Current Developments in Mathematical Biology: Proceedings of the Conference on Mathematical Biology and Dynamical Systems, the University of Texas at Tyler, ... 2005 (Series on Knots and Everything)

This volume is a collection of papers on various areas of current interest in mathematical biology, such as epidemic disease modeling, including the effects of vaccination and strain replacement; immunology, such as T-Cell dynamics and the mechanism of phagocytosis; knot theory; DNA computation; and Boolean networks. CONTENTS......Page 8 Introduction......Page 6 Methylation of DNA may be Useful as a Computational Tool: Experimental Evidence Susannah Gal, Nancy Monteith, Sara Shkalim, Hu Huang and Tom Head......Page 10 1. Introduction......Page 11 2. Materials and Methods......Page 12 3.1. Explanation of the system set-up and approach......Page 13 3.2. Testing available methylases and restriction enzymes......Page 15 3.3. 3-variable SAT computation with plasmid DNA and PCR product......Page 17 4. Discussion......Page 20 References......Page 23 1. Introduction......Page 24 2. Definitions......Page 30 3. Local Structure of Networks......Page 32 4. Branching Processes......Page 34 5. Ineffective Gates......Page 36 6. Frozen Gates......Page 41 7. Networks of 2-Input Gates......Page 43 8. Future Work and Open Problems......Page 45 References......Page 46 0. Introduction......Page 48 1. Culprits......Page 50 2. Rational Tangles, Rational Knots and Continued Fractions......Page 52 2.1. Rational Tangles and their Invariant Fractions......Page 54 2.2. Rational Knots and Continued Fractions......Page 60 3. Sums of Two Rational Tangles......Page 63 4. Continued Fractions, Convergents and Lots of Unknots......Page 64 5. Constructing Hard Unknots......Page 66 6. The Smallest Hard Unknots......Page 69 7. The Goeritz Unknot......Page 71 8. Stability in Processive DNA Recombination......Page 72 References......Page 75 1. Introduction......Page 78 2. Efficacy of Phagocytosis......Page 79 3. Modeling phagocytosis......Page 80 4. Modeling Goals......Page 83 References......Page 85 1. Biological Background......Page 88 2. The Model and Analysis......Page 90 2.1. Derivation of the age-structured PDE system......Page 93 2.2. Age independent ODE problem......Page 96 2.3. Simplified case: No net population change......Page 97 2.4. Formulation of the Cauchy problem......Page 98 2.5. Solution using Laplace transforms......Page 100 2.6. Extending the solution: Allowing population growth......Page 104 3.1. Parameter selection and assumptions......Page 107 3.2. Single clone results......Page 108 3.3. Scarcity of antigen......Page 109 3.4. Memory......Page 111 3.5. Two clones competing for antigen sites......Page 113 References......Page 115 1. Introduction......Page 116 2. General Problem......Page 117 3.1. Moving-grid Galerkin method in age......Page 120 3.2. Relationship to Leslie matrix models......Page 125 3.3. Time and space integration......Page 127 3.4. Convergence results......Page 128 4. Example Systems......Page 130 4.1. Proteus mirabilis swarm-colony development......Page 131 4.2. Tumor invasion......Page 134 5. Conclusions......Page 136 References......Page 137 1. Introduction......Page 140 2. The Nutrient-Phytoplankton-Zooplankton Model......Page 141 3. A Nutrient-Plankton-Toxin Model with Inhibition of the Phytoplankton......Page 147 4. Discussion......Page 153 References......Page 155 1. Introduction......Page 158 2. The Mechanism of Strain Replacement......Page 160 3. Differential Effectiveness of the Vaccine and Strain Replacement......Page 162 4. Coinfection, Perfect Vaccination and Strain Replacement......Page 168 5. Cross-Immunity and Strain Replacement......Page 175 6. Discussion......Page 177 References......Page 180