A discrete dynamical system arising in molecular biology

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September 2012, 17(6): 2091-2151. doi: 10.3934/dcdsb.2012.17.2091

A discrete dynamical system arising in molecular biology

Howard A. Levine ^1, , Yeon-Jung Seo ^2, and Marit Nilsen-Hamilton ^3,

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

Received May 2011 Revised April 2012 Published May 2012

Abstract
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SELEX (Systematic Evolution of Ligands by EXponential Enrichment) is an iterative separation process by which a pool of nucleic acids that bind with varying specificities to a fixed target molecule or a fixed mixture of target molecules, i.e., single or multiple targets, can be separated into one or more pools of pure nucleic acids. In its simplest form, as introduced in [6], the initial pool is combined with the target and the products separated from the mixture of bound and unbound nucleic acids. The nucleic acids bound to the products are then separated from the target. The resulting pool of nucleic acids is expanded using PCR (polymerase chain reaction) to bring the pool size back up to the concentration of the initial pool and the process is then repeated. At each stage the pool is richer in nucleic acids that bind best to the target. In the case that the target has multiple components, one obtains a mixture of nucleic acids that bind best to at least one of the components. A further refinement of multiple target SELEX, known as alternate SELEX, is described below. This process permits one to specify which nucleic acids bind best to each component of the target.
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].

Keywords: nucleic acids, SELEX., global attractors, polymerase chain reaction, Discrete dynamical systems.

Mathematics Subject Classification: Primary: 37B25, 92C40; Secondary: 37B55, 92D2.

Citation: Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091

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 non-linear analysis,, Journal of Molecular Biology, 222 (1991), 739. doi: 10.1016/0022-2836(91)90509-5. Google Scholar
[7]	S. Klug and M. Famulok, All you wanted to know about SELEX,, Molecular Biology Reports, 20-2 (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. Nilsen-Hamilton, 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. Nilsen-Hamilton, A mathematical analysis of multiple-target SELEX,, Bulletin of Mathematical Biology, 72 (2010), 1623. doi: 10.1007/s11538-009-9491-x. Google Scholar
[12]	Y.-J. Seo, "A Mathematical Analysis of Multiple-Target SELEX,", Ph.D thesis, (2010). Google Scholar
[13]	Y. Seo, H. A. Levine and M. Nilsen-Hamilton, A mathematical analysis of alternate SELEX,, in preparation., (). Google Scholar
[14]	R. Stoltenburg, C. Reinemann and B. Strehlitz, SELEX-A (r)evolutionary method to generate high-affinity 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. Vant-Hull, A. Payano-Baez, 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 non-linear analysis,, Journal of Molecular Biology, 222 (1991), 739. doi: 10.1016/0022-2836(91)90509-5. Google Scholar
[7]	S. Klug and M. Famulok, All you wanted to know about SELEX,, Molecular Biology Reports, 20-2 (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. Nilsen-Hamilton, 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. Nilsen-Hamilton, A mathematical analysis of multiple-target SELEX,, Bulletin of Mathematical Biology, 72 (2010), 1623. doi: 10.1007/s11538-009-9491-x. Google Scholar
[12]	Y.-J. Seo, "A Mathematical Analysis of Multiple-Target SELEX,", Ph.D thesis, (2010). Google Scholar
[13]	Y. Seo, H. A. Levine and M. Nilsen-Hamilton, A mathematical analysis of alternate SELEX,, in preparation., (). Google Scholar
[14]	R. Stoltenburg, C. Reinemann and B. Strehlitz, SELEX-A (r)evolutionary method to generate high-affinity 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. Vant-Hull, A. Payano-Baez, 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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