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Intersecting Symplectic Topology and Dynamics with Biological Systems

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Taken from lecture notes on “Planar maps, random walks and circle packing” by Asaf Nachmias, 2018. The He-Schramm Theorem establishes an elegant link between the recurrence of a random walk on a plane triangulation and wether it may be circle packed into the disk or the plane.

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The hourglass model of vertebrate development, 1994. While traditionally explained away by the prevalence of Hox genes constraining mechanism at the meta-cis level,  a modernized explanation from Rudolf A. Raff’s A Shape of Life aims to understand the convergence as a product of interaction dynamics between tissues in organogenesis.

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Original sketches by inventor of PCR (Polymerase Chain Reaction) Kary Mullis.

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Intersecting lineage tracing data with transcriptional manifolds. Trying to understands how cells move through their fate space, inferring dynamics based on energy conservation coupled with causal, plural data.

Taken from “Lineage tracing meets single-cell omics: opportunities and challenges”, a review byDaniel E. Wagner and  Allon M. Klein, 2020.

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Intelligent simulations perform better when their knowledge is situated with respect to a sense of self or a directionality of perception.

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Figure from James Meinhardt’s 1982 Book on Models of Biological Pattern Formation. The figure depicts various non-living, self-enhancing processes and their structures emerging from auto catalysation and its lateral inhibition.

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Example and Counterexample for the structural stability of a dynamical system. A dynamical system is said to be structurally stable if for a sufficiently small perturbation of the system, the resulting system is topologically isomorphic to the unperturbed system. 

Taken from Rene Thom’s “Topological Models in Biology”: https://core.ac.uk/download/pdf/82643879.pdf

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Braid Diagram. Used to prove the Long Exact Homology Sequence for three nested sets.

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Diagram of some 2000 metabolic pathways in human cells, glycolysis and citric acid cycle center right. 

Taken from Bruce Alberts’ Molecular Biology of the Cell, 6th Edition - adapted with permission from Kanehisa Laboratories.

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Ford Circle Packings and Farey Sequences.

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KAM twist map. As the perturbation increases, the closed, periodic orbits split into chains of smaller closed orbits mixed with chaotic behavior.

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Torus Shading in Blender