October  2020, 7(4): 253-268. doi: 10.3934/jdg.2020018

Foundations of semialgebraic gene-environment networks

1. 

Christerweg, 12, 83624 Otterfing, Germany

2. 

Faculty of Engineering Management, Poznan University of Technology, Poznan, Poland

3. 

Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

4. 

Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran

5. 

Department of Industrial and Systems Engineering, Istinye University, Istanbul, Turkey

* Corresponding author: Erik Kropat

Received  December 2019 Published  July 2020

Gene-environment network studies rely on data originating from different disciplines such as chemistry, biology, psychology or social sciences. Sophisticated regulatory models are required for a deeper investigation of the unknown and hidden functional relationships between genetic and environmental factors. At the same time, various kinds of uncertainty can arise and interfere with the system's evolution. The aim of this study is to go beyond traditional stochastic approaches and to propose a novel framework of semialgebraic gene-environment networks. Foundation is laid for future research, methodology and application. This approach is a natural extension of interconnected systems based on stochastic, polyhedral, ellipsoidal or fuzzy (linguistic) uncertainty. It allows for a reconstruction of the underlying network from uncertain (semialgebraic) data sets and for a prediction of the uncertain futures states of the system. In addition, aspects of network pruning for large regulatory systems in genome-wide studies are discussed leading to mixed-integer programming (MIP) and continuous programming.

Citation: Erik Kropat, Gerhard-Wilhelm Weber, Erfan Babaee Tirkolaee. Foundations of semialgebraic gene-environment networks. Journal of Dynamics & Games, 2020, 7 (4) : 253-268. doi: 10.3934/jdg.2020018
References:
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[2]

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A. Mardani, D. Kannan, R. E. Hooker, S. Ozkul, M. Alrasheedi and E. B. Tirkolaee, Evaluation of green and sustainable supply chain management using structural equation modelling: A systematic review of the state of the art literature and recommendations for future research, Journal of Cleaner Production, 249 (2020), 119383. doi: 10.1016/j.jclepro.2019.119383.  Google Scholar

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M. C. Mills and C. Rahal, A scientometric review of genome-wide association studies, Communications Biology, 2 (2019), 11 pp. doi: 10.1038/s42003-018-0261-x.  Google Scholar

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show all references

References:
[1]

M. U. AkhmetM. KiraneM. A. Tleubergenova and G. Weber, Control and optimal response problems for quasilinear impulsive integrodifferential equations, European J. Oper. Res., 169 (2006), 1128-1147.  doi: 10.1016/j.ejor.2004.10.030.  Google Scholar

[2]

B. Akteke-ÖztürkG.-W. Weber and G. Köksal, Optimization of generalized desirability functions under model uncertainty, Optimization, 66 (2017), 2157-2169.  doi: 10.1080/02331934.2017.1371167.  Google Scholar

[3]

A. Arnau-Soler, et al., Genome-wide by environment interaction studies of depressive symptoms and psychosocial stress in UK Biobank and Generation Scotland, Translational Psychiatry, 9 (2019), 13 pp. doi: 10.1214/19-AOAS1291.  Google Scholar

[4]

E. AyyildizV. Purutçuoglu and G. W. Weber, Loop-based conic multivariate adaptive regression splines is a novel method for advanced construction of complex biological networks, European J. Oper. Res., 270 (2018), 852-861.  doi: 10.1016/j.ejor.2017.12.011.  Google Scholar

[5]

J. Bochnak, M. Coste and M.-F. Roy, Semi-algebraic Sets, A Series of Modern Surveys in Mathematics, 36, Springer Berlin Heidelberg, Berlin, Heidelberg, 1998, 23-58. doi: 10.1007/978-3-662-03718-8_3.  Google Scholar

[6]

A. ÇevikG. WeberB. M. Eyüboglu and K. K. Oguz, The Alzheimer's disease neuroimaging initiative Voxel-MARS: A method for early detection of Alzheimer's disease by classification of structural brain MRI, Ann. Oper. Res., 258 (2017), 31-57.  doi: 10.1007/s10479-017-2405-7.  Google Scholar

[7]

H. Delfs and M. Knebusch, Locally Semialgebraic Spaces, Lecture Notes in Mathematics, Vol. 1173, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0074551.  Google Scholar

[8]

R. Dobson and G. Giovannoni, Multiple sclerosis - A review, European Journal of Neurology, 26 (2019), 27-40.  doi: 10.1111/ene.13819.  Google Scholar

[9]

X. Dong and Y. Yang, Network pruning via transformable architecture search, NeurIPS, 32 (2019), 760-771.   Google Scholar

[10]

A. EidI. Mhatre and J. R. Richardson, Gene-environment interactions in Alzheimer's disease: A potential path to precision medicine, Pharmacology & Therapeutics, 199 (2019), 173-187.  doi: 10.1016/j.pharmthera.2019.03.005.  Google Scholar

[11]

S. Z. A. Gök and G.-W. Weber, On dominance core and stable sets for cooperative ellipsoidal games, Optimization, 62 (2013), 1297-1308.  doi: 10.1080/02331934.2013.793327.  Google Scholar

[12]

A. GoliH. K. ZareR. Tavakkoli-Moghaddam and A. Sadegheih, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-36.  doi: 10.1111/coin.12228.  Google Scholar

[13]

A. Goli, H. Khademi Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem case study: The dairy products industry, Computers and Industrial Engineering, 137 (2019), 106090. doi: 10.1016/j.cie.2019.106090.  Google Scholar

[14]

G. KaraA. Özmen and G.-W. Weber, Stability advances in robust portfolio optimization under parallelepiped uncertainty, CEJOR Cent. Eur. J. Oper. Res., 27 (2019), 241-261.  doi: 10.1007/s10100-017-0508-5.  Google Scholar

[15]

E. KropatA. ÖzmenG.-W. WeberS. Meyer-Nieberg and O. Defterli, Fuzzy prediction strategies for gene-environment networks- fuzzy regression analysis for two-modal regulatory systems, RAIRO-Oper. Res., 50 (2016), 413-435.  doi: 10.1051/ro/2015044.  Google Scholar

[16]

E. Kropat and G.-W. Weber, Fuzzy target-environment networks and fuzzy-regression approaches, Numer. Algebra, Control Optim., 8 (2018), 135-155.  doi: 10.3934/naco.2018008.  Google Scholar

[17]

E. Kropat, G.-W. Weber and C. Pedamallu, Regulatory networks under ellipsoidal uncertainty - data analysis and prediction by optimization theory and dynamical systems, in Data Mining: Foundations and Intelligent Paradigms Vol. 2, Intell. Syst. Ref. Libr., 24, Springer, Berlin, 2012, 27-56. doi: 10.1007/978-3-642-23241-1_3.  Google Scholar

[18]

E. KropatG.-W. Weber and J.-J. Rückmann, Regression analysis for clusters in gene-environment networks based on ellipsoidal calculus and optimization, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 17 (2010), 639-657.   Google Scholar

[19]

E. KürümG.-W Weber and C. Iyigun, Early warning on stock market bubbles via methods of optimization, clustering and inverse problems, Ann. Oper. Res., 260 (2018), 293-320.  doi: 10.1007/s10479-017-2496-1.  Google Scholar

[20]

A. Mardani, D. Kannan, R. E. Hooker, S. Ozkul, M. Alrasheedi and E. B. Tirkolaee, Evaluation of green and sustainable supply chain management using structural equation modelling: A systematic review of the state of the art literature and recommendations for future research, Journal of Cleaner Production, 249 (2020), 119383. doi: 10.1016/j.jclepro.2019.119383.  Google Scholar

[21]

K. McAllister, Current challenges and new opportunities for gene-environment interaction studies of complex diseases, American Journal of Epidemiology, 186 (2017), 753-761.  doi: 10.1093/aje/kwx227.  Google Scholar

[22]

M. C. Mills and C. Rahal, A scientometric review of genome-wide association studies, Communications Biology, 2 (2019), 11 pp. doi: 10.1038/s42003-018-0261-x.  Google Scholar

[23]

R. J. Musci, J. L. Augustinavicius and H. Volk, Gene-environment interactions in psychiatry: Recent evidence and clinical implications, Current Psychiatry Reports, 21 (2019), 81 pp. doi: 10.1007/s11920-019-1065-5.  Google Scholar

[24]

Ö. N. Onak, Y. S. Dogrusoz and G. Weber, Robustness of reduced order nonparametric model for inverse ECG solution against modeling and measurement noise, in Computing in Cardiology, CinC 2018, Maastricht, The Netherlands, September 23-26, 2018, 2018, 1-4. Google Scholar

[25]

A. ÖzmenE. Kropat and G.-W. Weber, Spline regression models for complex multi-modal regulatory networks, Optim. Methods Softw., 29 (2014), 515-534.  doi: 10.1080/10556788.2013.821611.  Google Scholar

[26]

A. ÖzmenE. Kropat and G.-W. Weber, Robust optimization in spline regression models for multi-model regulatory networks under polyhedral uncertainty, Optimization, 66 (2017), 2135-2155.  doi: 10.1080/02331934.2016.1209672.  Google Scholar

[27]

A. ÖzmenG.-W. Weber and E. Kropat, Robustification of conic generalized partial linear models under polyhedral uncertainty, International IFNA-ANS Scientific Journal "Problems of Nonlinear Analysis in Engineering Systems", 38 (2012), 104-113.   Google Scholar

[28]

S. Özöǧür-Akyüz and G.-W. Weber, On numerical optimization theory of infinite kernel learning, J. Global Optim., 48 (2010), 215-239.  doi: 10.1007/s10898-009-9488-x.  Google Scholar

[29]

S. Özögür-Akyüz and G.-W. Weber, Infinite kernel learning via infinite and semi-infinite programming, Optim. Methods Softw., 25 (2010), 937-970.  doi: 10.1080/10556780903483349.  Google Scholar

[30]

T. PaksoyE. Özceylan and G.-W. Weber, Profit oriented supply chain network optimization, CEJOR Cent. Eur. J. Oper. Res., 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0.  Google Scholar

[31]

J. RogersT. Renoir and A. J. Hannan, Gene-environment interactions informing therapeutic approaches to cognitive and affective disorders, Neuropharmacology, 145 (2019), 37-48.  doi: 10.1016/j.neuropharm.2017.12.038.  Google Scholar

[32]

S. K. RoyG. Maity and G.-W Weber, Multi-objective two-stage grey transportation problem using utility function with goals, CEJOR Cent. Eur. J. Oper. Res., 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5.  Google Scholar

[33]

E. Savku and G. -W Weber, A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to finance, J. Optim. Theory Appl., 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3.  Google Scholar

[34]

A. J. Schork, A genome-wide association study of shared risk across psychiatric disorders implicates gene regulation during fetal neurodevelopment, Nature Neuroscience, 22 (2019), 353-361.  doi: 10.1038/s41593-018-0320-0.  Google Scholar

[35]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973433.  Google Scholar

[36]

O. Stein, Bi-Level Strategies in Semi-Infinite Programming, Nonconvex Optimization and its Applications, 71, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-9164-5.  Google Scholar

[37]

P. TaylanG.-W. Weber and E. Kropat, Approximation of stochastic differential equations by additive models using splines and conic programming, International Journal of Computing Anticipatory Systems, 21 (2008), 341-352.   Google Scholar

[38]

E. B. Tirkolaee, A. Goli and G.-W. Weber, Multi-objective Aggregate Production Planning Model Considering Overtime and Outsourcing Options Under Fuzzy Seasonal Demand, Springer, Cham, 2019, 81-96. Google Scholar

[39]

E. B. TirkolaeeS. HadianG.-W. Weber and I. Mahdavi, A robust green traffic-based routing problem for perishable products distribution, Computational Intelligence, 36 (2020), 80-101.  doi: 10.1111/coin.12240.  Google Scholar

[40]

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Figure 1.  Gene-environment networks: Interactions between genes (red), environmental factors and genes (green) and between environmental factors (blue).
Figure 2.  Gene-environment networks with cluster partition: Interactions between genetic clusters (red), environmental clusters and genetic clusters (green) and between environmental clusters (blue).
Figure 3.  Gene-environment network under semialgebraic uncertainty: Interactions between genetic clusters and/or environmental clusters as well as the corresponding semialgebraic uncertainty sets.
Figure 4.  Comparison of measurements and predictions. The computation of the parameter vector $\theta \in \Theta$ depends on $K$ direct comparisons of the intersection between semialgebraic measurements and predictions
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