
Previous Article
Global injectivity and multiple equilibria in uni and bimolecular reaction networks
 DCDSB Home
 This Issue

Next Article
Some $L_{p}$estimates for elliptic and parabolic operators with measurable coefficients
A discrete dynamical system arising in molecular biology
1.  Department of Mathematics, Iowa State University, Ames, Iowa 50011, United States 
2.  Department of Statistics, Iowa State University, Ames, Iowa 50010, United States 
3.  Department of Biochemistry, Biophysics and Molecular Biology, Iowa State University, Ames, Iowa 50010, United States 
These processes give rise to discrete dynamical systems based on consideration of statistical averages (the law of mass action) at each step. A number of interesting questions arise in the mathematical analysis of these dynamical systems. In particular, one of the most important questions one can ask about the limiting pool of nucleic acids is the following: Under what conditions on the individual affinities of each nucleic acid for each target component does the dynamical system have a global attractor consisting of a single point? That is, when is the concentration distribution of the limiting pool of nucleic acids independent of the concentrations of the individual nucleic acids in the initial pool, assuming that all nucleic acids are initially present in the initial pool? The paper constitutes a summary of our theoretical and numerical work on these questions, carried out in some detail in [9], [11], [13].
References:
[1] 
E. Akin, "The General Topology of Dynamical Systems,", Graduate Studies in Mathematics, (1993). Google Scholar 
[2] 
C. Chen, Complex SELEX against target mixture: Stochastic computer model, simulation and analysis,, Computer Methods and Programs in Biomedicine, 8 (2007), 189. Google Scholar 
[3] 
C. Chen, T. Kuo, P. Chan and L. Lin, Subtractive SELEX against two heterogeneous target samples: Numerical simulations and analysis,, Computers in Biology and Medicine, 37 (2007), 750. doi: 10.1016/j.compbiomed.2006.06.015. Google Scholar 
[4] 
A. D. Ellington and J. W. Szostak, In vitro selection of RNA molecules that bind specific ligands,, Nature, 346 (1990), 818. doi: 10.1038/346818a0. Google Scholar 
[5] 
W. Hahn, "Stabilty of Motion,", Die Grundlehren der mathematischen Wissenschaften, (1967). Google Scholar 
[6] 
D. Irvine, C. Tuerk and L. Gold, SELEXION: Systemic evolution of nucleic acids by exponetial enrichment with integrated optimization by nonlinear analysis,, Journal of Molecular Biology, 222 (1991), 739. doi: 10.1016/00222836(91)905095. Google Scholar 
[7] 
S. Klug and M. Famulok, All you wanted to know about SELEX,, Molecular Biology Reports, 202 (1994), 20. doi: 10.1007/BF00996358. Google Scholar 
[8] 
T. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, Journal of Chemical Physics, 57 (1972), 2976. doi: 10.1063/1.1678692. Google Scholar 
[9] 
H. A. Levine and M. NilsenHamilton, A Mathematical Analysis of SELEX,, Computational Biology and Chemistry, 31 (2007), 11. doi: 10.1016/j.compbiolchem.2006.10.002. Google Scholar 
[10] 
J. Pollard, S. D. Bell and A. D. Ellington, Generation and use of combinatorial libraries,, in, (2000). Google Scholar 
[11] 
Y.J. Seo, S. Chen, H. A. Levine and M. NilsenHamilton, A mathematical analysis of multipletarget SELEX,, Bulletin of Mathematical Biology, 72 (2010), 1623. doi: 10.1007/s115380099491x. Google Scholar 
[12] 
Y.J. Seo, "A Mathematical Analysis of MultipleTarget SELEX,", Ph.D thesis, (2010). Google Scholar 
[13] 
Y. Seo, H. A. Levine and M. NilsenHamilton, A mathematical analysis of alternate SELEX,, in preparation., (). Google Scholar 
[14] 
R. Stoltenburg, C. Reinemann and B. Strehlitz, SELEXA (r)evolutionary method to generate highaffinity nucleic acid ligands,, Biomedical Engineering, 24 (2007), 381. Google Scholar 
[15] 
C. Tuerk and L. Gold, Systemic evolution of ligands by exponetial enrichment: RNA ligands to bacteriophage T4 DNA polymerase,, Science, 249 (1990), 505. doi: 10.1126/science.2200121. Google Scholar 
[16] 
B. VantHull, A. PayanoBaez, R. H. Davis and L. Gold, The mathematics of SELEX against complex targets,, Journal of Molecular Biology, 278 (1998), 579. doi: 10.1006/jmbi.1998.1727. Google Scholar 
[17] 
F. T. Wall, "Chemical Thermodynamics,", W. H. Freeman, (1958). Google Scholar 
show all references
References:
[1] 
E. Akin, "The General Topology of Dynamical Systems,", Graduate Studies in Mathematics, (1993). Google Scholar 
[2] 
C. Chen, Complex SELEX against target mixture: Stochastic computer model, simulation and analysis,, Computer Methods and Programs in Biomedicine, 8 (2007), 189. Google Scholar 
[3] 
C. Chen, T. Kuo, P. Chan and L. Lin, Subtractive SELEX against two heterogeneous target samples: Numerical simulations and analysis,, Computers in Biology and Medicine, 37 (2007), 750. doi: 10.1016/j.compbiomed.2006.06.015. Google Scholar 
[4] 
A. D. Ellington and J. W. Szostak, In vitro selection of RNA molecules that bind specific ligands,, Nature, 346 (1990), 818. doi: 10.1038/346818a0. Google Scholar 
[5] 
W. Hahn, "Stabilty of Motion,", Die Grundlehren der mathematischen Wissenschaften, (1967). Google Scholar 
[6] 
D. Irvine, C. Tuerk and L. Gold, SELEXION: Systemic evolution of nucleic acids by exponetial enrichment with integrated optimization by nonlinear analysis,, Journal of Molecular Biology, 222 (1991), 739. doi: 10.1016/00222836(91)905095. Google Scholar 
[7] 
S. Klug and M. Famulok, All you wanted to know about SELEX,, Molecular Biology Reports, 202 (1994), 20. doi: 10.1007/BF00996358. Google Scholar 
[8] 
T. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, Journal of Chemical Physics, 57 (1972), 2976. doi: 10.1063/1.1678692. Google Scholar 
[9] 
H. A. Levine and M. NilsenHamilton, A Mathematical Analysis of SELEX,, Computational Biology and Chemistry, 31 (2007), 11. doi: 10.1016/j.compbiolchem.2006.10.002. Google Scholar 
[10] 
J. Pollard, S. D. Bell and A. D. Ellington, Generation and use of combinatorial libraries,, in, (2000). Google Scholar 
[11] 
Y.J. Seo, S. Chen, H. A. Levine and M. NilsenHamilton, A mathematical analysis of multipletarget SELEX,, Bulletin of Mathematical Biology, 72 (2010), 1623. doi: 10.1007/s115380099491x. Google Scholar 
[12] 
Y.J. Seo, "A Mathematical Analysis of MultipleTarget SELEX,", Ph.D thesis, (2010). Google Scholar 
[13] 
Y. Seo, H. A. Levine and M. NilsenHamilton, A mathematical analysis of alternate SELEX,, in preparation., (). Google Scholar 
[14] 
R. Stoltenburg, C. Reinemann and B. Strehlitz, SELEXA (r)evolutionary method to generate highaffinity nucleic acid ligands,, Biomedical Engineering, 24 (2007), 381. Google Scholar 
[15] 
C. Tuerk and L. Gold, Systemic evolution of ligands by exponetial enrichment: RNA ligands to bacteriophage T4 DNA polymerase,, Science, 249 (1990), 505. doi: 10.1126/science.2200121. Google Scholar 
[16] 
B. VantHull, A. PayanoBaez, R. H. Davis and L. Gold, The mathematics of SELEX against complex targets,, Journal of Molecular Biology, 278 (1998), 579. doi: 10.1006/jmbi.1998.1727. Google Scholar 
[17] 
F. T. Wall, "Chemical Thermodynamics,", W. H. Freeman, (1958). Google Scholar 
[1] 
Wei Feng, C. V. Pao, Xin Lu. Global attractors of reactiondiffusion systems modeling food chain populations with delays. Communications on Pure & Applied Analysis, 2011, 10 (5) : 14631478. doi: 10.3934/cpaa.2011.10.1463 
[2] 
Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems  A, 2008, 20 (4) : 10391056. doi: 10.3934/dcds.2008.20.1039 
[3] 
David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96101. doi: 10.3934/proc.2001.2001.96 
[4] 
B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537550. doi: 10.3934/cpaa.2006.5.537 
[5] 
Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reactiondiffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 18911906. doi: 10.3934/cpaa.2014.13.1891 
[6] 
Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for nonautonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems  B, 2017, 22 (5) : 20532065. doi: 10.3934/dcdsb.2017120 
[7] 
Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems  A, 2011, 29 (1) : 91107. doi: 10.3934/dcds.2011.29.91 
[8] 
Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems  A, 2006, 16 (3) : 587614. doi: 10.3934/dcds.2006.16.587 
[9] 
Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems  B, 2005, 5 (2) : 215238. doi: 10.3934/dcdsb.2005.5.215 
[10] 
Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete & Continuous Dynamical Systems  A, 2002, 8 (3) : 585597. doi: 10.3934/dcds.2002.8.585 
[11] 
Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regularity of global attractors for reactiondiffusion systems with no more than quadratic growth. Discrete & Continuous Dynamical Systems  B, 2017, 22 (5) : 18991908. doi: 10.3934/dcdsb.2017113 
[12] 
Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous ReactionDiffusion systems without uniqueness. Discrete & Continuous Dynamical Systems  S, 2009, 2 (1) : 5566. doi: 10.3934/dcdss.2009.2.55 
[13] 
Noriaki Yamazaki. Global attractors for nonautonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935944. doi: 10.3934/proc.2003.2003.935 
[14] 
Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 19892004. doi: 10.3934/cpaa.2014.13.1989 
[15] 
Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete & Continuous Dynamical Systems  A, 1998, 4 (1) : 4354. doi: 10.3934/dcds.1998.4.43 
[16] 
Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems  A, 2003, 9 (3) : 727744. doi: 10.3934/dcds.2003.9.727 
[17] 
Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803813. doi: 10.3934/cpaa.2009.8.803 
[18] 
P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems  A, 2005, 12 (2) : 213232. doi: 10.3934/dcds.2005.12.213 
[19] 
Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems  A, 2011, 31 (2) : 445467. doi: 10.3934/dcds.2011.31.445 
[20] 
Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems  A, 2015, 35 (9) : 46654681. doi: 10.3934/dcds.2015.35.4665 
2018 Impact Factor: 1.008
Tools
Metrics
Other articles
by authors
[Back to Top]