE IM IEk k k=

( )( )

0kkM

u

= +

E E EM M M

( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( )1E (E , ) (E , ) ( )(E , ) 1

2

k k kk k k k k ki i

i i i i ik k

W Wa t b t b'b t

h h

∆ ∆≈ + + −

( ) ( ) ( ) ( )E (E , ) (E , )t t t ti i i id a t dt b t dW= +

Gene-Environmental and Financial Dynamics

.

( +1) ( ) ( ) IE IM IEk k k=

( ) ( )2i i i ik kh h

Modeling

Testing

Prediction

Stability

•

•

•

•

( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( )1E (E , ) (E , ) ( )(E , ) 1

2

k k kk k k k k ki i

i i i i ik k

W Wa t b t b'b t

h h

∆ ∆≈ + + −

( ) ( ) ( ) ( )E (E , ) (E , )t t t ti i i id a t dt b t dW= +

Gene-Environmental and Financial Dynamics

.

( +1) ( ) ( ) IE IM IEk k k=

( ) ( )2i i i ik kh h

Modeling

Testing

Prediction

Stability

•

•

•

•

Regulatory Networks: Errors and Uncertainty

Errors uncorrelated Errors correlated Fuzzy values

Interval arithmetics Ellipsoidal calculus Fuzzy arithme tics

θθ11

θθ22

Regulatory Networks and and

Ellipsoidal Calculus

Assumption:Interacting groups (clusters) of genetic and environmental variables

Clustered variables (errors) are correlated••

Regulatory Networks ― Ellipsoidal Calculus

How can we model the time-discrete dynamics of the ellipsoidal states of clusters?

1. Clustering (Groups of genes / groups of environmental items)

2. Assign ellipsoids(Center = measurement value, configuration matrix = covariances)

3. Regulatory system


3. Regulatory system(Interaction of clusters defined by affine-linear coupling rules)

4. Parameter identification

1) Clustering

Identify groups (clusters) of jointly acting genetic and environmental variables


disjoint

overlapping

2) Interaction of Genetic Clusters


3) Interaction of Environmental Clusters


3) Interaction of Genetic & Environmental Clusters


Determine the degree of connectivity⇒⇒⇒⇒

Task:

Identify and analyze highly interconnected systems of clusters ofgenetic and environmental data based on ellipsoidal measurement data.

Calculate predictions of the ellipsoidal states.

••


Assume: Affine-linear coupling rules.

••

⇒⇒⇒⇒ Ellipsoidal Calculus

Clusters and Ellipsoids:

Genetic clusters: C1,C2,…,CREnvironmental clusters: D1,D2,…,DS

Genetic ellipsoids: X1,X2,…,XR Xi = E (µi,Σi) Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)


Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)



r=1

The Regression Problem:


Maximize (overlap of ellipsoids)

∑ ∑ ∑= = =

∩+∩T R

r

R

rrrrr EEXX

1 1 1

)()()()( ˆˆκ

κκκκ

measurement

prediction

Measures for the size of intersection:

• Volume (→ ellipsoid matrix determinant)

• Sum of squares of semiaxes (→ trace of configuration matrix)

• Length of largest semiaxes (→ eigenvalues of configuration matrix)


( )rr Π,µE

rµ

Thank you very much for your attention!

References:

[email protected]

http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf

Education

Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks