Recent Patents on Biomedical Engineering 2011, 4, 000-000
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Chaos and Gene expression:A Theoretical Approach Athanasios Tsatsaris1,2, Dimitrios Triantaffylou3, Antonios Baldoukas4, George Tsiolis2 and Despina 1 Perrea 1
Laboratory for Experimental Surgery and Surgical Research, School of Medicine, University of Athens, Greece, Laboratory of Sanitary Police Department of Grevena, Police Academy, Greece, 3Laboratory of Applied Mathematics, Department of Mathematics, University of Athens, Greece, 4Laboratory of Mechanical Engineering Research, Technical Institute of Halkida, Greece 2
Received: February 3, 2011; Accepted: February 25, 2011; Revised: February 28, 2011
Abstract: Intoduction: The aim of the study was to investigate whether chaotic phenomena (chaos theory) affects the process of Gene Expression. Methods: Modeling the genes X, Y and Z-which encode a certain protein P-a set of three first order differential equations has been developed and studied in phase-space (x,y,z). Results: The elementary equilibrium points in three dimensional phase portrait analysis, include attractors, saddles and repellors. Conclusions: Attractors indicate a stable equilibrium point which attenuates the production of the protein P, while the saddles and particularly the repellors correspond to an unstable dynamic system, which promotes either the production of a P-flaw protein or totally inhibits gene expression. Among other mechanisms-e.g. patents for gene treatment (US2007000515344) and modulation(US20040014083A1)-chaotic phenomena also seem to regulate in a particular way the DNA encoding , that calls for further theoretical and experimental research(e.g. cardiovascular disease, oncology).
Keywords: DNA, chaos, phase-space, gene expression, cardiovascular disease. 1. INTRODUCTION Since the first published full DNA sequence [1], DNA sequencing technology has made enormous steps. DNA sequencing leads to the definition of genotype which in turn, determines under certain biochemical prerequisites the resulting phenotype. Extensive research has been concentrated on how the phenotype can be predicted from the genotype. Initially, a linear relationship between genes and cellular functions was hypothesized [2], and despite the discovery of some disease-dependent-genes (i.e. breast cancer gene), indisputable scientific evidence has revealed the non-linear relationship between genotype and phenotype [3] . Additionally, it has been acknowledged that most cellular functions require simultaneous interactions among certain gene products, interactions that are usually complicated and extremely non-linear [4-6]. Over the last decades Dynamical System Theory has found application in a great variety of disciplines, including chemistry, physics, biology, medicine and bioengineering revealing very often the underlying chaotic, non-linear behavior of the observed systems [7-9]. Especially, Drug Delivery Systems releasing drug at a controlled rate may be affected by gene-protein response either linearly or no-linearly-Drug Discovery,1,32-34(2006);Current Nanoscience, *Address correspondence to this author at the 4 Ag.Kosma street, 51100, Grevena, Greece; Tel/Fax: 003-24620-28285; E-mail:
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5,134-140(2009);International J. of Nanotechnology,3,416461(2006);Journal of Nanoscience and Nanotechnology, 6,2320-2328(2006);Current Drug Safety,4,79-83(2009)-. In the present study, the genes X,Y and Z that encode a certain protein P , are combined appropriately to create a Three-Dimensional Dynamic system (3-D) , with a view to extracting hidden information of gene expression by exploring the resulting phase-space. 2. METHODS Taking into account up to date experimental results [1012] , it is obvious that we may hypothesize that the genes X , Y and Z-which encode a certain protein P-are related to one another in a particular way. Moreover, attempting to formulate a 3-D first order differential system [13], the axioms below must be stated: Axiom One, the presence of gene X through time-T (denoted as X) may be affected either positively or negatively by the gene Y and vice-versa; Axiom Two, the presence of gene Y through time may be, as well, affected positively or negatively by the gene Z and viceversa; Axiom Three, the presence of gene Z (denoted as Z ) may too , be affected positively or negatively by the gene Xwhere positive or negative influence means a positive or negative feedback. So, a negative or a positive feedback is stipulated among genes X,Y and Z . In more general expression the system (SYS) could be written as:
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2 Recent Patents on Biomedical Engineering, 2011, Vol. 4, No. 1
DX = FX (X,Y , Z ) DT DY = FY (X,Y , Z ) DT DZ = FZ (X,Y , Z ) DT
Tsatsaris et al.
(1)
Where the first part of the equations represent the first Derivatives with respect to time of X,Y,Z and the second part contains non linear Functions (FX ,Y,Z) of X,Y,Z . On the basis of local linearization technique applied closely to an equilibrium point, the SYS can be converted to : DX = a11 X + a12Y + a13 Z + b1 DT DY = a21 X + a22Y + a23 Z + b2 DT DZ = a31 X + a32Y + a33 Z + b3 DT
(2)
cally stable; b) Situation Two , if at least one eigenvalue has positive Real part then zero solution is unstable; c) Situation Three, if for all eigenvalues with Real part
