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Global injectivity and multiple equilibria in uni and bimolecular reaction networks
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 
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