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New Advances in Prediction of
Gene-Environment Networks by Applied Mathematics Tools
Gerhard-Wilhelm Weber *
Institute of Applied Mathematics, METU, Ankara, Turkey
Ozlem Defterli Department of Mathematics and Computer Science,
Çankaya University, Ankara, Turkey
Armin Fügenschuh Department for Optimization, Zuse Institute Berlin,
Germany
* Faculty of Economics, Management Science and Law, University of Siegen, Germany;
Center for Research on Optimization and Control, University of Aveiro, Portugal;
Universiti Teknologi Malaysia, Skudai, Malaysia
METU
5th International Summer School
Achievements and Applications of Contemporary Informatics,
Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 3-15, 2010
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• Motivation: Bio-Systems
• Computational Biology
• Modeling and Prediction of Gene Patterns
• Genetic Networks
• Gene-Environment Networks
• The Model Class
• The Time-Discretized Model
• Optimization Problem
• The Mixed-Integer Problem
• Numerical Example
• Conclusion
Outline
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Computational Biology
DNA microarray chip experiments
prediction of gene patterns based on
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. Yilmaz, B. Akteke-Öztürk
S. Özöğür, Z. Alparslan-Gök
A. Soyler, B. Soyler, M. Çetin
S. Özöğür-Akyüz, Ö. Defterli
N. Gökgöz
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Sequence Data(cDNA, Genome,Genbank, etc.)
Selection or Design andSynthesis of the Probes
Array Production
Laser Scan of the Array
Picture Analysis
Test Material Control Material
mRNA-Isolation
cDNA-Synthesisand Labeling
Hybridization
Array Preparation Sample Preparation Data Analysis
DNA experiments
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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/
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E0 : metabolic state of a cell at t0 (:=gene expression pattern),
ith-element of the vector E0 :=expression level of gene i,
Mk := I + hkM(Ek) , Ek (k є N0) is recursively defined as Ek+1 := MkEk.
Scheme of metabolic shift.
Gebert et al. 2006
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Modeling & Prediction
)(: nE
0)0(,)( EEEEME
)(: nnM
prediction, anticipationleast squares – max likelihood
statistical learning
(a) time-continuous:
expression data
matrix-valued function – metabolic reaction
T
n tetetetEE ))(,...,)(,)(()( 21
Expression data
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Modeling & Prediction
kkk EE M1
1 ( ( )) ,k k k kE I h M E E2
2
1 2( )k
k k k
hE I h M M E
(b) time-discrete:
Ex.:
)(Μ jik em M
We analyze the influence of em -parameters
on the dynamics (expression-metabolic).
Ex.: Euler, Runge-Kutta
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• For which parameters, i.e., for which set M(hence, dynamics), is stability guaranteed ?
Mstable
unstable
metabolic reaction
feasible
unfeasible
Stability:
Def.: M is stable : B : (complex) bounded neighbourhood of
0 1 1, M ,M ,..., Mkk ΙΝ M :
1 2 0(M M ... M ) .k k
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Genetic Network
, 1E M E h
)()()()(
)()()()(
)()()()(
)()()()(
34333231
24232221
14131211
04030201
tEtEtEtE
tEtEtEtE
tEtEtEtE
tEtEtEtE
080170255
25570180255
050200255
2550250255
2001
039.02.00
0061.04.0
0000
M
Ė 4Ė 2
Ė 0
Ė 5
Ė 1
Ė 3
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8
Time, t
Ex
pre
ss
ion
le
ve
l, Ė
Ex. :
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gene2
gene3
gene1
gene4
0.4 x1
0.2 x2 1 x1
Genetic Network
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Gene-Environment Networks
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Gene-Environment Networks
1:
0i j
if gene j regulates gene i
otherwise
,i i
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The Model Class
d vector of positive concentration levels
of proteins and of certain levels of environmental factors
continuous change in the gene-expression data in time
is the firstly introduced time-autonomous form, where
initial values of the gene-exprssion levels
: experimental data vectors obtained from microarray experiments
and environmental measurements, at the sample times
: the gene-expression level (concentration rate) of the i th gene at time t
denotes anyone of the first n coordinates in the
d vector of genetic and environmental states.
is the set of genes.
Weber et al. (2008c), Chen et al. (1999),
Gebert et al. (2004a),
Gebert et al. (2006), Gebert et al. (2007),
Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005),
Sakamoto and Iba (2001), Tastan et al. (2005)
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(i) is an nxn constant matrix
is an nx1 vector of gene-expression levels
(ii) represents and th dynamical system of the n genes
and their interaction alone.
: nxn matrix with entries as functions of polynomials, exponential, trigonometric,
splines or wavelets containing some parameters to be optimized.
(iii)
environmental effects
n genes , m environmental effects
are (n+m) vector and
(n+m)x(n+m) matrix, respectively.
Weber et al. (2008c), Tastan (2005),
Tastan et al. (2006),
Ugur et al. (2009), Tastan et al. (2005),
Yilmaz (2004), Yilmaz et al. (2005),
Weber et al. (2008b), Weber et al. (2009b)
(*)
The Model Class
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The Model Class
• In general, in the d dimensional extended space
with
: dxd matrix
: dx1 vectors
Ugur and Weber (2007),
Weber et al. (2008c),
Weber et al. (2008b),
Weber et al. (2009b)
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The Time-Discretized Model
- Euler’s method,
- Runge-Kutta methods, e.g., 2nd-order Heun's method
3rd-order Heun's method is newly introduced: Defterli et al. (2009)
we rewrite as
where
Ergenc and Weber (2004),
Tastan (2005), Tastan et al. (2006),
Tastan et al. 2005)
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The Time-Discretized Model
in the extended space denotes the DNA microarray
experimental data and the data of environmental items
obtained at the time-level
the approximations obtained
by the iterative formula above
initial values
k th approximation or prediction is calculated as:
(**)
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Matrix Algebra:
are nxn and nxm matrices, respectively
(n+m) x (n+m) matrix
are (n+m) vectors
Applying the 3rd-order Heun’s method to the eqn. (*) gives the iterative formula (**), where
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Final canonical block form of : .
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Optimization Problem
mixed-integer least-squares optimization problem :
subject to
Ugur and Weber (2007),
Weber et al.(2008c),
Weber et al. (2008b),
Weber et al. (2009b),
Gebert et al. (2004a),
Gebert et al. (2006),
Gebert et al. (2007).
Boolean variables
, : th : the numbers of genes regulated by gene (its outdegree),
by environmental item , or by the cumulative environment, resp..
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The mixed-integer problem
: nxn constant matrix with entries representing the effect
which the expression level of gene has on the change of expression of gene
Genetic regulation network
mixed-integer nonlinear optimization problem (MINLP):
subject to
: constant vector representing the lower bounds for the decrease of the transcript concentration.
in order to bound the indegree of each node, introduce
binary variables :
is a given parameter.
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Numerical Example:
Consider our MINLP for the data:
Gebert et al. (2004a)
Apply 3rd-order Heun method
Take
using the modeling language Zimpl 3.0, we solve
by SCIP 1.2 as a branch-and-cutframework,
together with SOPLEX 1.4.1 as our LP-solver
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Numerical Example:
Apply 3rd-order Heun’s time discretization :
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Results of Euler Method for all genes:
____ gene A
........ gene B
_ . _ . gene C
- - - - gene D
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Results of 3rd-order Heun Method for all genes:
____ gene A
........ gene B
_ . _ . gene C
- - - - gene D
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Conclusion
The structural behavior of the obtained results is almost the same
(constant first column, decreasing second column and increasing third column)
with the given data in Table 1.
For the values presented in the last column of Table 2, instead of
an alternating behavior, we obtain a much more smooth behavior
by using the 3rd-order Heun's discretization scheme.
The above generated time series results are convergent
and we reach the stable values after a few time steps.
This shows the speed of the new discretization technique.
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Conclusion
In this presentation, we gave a contribution to an improved modeling
of gene-environment networks, including their rarefication
which may be regarded as a regularization,
and to the numerical solution of their dynamics.
By this, we supported to a better future prediction of how such networks
can develop in time, with important consequences in
health care, environmental protection, in the financial sector
and the field of education.
In our future challenges,
we will work on the further improvements of the algorithms,
different kinds of rarefications and combined methods
together with comparative studies.
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