CHAPTER SIX Mathematical Modeling of Neuronal Polarization During Development Honda Naoki * , Shin Ishii † * Imaging Platform for Spatio-Temporal Information, Graduate School of Medicine, Kyoto University, Kyoto, Kyoto, Japan † Graduate School of Informatics, Kyoto University, Uji, Kyoto, Japan Contents 1. Biological Background 128 2. Biophysical Model 129 2.1 Model for active transport, diffusion, and degradation of factor X 130 2.2 Model for regulation of neurite growth by factor Y 131 3. Mathematical Analysis 132 3.1 Factor X along neurite 132 3.2 Factor X in the soma 133 4. Mechanism of Neuronal Polarization 134 4.1 Local activation 134 4.2 Global inhibition 134 4.3 Logic of neuronal polarization 135 5. Discussion 138 5.1 Candidates for two factors 138 5.2 Comparisons with experiments 139 Acknowledgments 140 References 140 Abstract During development of the brain, morphogenesis of neurons is dynamically organized from a simple rounded shape to a highly polarized morphology consisting of soma, one axon, and dendrites, which is a basis for establishing the unidirectional transfer of elec- tric signals between neurons. The mechanism of such polarization is thought to be “local activation–global inhibition”; however, globally diffusing inhibitor molecules have not been identified. In this chapter, we present a theoretical modeling approach of such neuronal development. We first summarize biological research on neuronal polarization and then develop a biophysical model. Through mathematical analysis, principles of local activation–global inhibition are illustrated based on active transport, protein deg- radation, and neurite growth, but not on globally diffusing inhibitor. Progress in Molecular Biology and Translational Science, Volume 123 # 2014 Elsevier Inc. ISSN 1877-1173 All rights reserved. http://dx.doi.org/10.1016/B978-0-12-397897-4.00003-6 127 Author's personal copy