Modeling Morphogenesis in Multicellular Structures with Cell Complexes and L-systems

We consider computational modeling of biological systems that consist of discrete components arranged into linear structures. As time advances, these components may process information, communicate and divide. We show that: (1) the topological notion of cell complexes provides a useful framework for simulating information processing and flow between components; (2) an index-free notation exploiting topological adjacencies in the structure is needed to conveniently model structures in which the number of components changes (for example, due to cell division); and (3) Lindenmayer systems operating on cell complexes combine the above elements in the case of linear structures. These observations provide guidance for constructing L-systems and explain their modeling power. L-systems operating on cell complexes are illustrated by revisiting models of heterocyst formation in Anabaena and by presenting a simple model of leaf development focused on the morphogenetic role of the leaf margin.

[1]  Aristid Lindenmayer,et al.  Adding Continuous Components to L-Systems , 1974, L Systems.

[2]  Przemyslaw Prusinkiewicz,et al.  Developmental Computing , 2009, UC.

[3]  B. Söderberg,et al.  Regulated transport as a mechanism for pattern generation: capabilities for phyllotaxis and beyond. , 2009, Journal of theoretical biology.

[4]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[5]  S. Shostak.  Philosophy of Mathematics and Natural Science , 2013 .

[6]  Vadim Shapiro,et al.  A multivector data structure for differential forms and equations , 2000 .

[7]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[8]  Ottoline Leyser,et al.  Auxin, Self-Organisation, and the Colonial Nature of Plants , 2011, Current Biology.

[9]  J. Friml,et al.  Control of leaf vascular patterning by polar auxin transport. , 2006, Genes & development.

[10]  Przemyslaw Prusinkiewicz,et al.  Visualization of Developmental Processes by Extrusion in Space-time , 1996, Graphics Interface.

[11]  Przemyslaw Prusinkiewicz,et al.  Solving Differential Equations in Developmental Models of Multicellular Structures Expressed Using L-systems , 2004, International Conference on Computational Science.

[12]  R. W. Baker,et al.  Simulation of organisms using a developmental model. 1. Basic description. , 1972, International journal of bio-medical computing.

[13]  P. Prusinkiewicz,et al.  A plausible model of phyllotaxis , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[14]  P. Prusinkiewicz,et al.  Model for the regulation of Arabidopsis thaliana leaf margin development , 2011, Proceedings of the National Academy of Sciences.

[15]  P. Prusinkiewicz,et al.  NUMERICAL METHODS FOR TRANSPORT-RESISTANCE SOURCE–SINK ALLOCATION MODELS , 2007 .

[16]  Miltos Tsiantis,et al.  ASYMMETRIC LEAVES1 and auxin activities converge to repress BREVIPEDICELLUS expression and promote leaf development in Arabidopsis , 2006, Development.

[17]  H. Yoon,et al.  Heterocyst pattern formation controlled by a diffusible peptide. , 1998, Science.

[18]  A. Lindenmayer,et al.  Discrete and continuous models for heterocyst differentiation in growing filaments of blue-green bacteria , 1987 .

[19]  S. Parter.  The Use of Linear Graphs in Gauss Elimination , 1961 .

[20]  Vadim Shapiro,et al.  Chain models of physical behavior for engineering analysis and design , 1993 .

[21]  A. Lindenmayer.  Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. , 1968, Journal of theoretical biology.

[22]  P. Prusinkiewicz,et al.  The genetics of geometry. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[23]  A. Lindenmayer.  Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. , 1968, Journal of theoretical biology.

[25]  Philip Haves,et al.  Numerical performance of the SPARK graph-theoretic simulation program , 1999 .

[26]  A. Müller,et al.  Regulation of polar auxin transport by AtPIN1 in Arabidopsis vascular tissue. , 1998, Science.

[27]  Gabor T. Herman,et al.  Simulation of organisms using a developmental model part 1: Basic description , 1972 .

[28]  Przemyslaw Prusinkiewicz,et al.  Graphical applications of L-systems , 1986 .

[29]  Yiying Tong,et al.  Discrete differential forms for computational modeling , 2005, SIGGRAPH Courses.

[30]  R. Haselkorn.  How Cyanobacteria Count to 10 , 1998, Science.

[31]  Przemyslaw Prusinkiewicz,et al.  Numerical methods for transport-resistance sink-source allocation models , 2007 .

[32]  J. Broeckhove,et al.  Pattern formation in a cell based auxin transport model with numerical bifurcation analysis , 2012 .

[33]  Leila De Floriani,et al.  Shape Representations Based on Simplicial and Cell Complexes , 2007, Eurographics.

[34]  Leah Edelstein-Keshet,et al.  Mathematical models in biology , 2005, Classics in applied mathematics.

[35]  E. Mjolsness,et al.  An auxin-driven polarized transport model for phyllotaxis , 2006, Proceedings of the National Academy of Sciences of the United States of America.