4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Modelling, Dynamics and Development
of Gene-Environment
and Eco-Finance Networks
Gerhard-Wilhelm Weber *, Ba şak Akteke-Öztürk
Zeynep 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, Turkey
* Faculty of Economics, Management Science and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal

Bio-Systems

Bio-Systems

Bio-Systems
sustainability

Bio-Systems
sustainability

Outline
• Computational Biology, Medicine, Health Care and Environment
• Gene-Environment Networks and Eco-Finance Networks
• Dynamical Systems
• Hybrid and Anticipatory Systems
• Stability
• Optimization and Control Theory
• Regression and Clustering
• Financial Mathematics and Risk Management
• Regulatory Networks under Uncertainty and Ellipsoidal Calculus
• Conclusion

Comp. Bio. & Med.
prediction of gene patterns based on
DNA microarray chip experiments
with
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.
Ex.: yeast data
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
'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.

Gene Patterns Modeling & Prediction
least squares – ML
statistical learning
time-contin.
Expression data
•
E = M (E) E + C(E) E ( 0 ) = E0
{
environmental effects
time-discr.
Ex.: Euler, Runge-Kutta
E k +1 = M k E k
Μk = (emi j ) ∈ M

Gene Patterns Modeling & Prediction
 
 
M ( E ), Μ k =  
 
 
 
 
 
E = 
 
 
Uğur, W. 2006, W., Taylan, Alparsan-Gök, Özöğür, Akteke-Öztürk 2006

Gene Patterns Modeling & Prediction
 
 
M ( E ), Μ k =  
 
 
 
 
 
E = 
 
 
Kropat, W. , Tezel, Özöğür-Akyüz 2008

Gene Patterns Model. & Pred.
For which parameters, i.e., for which set M (or: dynamics),
is stability guaranteed ?
Def.: M is stable : ⇔ ∃ B: (complex) bounded neighbourhood of Οn ,
∀ k ∈ ΙΝ , M 0, M1 ,..., M k −1 ∈ M :
(M k −1 M k − 2 ... M 0 ) Β ⊆ Β .

Stability Analysis
Bk
Yılmaz 2004 Yılmaz, Öktem, W. 2005
Gebert, Radde, W. 2005
Akhmet, Gebert, Pickl, Öktem, W. 2005
Öktem 2005, Akçay 2005
Uğur, Pickl, Taştan, W. 2005
Bk +1 Weber, Tezel 2006 Uğur, W. 2006
W, Tezel, Taylan, Soyler, Çetin 2007
W, Ugur, Taylan, Tezel 2007
Stability Theorems Uğur, Pickl, W, Wünschiers 2007

Analysis with Polytopes
Theorem (Brayton, Tong 1979) :
Given a set M : = {M 0 , ... , M m −1 } of m distinct complex matrices.
Then,
∞
M is stable ⇔ B* = U
k =0
Bk is bounded .
Here, B0 is a bounded neighbourhood of 0n ,
and for k > 0 ∞
 
Bk := H  U M i
B k −1  ,
 i=0
k' 
where  
k ′ := ( k − 1) mod m , H: convex hull .

Extremal Points
The ‘‘discrete” power of the algorithm is based on using
polyhedra Bk and focussing on the extremal points
of the sets Bk .
Theorem 1: If z is an extremal point of B k , then
there exists a j ∈ ΙΝ 0 and an extremal point u of B k − 1 ,
in short : u ∈ E ( B k −1 ) , such that
z = M k′ u .
j

Construction Principle

Stopping Criterion
Theorem 2:
Let zi = M kj ′ ui ( as above, i ∈ {1,2,..., r}).
Then,
H {z1,...,zr } = Bk ⇔ M k ′ zi ∈ H {z1,...,zr } ∀ i = 1,2,..., r.

Construction Principle
 " stopping " (at step k0 ),
 ∞

• Bk = Bk0 (k ≥ k0 ) ⇒  B∗ = U Bi bounded,
i =0

 stability of matrices / dynamics

 B ∗ unbounded
• ∂B0 ∩ ∂Bk = 0
/ ⇒ 
 instability
Gebert, Laetsch, Pickl, We. Wünschiers 2004
Ergenc, We. 2004

Stability Analysis
Ex.:
 a 1  0 1
M = {M 0 , M1} M0 = 
 b 0 ,
 M1 = 
 b 0

   
region of stability
algorithm
instability

Genetic Network
•
Ex. : E = M E, h = 1,
κ
scalar - valued case
 E1 (t 0 ) E 2 (t 0 ) E 3 (t 0 ) E 4 (t 0 )   255 250 0 255 
   
 E1 (t1 ) E 2 (t1 ) E 3 (t1 ) E 4 (t1 )   255 200 50 0 
 E (t ) =
E 2 (t 2 ) E 3 (t 2 ) E 4 (t 2 )   255 180 70 255 
 1 2   
 E (t ) E 2 (t 3 ) E 3 (t 3 ) E 4 (t 3 )   255 0 
 1 3   170 80 
9  0 0 0 0 
8 ö5
 
Expression level, ö
7 ö3
 0 .4 − 0 . 61 0 0 
M =
6 ö1
0 
5
4 ö2 ö4
0 0 .2 − 0 . 39
3
 
2 ö0
 1 − 2
1
0  0 0 
0 2 4 6 8
Time, t

Genetic Network
0.4 x1
gene1 gene2
0.2 x2 1 x1
gene3 gene4

Gene-Environment Networks - Hybrid Systems
E (k + 1) = M s ( k ) E (k ) + Cs ( k )
s (k ) := FB Q( E (k − 1))
1 if Ei (k ) > Ωi
Qi ( E (k )) := 
0 else
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

Gene-Environment Networks
E (k + 1) = M s ( k ) E (k ) + Cs ( k )
IE (k + 1) = IM k IE (k )
•
IE (t) = IM IE (t) locally
• )
E(t ) = Ms(t ) E(t ) + Cs(t ) E(t ) + Ds(t )

Gene-Environment Networks
)
E (k + 1) = M s ( k ) { (k ) + Cs ( k ) { (k ) + Ds ( k )
E E
IE (k + 1) = IM k IE (k )
•
IE (t) = IM IE (t) locally
• )
E(t ) = Ms(t ) E(t ) + Cs(t ) E(t ) + Ds(t )

Gene-Environment Networks



 IE (k + 1) = IM k IE (k )
 
 
 
  •
IE (t) = IM IE (t)
modules

Gene-Environment Networks
• )
E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + Ds ( t )
where
s (t ) := F (Q ( E (t )))
Q( E (t )) = (Q1 ( E (t )),..., Qn ( E (t )))
0 for Ei (t ) < θ i ,1
1 for θ i ,1 < Ei (t ) < θ i , 2

Qi ( E (t )) := 
...
di for
 θ i ,d < Ei (t )
i

Gene-Environment Networks
• )
E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + Ds ( t )
where
s (t ) := F (Q ( E (t )))
Q( E (t )) = (Q1 ( E (t )),..., Qn ( E (t )))
parameter estimation:
0 for Ei (t ) < θ i ,1
1 for θ i ,1 < Ei (t ) < θ i , 2
 (i) estimation of thresholds
Qi ( E (t )) := 
...
di for
 θ i ,d < Ei (t )
i
(ii) calculation of matrices and vectors
describing the system in between
thresholds

Gene-Environment Networks
2
l ∗ −1 )
∗
min
∗ ∗
∑
α =0
&
M Eκα + C E κα + D∗ − Eκα
∗ ∗
(mij ), (cil ), (di ) ∞
Chebychev (maximum) norm

Gene-Environment Networks

Gene-Environment Networks
1 if gene j regulates gene i
χi j := 
0 otherwise
ξi l , ζ i

Gene-Environment Networks
mixed integer programming
2
l ∗ −1 )
min ∑
α =0
∗ &
M Eκα + C E κα + D∗ − Eκα
∗
(mij ∗ ), (cil∗ ), (di ∗ ), ( χ ij ), (ξil ), (ζ i ) ∞
subject to
( j = 1, 2,..., n)
n
∑χ
i =1
ij ≤αj
n
∑ξ il ≤ βl (l = 1, 2,..., m)
i =1
n
∑ζ
i =1
i ≤γ
mii ≥ δ i ,min (i = 1, 2,..., n)
& overall box constraints

Gene-Environment Networks
TPS3 GSY2
Trehalose UDP-Glucose
GLC3
Glycogen
UGP1
NTH2 Glucose-1-Phosphate GPH1
PGM1
HXK1
Glucose Glucose-6-Phosphate
Glycolysis pathway
knockout
glycogen metabolism pathway in yeast Saccharomyces cerevisiae

Gene-Environment Networks
GSIP relaxation
2
l ∗ −1 )
min ∑
α =0
∗ &
M Eκα + C E κα + D∗ − Eκα
∗
∞
(mij ∗ ), (cil∗ ), (di ∗ )
subject to
n
∑i =1
p ij ( m ij ∗ , y ) ≤ α j ( y ) ( j = 1, ..., n )
n
∑i =1
q il ( c il ∗ , y ) ≤ β l ( y ) ( l = 1, ..., m ) ( y ∈ Y (C ∗ , D∗ ))
n
∑ ζ i ( d i∗ , y ) ≤ γ ( y ) set of combined environmental effects :
i =1 Y (C ∗ , D∗ ) :=
m ii ≥ δ i , m in ( i = 1, . . . , n ) ( ∏
i =1,..., n
0, ci∗l  ) × (
  ∏
i =1,..., n
0, d i∗  )
 
& o v e r a ll b o x c o n s t r a in t s l =1,..., m

General. Semi-Infinite Programming
C2
I, K, L finite

GSIP – Structural Stability
∃ ∀ ∈ ∃ ψ (⋅), ϕ (⋅,⋅)∈ C 0 :
ψ (τ )
τ
Jongen, W.
ψ
ϕ (⋅,τ )
homeom.
⇔
asymptotic
: structurally stable
effect
ε (⋅)
IR n
global local global

GSIP – Structural Stability
Thm. (W. 1999/2003, 2006):
⇔
ξ

GSIP – Structural Stability

Spline Approximation
l −1 . 2
)
min PRSS ( M , C, D) := ∑
κ =0
M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ
∞
+ penalty term
 n 1,i , j ′′ dE 
2
∫ (f )
n n U 2
penalty term := ∑∑  ∑ λα
Eα

1,i , j
L α ( Eα ) E j α
i=1 i=1  α=1 ∞
Eα Eα
 
 n 2,i ,l Eα 2,i ,l ) ′′2
2

( )
n m U
+ ∑∑  ∑ µα ∫ L f α ( Eα ) El dEα 
i=1 l=1  α=1 ∞
Eα Eα
 
n  n
′′ dE 
2
( )
U 2
+ ∑  ∑ ςα ∫ L f α ( Eα )
Eα

3,i 3,i
α
i=1  α=1 ∞
Eα Eα
 
Tikhonov regularization (ridge regression)

Spline Approximation
. 2
l −1 )
min PRSS ( M , C, D) := ∑
κ =0
M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ
∞
+ penalty term
 n 1,i , j ′′ dE 
2
∫ (f )
n n U 2
penalty term := ∑∑  ∑ λα
Eα

1,i , j
L α ( Eα ) E j α
i=1 i=1  α=1 ∞
Eα Eα
 
 n 2,i ,l Eα 2,i ,l ) ′′2
2

( )
m in t n m U
2
+ ∑∑  ∑ µα ∫ L f α ( Eα ) El dEα 
U (θ ,θ ,θ ) ≤ t i=1 l=1  α=1 ∞
1 2 3 2 Eα Eα
s .t.
∞
 
n  n
′′ dE 
2
( )
2 2
V i α j (θ 1 )
U
≤ M
2
+ ∑  ∑ ςα ∫ L f α ( Eα )
Eα

3,i 3,i
∞ iα j α
i=1  α=1 ∞
Eα Eα
2  
W i α l (θ 2 ) ≤ N
2
∞ iα l
2
Z i α (θ 3 ) ≤ R
2
conic quadratic programming
∞ iα
0 ≤ t interior points methods

Spline Approximation
. 2
l −1 )
min PRSS ( M , C, D) := ∑
κ =0
M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ
∞
+ penalty term
m in t
2
s .t. U (θ ,θ ,θ )
1 2 3
≤ t2
∞
2
V i α j (θ 1 ) ≤ M
2
∞ iα j
2
W i α l (θ 2 ) ≤ N
2
∞ iα l
2
Z i α (θ 3 ) ≤ R
2
conic quadratic programming
∞ iα
0 ≤ t interior points methods

TEM Model

TEM Model
Example of stability
Article 2, Kyoto Protocol

Dynamics and Control in CO2-Emission Reduction
( +1) () ()
 0
(k
     ( k
(k
  = M ( k )   0   0
     
 
( k +1)
    ( k )   ( k )  0 
  = M (k )       +  (k ) 
      u 
 
IE ( k +1) = IM ( k ) IE ( k )

Dynamics and Control in CO2-Emission Reduction
( +1) () ()
 0
(k
     ( k
(k
  = M ( k )   0   0
     
 
( k +1)
    ( k )   ( k )  0 
  = M (k )       +  (k ) 
      u 
 
IE ( k +1) = IM ( k ) IE ( k )

Gene-Environmental and Financial Dynamics
d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t )
. (k )
∆Wi ( k ) 1 ( k )  ( ∆Wi
(k ) 2
) 
E i ≈ ai (E , t (k ) (k )
) + bi (E , t
(k ) (k )
) ( k ) + (b'bi )(E , t ) 
i
(k )
(k )
− 1
h 2  h 
• Modeling
• Testing
IE ( k +1) = IM ( k ) IE ( k ) • Prediction
• Stability

Gene-Environmental and Financial Dynamics
d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t )
. (k )
∆Wi ( k ) 1 ( k )  ( ∆Wi
(k ) 2
) 
E i ≈ ai (E , t (k ) (k )
) + bi (E , t
(k ) (k )
) ( k ) + (b'bi )(E , t ) 
i
(k )
(k )
− 1
h 2  h 
• Modeling
• Testing
IE ( k +1) = IM ( k ) IE ( k ) • Prediction
• Stability

Regulatory Networks: Errors and Uncertainty
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ2
θ1

Regulatory Networks
and
Ellipsoidal Calculus

Regulatory Networks ― Ellipsoidal Calculus
Assumption:
• Clustered variables (errors) are correlated
Interacting groups (clusters) of genetic and environmental variables
•
How can we model the time-discrete dynamics of
the ellipsoidal states of clusters?

Regulatory Networks ― Ellipsoidal Calculus
1. Clustering
(Groups of genes / groups of environmental items)
2. Assign ellipsoids
(Center = measurement value, configuration matrix = covariances)
3. Regulatory system
(Interaction of clusters defined by affine-linear coupling rules)
4. Parameter identification

Regulatory Networks ― Ellipsoidal Calculus
1) Clustering
Identify groups (clusters) of jointly acting genetic and environmental
variables
disjoint
overlapping

Regulatory Networks ― Ellipsoidal Calculus
2) Interaction of Genetic Clusters

Regulatory Networks ― Ellipsoidal Calculus
3) Interaction of Environmental Clusters

Regulatory Networks ― Ellipsoidal Calculus
3) Interaction of Genetic & Environmental Clusters
⇒ Determine the degree of connectivity

Regulatory Networks ― Ellipsoidal Calculus
Task:
• Identify and analyze highly data based on ellipsoidalofmeasurement data.
genetic and environmental
interconnected systems clusters of
• Calculate predictions of the ellipsoidal states.
• Assume: Affine-linear coupling rules.
⇒ Ellipsoidal Calculus

Regulatory Networks ― Ellipsoidal Calculus
Clusters and Ellipsoids:
Genetic clusters: C1,C2,…,CR
Environmental clusters: D1,D2,…,DS
Genetic ellipsoids: X1,X2,…,XR Xi = E (µi,Σi)
Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)

Regulatory Networks ― Ellipsoidal Calculus

Regulatory Networks ― Ellipsoidal Calculus
r=1

Regulatory Networks ― Ellipsoidal Calculus
The Regression Problem:
measurement
Maximize (overlap of ellipsoids)
T
 R ˆ (κ ) R
ˆ (κ ) ∩ E ( κ ) 
∑
κ =1
 ∑ X r ∩ X r + ∑ Er
 r =1
(κ )
r =1
r 

prediction

Regulatory Networks ― Ellipsoidal Calculus
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)
E (µ r , Π r )
µr

Thank you very much for your attention!
References:
http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf
gweber@metu.edu.tr