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[11] 25/JAN/24

Turing Symposium on Morphogenesis, 2024
--a Paranoma in Turing's Sight--

February 8 -- February 10, 2024


Venue:    Kawai Hall, Graduate School of Science, Tohoku University
                Aobayama, Aoba-ku, Sendai    
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February 8         February 9         February 10


Abstracts


Thursday, February 8, 2024

9:45     Opening


10:10 -- 11:00   Anna Marciniak-Czochra (Heidelberg University)
Increasing complexity of experimental data vs. mathematical models of developmental pattern formation

Abstract: The lecture will focus on Turing and non-Turing aspects of developmental pattern formation. It will discuss the role of mathematical models in studying biological mechanisms based on new experimental data.
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.

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11:00 -- 11:45   Masataka Kuwamura (Kobe University)
Periodic solutions of mass-conserving reaction-diffusion systems and their perturbed systems

Abstract: We show two examples of periodic solutions of mass-conserving reaction-diffusion systems and their perturbed systems. First, we consider the diffusion-driven destabilization (Turing instability) of a spatially homogeneous limit cycle, which leads to a spatially nonhomogeneous limit cycle. Next, we consider two types of oscillatory patterns, which can be useful for understanding cell polarity oscillations with the reversal and non-reversal of polarity, respectively. This talk is based on joint works with H. Izuhara (University of Miyazaki).   (pdf file)
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lunch break

13:30 -- 14:15   Shin-Ichiro Ei (Hokkaido University)
Some researches on the physics of pattern formation

Abstract: Turing patterns appearing in the activator-inhibitor model is believed to be due to the effect of local activation and long range inhibition (LALI), which corresponds to an integral kernel with the Mexican hat profile. In this talk, we propose a new method to theoretically derive the Mexican hat type integral kernel from two component activator-inhibitor systems. The method is extend to the one to derive an essential integral kernel from any given network system. In practice, we can show that any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations including the reduced integral kernel called ``effective kernel'' in the convolution type. Applications including skin patterns and wave patterns on fly brain will be mentioned.
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14:30 -- 15:15   Kousuke Kuto (Waseda University)
Bifurcation structure of cross-diffusion limit in the SKT model

Abstract: This talk is concerned with the Lotka-Volterra competition system with cross-diffusion terms, which was proposed by Shigesada, Kawasaki and Teramoto (1979). We focus on the asymptotic behavior of positive steady-states when one or both of the cross-diffusion coefficients tend to infinity (the unilateral or full cross-diffusion limit). Each of the two types of cross-diffusion limits is considered with two types of boundary conditions, Neumann and Dirichlet types, and a total of four types of global bifurcation structures are presented. Among other things, in the perturbation of the full cross-diffusion limit under the Dirichlet boundary condition, a branch of small coexistence solutions bifurcates from the trivial solution, and many branches of segregating solutions bifurcate from the branch of small coexistence solutions.

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15:30 -- 16:15   Sohei Tasaki (Hokkaido University)
Multilevel mathematical modeling methods for morphogenesis of bacterial cell populations

Abstact: Bacterial cell populations exhibit diverse growth morphologies and collectively form a robust system that can withstand environmental fluctuations. The diversity of macroscopic spatiotemporal patterns and flexible environmental responses in morphogenesis are supported by a variety of cellular states. Therefore, to understand the morphogenesis of bacterial populations, it is necessary to construct and analyze multilevel mathematical models that connect the cellular and tissue levels. Here we propose two multilevel modeling methods.

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16:30 -- 17:15   Discussion
How to get started with interdisciplinary studies

Agenda: Sometimes, interdisciplinary research has a decisive impact on each field of sciences, triggering its new development.
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.

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17:30 -- 19:00     Welcome



Friday, February 9, 2024


10:00 -- 11:00   Frits Veerman (Leiden University)
Beyond Turing: far-from-equilibrium patterns in a mechanochemical model

Abstract: The appearance of Turing patterns is generally believed to depend on an underlying activator-inhibitor mechanism. However, in a number of biological applications, the experimental identification of these components has been problematic. The hypothesis of mechano-chemical interaction, where the morphogen and the surface dynamically interact, provides an alternative to the activator-inhibitor paradigm. We present a mechano-chemical model, where the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replaces the inhibitor. We show how existing ideas and techniques for the rigorous analysis of far-from-equilibrium patterns can be extended to the mechano-chemical context, and demonstrate the use of geometric singular perturbation theory in the construction of patterns on (and of) a planar curve. We highlight and discuss mathematical challenges posed by this particular interplay of partial differential equations and differential geometry.
Joint work with Anna Marciniak-Czochra, Moritz Mercker (U. Heidelberg), and Daphne Nesenberend (U. Leiden).

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11:00 -- 11:45   Kanako Suzuki (Ibaraki University)
Stability of stationary solutions to reaction-diffusion-ODE systems

Abstract: We give a survey of stability results of reaction-diffusion-ODE systems, which consists of a single reaction-diffusion equation coupled with ordinary differential equations. Reaction-diffusion-ODE systems arise, for example, from modeling of interactions between cellular processes and diffusing growth factors.
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). (
pdf file)

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lunch break

13:30 -- 14:15   Ken-Ichi Nakamura (Meiji University)
A remark on the speed of bistable traveling waves for the Lotka-Volterra competion-diffusion system

Abstract: This talk concerns traveling waves for the classical 2-component Lotka-Volterra competition-diffusion system. It is well-known that under strong competition conditions, a unique bistable traveling front connecting two stable states exists (up to translation) and is asymptotically stable (in an appropriate sense). However, the propagation direction of the traveling fronts has not been well-studied. Recently, Ou and his group proposed a method for determining the sign of the speed of traveling fronts by constructing suitable comparison functions.
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).
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14:30 -- 15:15   Kentaro Fujie (Tohoku University)
Global dynamics of chemotaxis models with local sensing

Abstract: In Liu et al. (Science, 2011), it was proposed that a density-suppressed motility could lead to patterns via the so-called "self-trapping" mechanism. Some Keller-- Segel-type reaction-diffusion model with density-suppressed motility was proposed to describe the processing of stripe pattern formation through self-trapping. In this talk, we will focus on some simplified model and study global dynamics of this system. Especially, from a mathematical viewpoint, we will see that this system has a very similar structure to the Keller--Segel system, and we will discuss the similarities and differences in the behaviours of the solutions of both systems. This talk is based on joint works with Jie Jiang (Chinese Academy of Sciences) and Takasi Senba (Fukuoka University).
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15:30 -- 17:30    Poster Session

Goro Akagi (Tohoku University)   
Asymptotic profiles for fast diffusion in domains

Abstract: This poster is concerned with asymptotic profiles of energy solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed in bounded domains. Early studies on this topic date back to the celebrated Berryman and Holland's works, which were motivated from an anomalous diffusion of hydrogen plasma across a purely poloidal octupole magnetic field. Thereafter this issue has been vigorously studied from qualitative and quantitative points of view, and in particular, recent developments in quantitative analysis since Bonforte and Figalli's seminal work are remarkable. In this poster, some of such recent results are overviewed.

Szymon Cygan (Heidelberg University & Wroclaw University)   
Discontinuous stationary solutions to reaction-diffusion-ODE systems

Abstract: pdf file


Kotaro Hisa (Tohoku University)   
Optimal singularities of initial data of a fractional semilinear heat equation in open sets

Abstract: We consider necessary conditions and sufficient conditions on the solvability of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in open sets (possibly unbounded and disconnected) with a smooth boundary.
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.


Lingling Hou (Hong Kong Polytechnic University & Xinjjiang University)   
Existence of Traveling wave solutions to reaction-diffusion-ODE systems with hysteresis

Abstract: This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first ordinary differential equation with a small parameter $\varepsilon>0$. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem has a heteroclinic orbit with jump discontinuity, while the layer problem has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solutions is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breakds down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.


Yuki Kaneko (Kanto Gakuin University)   
Borderline behavior and propagating terrace for a free boundary problem of a reaction-diffusion equation

Abstract: We discuss the asymptotic behavior of a Stefan-type free boundary problem for a kind of multi-stable reaction-diffusion equations. This problem, modeling the spreading of biological species, was proposed by Y. Du and Z.G. Lin in 2010, where the spreading-vanishing dichotomy was proved for the free boundary problem of a monostable diffusion equation with a Neumann boundary condition. When we consider a multi-stable diffusion equation with a Dirichlet boundary condition, the asymptotic behaviors are classified into four cases: big spreading, small spreading, vanishing and a borderline behavior between big spreading and small spreading. Moreover, the big spreading solution converges to a propagating terrace when the Stefan coefficient is sufficiently large. This is based on a joint work with Professor Yoshio Yamada (Waseda University).


Jo Kubokawa (Meiji University)   
Pulsating Traveling Wave of the Mitchell-Schaeffer Model

Abstract: This study investigates the pulsating traveling wave of the Mitchell-Schaeffer model numerically. The Mitchell-Schaeffer model is a mathematical model that describes the dynamics of cardiac action potentials. Numerical simulation of this model under periodic boundary condition shows a pulsating traveling wave solution, which is a time-periodic deformation of a pulse-type traveling wave solution by pulse-to-pulse strong interactions. We investigate the singular limit problem of the Mitchell-Schaeffer model and present a pulsating traveling wave numerically. This is based on a joint work with H.Ninomiya.


Ryunosuke Mori (Meiji University)   
Blocking and propagation phenomena in spatially undulating cylindrical domains

Abstract: We consider blocking and propagation phenomena of free boundary problem for curve shortening flow with a driving force in spatially undulating cylindrical domains. In this free boundary problem, Matano, Nakamura and Lou in 2006, 2013 characterize the effect of the shape of the boundary to blocking and propagation of the solutions under some slop condition about the boundary that implies time global existence of the classical solutions. In this study, we consider the effect of the shape of the boundary to blocking and propagation of this free boundary problem under more general situation that the solutions may develop singularities near the boundary. As a result, we obtain that if the spatial undulation of the boundary is steep, then shorter interval of the undulation speeds up propagation because of the singularities of the solutions.


Finn Münnich (Heidelberg University)   
Nonlinear stability analysis and pattern formation in reaction-diffusion-ODE systems

Abstract: Reaction-diffusion-ODE systems are emerging in modeling of biological pattern formation based on the coupling of diffusive and non-diffusive nonlinear processes. They may exhibit patterns with singularities such as jump-discontinuities. We first establish a general framework for the stability analysis of steady states by providing nonlinear stability and instability conditions for bounded stationary solutions of reaction-diffusion-ODE systems. Then we consider Turing patterns. A necessary condition for the appearance of Turing patterns is diffusion-driven instability (DDI). We derive sufficient conditions for DDI in reaction-diffusion-ODE systems. Furthermore, we summarize when we cannot expect stable Turing patterns even if we have DDI.


Keita Nakajima (Meiji University)   
Influence of heterogeneous biological diffusion on propagation phenomena in reaction-diffusion systems

Abstract: The heart pumps blood through the body by regularly propagating the action potential. However, it is said that abnormal electrical signals are generated when the heart becomes spatially heterogeneous due to myocardial infarction.
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.


Ryosuke Nishide (University of Tokyo)   
Pattern propagation driven by surface curvature

Abstract: Pattern formation and dynamics on curved surfaces are ubiquitous, especially in biological systems. The behavior of Turing patterns on curved surfaces was first considered by Turing himself in 1952 and has been of interest ever since. In this talk, we report that the propagation of Turing patterns is driven by surface curvature: Turing patterns are static on a flat surface but can propagate on an axisymmetric surface. Numerical and theoretical analyses reveal that the conditions for the onset of pattern propagation depend on the symmetry of the surface and the pattern.


Hyunjoon Park (Meiji University)   
Interface motion of Allen-Cahn equation with anisotropic nonlinear diffusion

Abstract: We consider the singular limit problem of Allen-Cahn equation when the diffusivity depends on the solution(or density) with anisotropic diffusivity. It is known that the solution dependent diffusivity affects the speed of the mean curvature flow motion, whereas the anisotropic diffusivty leads to the anisotropic mean curvauture flow of the interface. In this study, we focus on understanding the interface motion of the Allen-Cahn equation when diffusivity has both the density dependent diffusion and the anisotropic diffusion. We first give a formal asymptotic expansion which allows us to expect the interface motion equation. We also give a brief sketch that the expected interface motion does related to the solution by constructing a pair of sub- and super-solutions.


Florian Salin (Tohoku University)   
Numerical Analysis of Fractional Nonlinear Diffusion Equation on Bounded Domain

Abstract: This poster will discuss a numerical method for a fractional nonlinear diffusion equation on bounded domains. This equation arises by combining fractional (in space) diffusion with a nonlinearity of porous medium or fast diffusion type. It is well-known that, in the porous medium case, the energy of the solutions to this equation decays algebraically, and in the fast diffusion case, solutions extinct in finite time. Based on a discretization of the fractional Laplacian recently introduced by Huang and Oberman, we will present a scheme that preserves the energy decay estimates as well as the finite time extinction phenomenon. For that, we will introduce an implicit Euler scheme that satisfies the same structural energy inequalities as the continuous equation, and introduce discrete functional analysis tools to adapt continuous arguments to the discrete level.


Kotaro Sato (Tohoku University)   
Vanishing-viscosity limit in rate-independent evolution equations with a degenerate and singular dissipation potential

Abstract: This poster is concerned with doubly-nonlinear evolution equations where the energy functional is not convex and the dissipation potential functional has singularity and degeneracy. Since the energy functional is not convex, the solutions are not necessarily continuous in time. In this poster, a parabolic regularized equation with a (strong) dissipation term is introduced and its vanishing-viscosity limit is considered via arc-length reparameterizations.


Izumi Takagi (Tohoku University)   
What is a "pattern"?

Abstract: There is no mathematical definition of "pattern". Even the "Turing pattern" is not defined uniquely. However, it is counterproductive to give a rigorous definition of pattern, because it contains a variety of phenomena. Instead, we discuss what kind of properties makes us recognize there is a pattern in specific examples.


Koichi Taniguchi (Tohoku University)   
Unconditional uniqueness and non-uniqueness of solutions of Hardy-H\'enon parabolic equations

Abstract: We study the problems of uniqueness for Hardy-H\'enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H\'enon type) in the nonlinear term. To deal with the Hardy-H\'enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. Our purpose is to prove unconditional uniqueness and non-uniqueness for Hardy-H\'enon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces. This is based on the joint work with Noboru Chikami (Nagoya Institute of Technology), Masahiro Ikeda (RIKEN/Keio University) and Slim Tayachi (Universit\'e de Tunis El Manar).


Conghui Zhang (Beijing University of Civil Engineering and Architecture)   
Existence and stability of discontinuous patterns to a receptor-based model

Abstract: I will consider a receptor-based model which arises from modeling of interactions between intracellular processes and diffusible signaling factors. We prove the existence of stationary solutions with jump discontinuity by a variational method and show that under a suitable topology, they are always stable. Then we consider the asymptotic behavior of stationary solutions as the diffusion coefficient tends to infinity and zero. In addition, some numerical simulations are presented to illustrate the theoretical results.

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Saturday, February 10, 2024



10:00 -- 10:45   Shigeru Kondo (Osaka University)
How to transfer the 2D pattern to 3D shape

Abstract: Turing's reaction-diffusion principle revealed how to create spatial order in the body of an organism, and is now accepted by experimental biologists as a fundamental principle in embryology. However, Turing's theory is generally only a two-dimensional, equally spaced pattern, and some additional mechanism is needed to create the characteristic three-dimensional structure of an organism. We believe that one such mechanism is the wrinkling of the cuticle during molting of exoskeletons.
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).

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11:00 -- 11:45   Hirokazu Ninomiya (Hokkaido University)
Turing's instability by equal diffusion

Abstract: In 1952, Turing proposed the mechanism of pattern formation in which a stable equilibrium of some kinetic system is destabilized by diffusion. In the case of two-component reaction--diffusion systems, however, the diffusion coefficients should be different for Turing's instability. I will give an example of a kinetic system with a asymptotically stable equilibrium, while the corresponding two-component reaction--diffusion system with equal diffusions has a family of unstable stationary solutions that is arbitrarily close to the homogeneous stationary solution.

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lunch break

13:00 -- 13:45   Hideo Ikeda (University of Toyama)
Stability of non-uniform solutions in mass-conserving reaction-diffusion systems with bistable non-linearity

Abstract: Mass-conserving reaction-diffusion systems with bistable nonlinearity are considered under general assumptions, which are useful models for studying cell polarity formation, whose process is key in cell division and differentiation.
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).
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14:00 -- 14:45   Yasumasa Nishiura (Hokkaido University)
Morphologies at nanoscale in materials science

Abstract: The focused investigation on the confined assembly of block copolymer particles (BCP) within emulsion droplets has garnered significant attention. This attention is attributed to the fact that self-assembly within flexible geometries serves as an efficient and promising approach for the preparation of unconventional polymer materials, which are not attainable in either bulk states or solutions. This presentation centers on elucidating the intricate process of forming polyhedral block copolymer particles (PBCP) at the nanoscale, characterized by cubic, octahedral, and various geometric structures. These structures represent a novel class, hitherto unexplored and unprecedented in fabrication.
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.
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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
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created on 10/jan/24