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[11] 25/JAN/24

Turing Symposium on Morphogenesis, 2024

--a Paranoma in Turing's Sight--

Venue:
Kawai Hall, Graduate School of Science, Tohoku University

Aobayama, Aoba-ku, Sendai
access

February 8 | February 9 | February 10 |

9:45 **Opening **

10:10 -- 11:00 **Anna Marciniak-Czochra** (Heidelberg University)

Turing patterns are specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor type interactions destabilised by diffusion. It is argued that this view is restrictive and problematic in terms of its consistency with biological observations. In this talk two alternatives to 'classical' Turing patterns are presented, based on different choices of fast and slow scale subsystems. The analysis includes far-from-equilibrium patterns arising from degenerate reaction-diffusion models and mechano-chemical patterns described by models defined in a dynamically deforming domain. The advantages of these two alternatives over 'classical' Turing analysis are highlighted. Recent results and future challenges for both approaches are presented. In particular, the problems posed by new experimental data on the dynamics and function of the Wnt signalling system in symmetry breaking and pattern formation in Hydra, a model organism in developmental biology, are discussed.

11:00 -- 11:45 **Masataka Kuwamura** (Kobe University)

lunch break

13:30 -- 14:15 **Shin-Ichiro Ei** (Hokkaido University)

14:30 -- 15:15 **Kousuke Kuto ** (Waseda University)

15:30 -- 16:15 **Sohei Tasaki** (Hokkaido University)

16:30 -- 17:15 **Discussion **

However, it is not easy for (pure) mathematicians to participate in interdisciplinary research. We will identify problems such as what are the obstacles and how to overcome them, and consider how we can start an interdisciplinary study involving mathematicians.

17:30 -- 19:00 Welcome

**Friday, February 9, 2024**

10:00 -- 11:00 **Frits Veerman** (Leiden University)

Joint work with Anna Marciniak-Czochra, Moritz Mercker (U. Heidelberg), and Daphne Nesenberend (U. Leiden).

11:00 -- 11:45 **Kanako Suzuki** (Ibaraki University)

We have showed that all regular stationary solutions are unstable, which implies that reaction-diffusion-ODE systems cannot exhibit spatial patterns and possible stable stationary solutions have to be singular or discontinuous. In this talk, we would like to show sufficient conditions for existence and stability of discontinuous stationary solutions.

These works have been obtained by joint works with A. Marciniak-Czochra (Heidelberg University), G. Karch (University of Wroclaw) and S. Cygan (Heidelberg University). (

lunch break

13:30 -- 14:15 **Ken-Ichi Nakamura** (Meiji University)

In this talk, we will present some results on determining the sign of the speed of traveling fronts by constructing new comparison functions based on the variational formulation of the speed of bistable traveling fronts. Using these results, we can show the uncontrollability of the propagation direction in the case where the interspecific competition coefficients are too different.

This talk is based on joint work with Toshiko Ogiwara (Josai University).

14:30 -- 15:15 **Kentaro Fujie** (Tohoku University)

15:30 -- 17:30 **Poster Session**

Our conditions enable us to identify the optimal strength of the admissible singularity of initial data for the local-in-time solvability and they differ in the interior of the set and on the boundary of the set.

I was interested in this phenomenon and conducted research on the mechanism of abnormal electrical signals using the Aliev-Panfilov model, which is a mathematical model of the heart. In this study, we assume that the cardiac cells damaged by myocardial infarction can be represented by small diffusion coefficients. Numerical simulations of the Aliev-Panfilov model in the case of spatial heterogeneity of diffusion show the complicated dynamics including spiral waves. To investigate the mechanism of the spontaneous generation of spiral waves, we investigate the effect of variable diffusion on propagation.

First, we go back to the derivation of the diffusion coefficient from the transition probability and generalize the repulsive/attractive transition. Then we derive the singular limit problem of the Allen-Cahn-Nagumo equation with the generalized diffusion term. We present the relationship between the diffusion coefficients and propagation. Applying this observation to the Aliev-Panfilov model, we discuss the mechanism of the generation of spiral waves. This is based on the joint work with H. Ninomiya.

**Saturday, February 10, 2024**

10:00 -- 10:45 **Shigeru Kondo** (Osaka University)

The horn precursors of beetles have a one-layered pouch-like structure. The pouch has a pattern of wrinkles just like those produced in Turing's simulation. During pupation, the wrinkles elongate, causing the precursor to grow into a giant, characteristic horn morphology, and the morphological changes are encoded by the shape (orientation and density) of the wrinkles. In my talk, I will report the results of my analysis of the principle of converting 2D patterns to 3D using biting insect horns as research material.

This talk is based on the joint work with Keisuke Matsuda (Osaka University).

11:00 -- 11:45 **Hirokazu Ninomiya** (Hokkaido University)

lunch break

13:00 -- 13:45 **Hideo Ikeda** (University of Toyama)

The existence of stationary solutions with a single internal transition layer is shown by using the analytical singular perturbation theory. Moreover, a stability criterion for the stationary solutions is provided by calculating the Evans function. Finally, we discuss the stability of double layer stationary solutions, which is an ongoing work. This talk is based on joint work with Masataka Kuwamura (Kobe University).

14:00 -- 14:45 **Yasumasa Nishiura** (Hokkaido University)

The emergence of these distinct cornered morphologies introduces an intriguing and counterintuitive phenomenon, intricately linked to process parameters, such as evaporation rates and initial concentration, while maintaining other variables as constants. Employing a system of coupled Cahn-Hillard (CCH) equations, we delve into uncovering the underlying mechanisms steering the formation of polyhedral particles. This exploration places emphasis on the critical role of controlling relaxation parameters for the shape variable 'u' and micro-phase separation 'v'.

While initially perceived as a macroscopic model serving as a metaphor or a qualitative representation to comprehend averaged behaviors, it is revealed that this model is not merely an abstraction. Instead, we establish a substantial correspondence between the experimental settings and the model parameters. This underscores the capability of the macroscopic model to encapsulate the microscopic details of particle shape, challenging previous perceptions and enhancing our understanding of the intricacies involved. This is a joint work with Hiroshi Yabu, Edgar Avalos, and Takashi Teramoto.

14:55 ** Closing **

Scientific Committee: Anna Marciniak-Czochra (Heidelberg), Yasumasa Nishiura (Sapporo)

Organizing Committee: Goro Akagi (Sendai), Kanako Suzuki (Mito), Izumi Takagi (Sendai)

Supported in part by

- JSPS Grant-in-Aid for Scientific Research (B) (No. 20H01812)
"Evolution equations describing non-standard irreversible processes
--Analysis on singularities emerging in the dynamics of solutions--"
Japan Society for the Promotion of Science, 2020-2023 and Grant-in-Aid for Challenging Research (Exploratory) (No. 21K18581) "Evolution equations with the coexistence of fractional derivatives and nonlinear structures --perturbation theory and asymptotic analysis--" (Akagi)

- JSPS Grant-in-Aid for Scientific Research (C) #18K03354 "Dynamics of reaction-diffusion-ODE systems exhibiting diffusion-driven instability" (Suzuki)

- JSPS Grant-in-Aid for Scientific Research (C) #19K03557 "Fundamental theory of reaction-diffusion equations with variable coefficients---a panorama in Turing's sight” and JSPS Grant-in-Aid for Scientific Research (C) #23K0317 "Behavior of nonstationary solutions to reaction-diffusion systems possessing continua of stationary solutions" (Takagi),

created on 10/jan/24