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updated on 27/Aug/14 [7]

Applications to Biology and Materials Science

Venue:
Sitzungszimmer, Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, Göttingen

**Serial Lectures**

1.

Topological approach is a new non-invasive mathematical measurement potentially applicable to various fields including materials science. In this talk I will present two case studies in which computational homology and its variants play a key role to extract essential information on morphological dynamics and its ordering. One is the diblock copolymer problem in a three-dimensional space. It is known that the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. Then we apply computational homology to characterize the topological changes during the morphology transition in three-dimensional space. It should be noted that even the “transient” morphologies can be detected during the time course of the transition, since integer-valued index (Betti numbers) can be observed for certain time before reaching the final state, for instance, transient perforated layers are found from layers to cylinders. The scaling law for Betti number in the phase ordering process is also presented. The other study is about a chracterization of glasses from topological point of view. The glass state is loosely speaking in between perfect crystal and liquid, namely it is not perfectly symmetric nor completely random. Using the method of persistent homology, we discuss about its characterization.

2.

Localized waves are one of the main carriers of information and the effect of heterogeneity of the media in which it propagates is of great importance for the understanding of signaling processes in biological and chemical problems. A typical and simple heterogeneity is a spatially localized bump or dent in 1D or 2D, which in general creates associated defects in the media. One of the main issues is how the geometry of heterogeneity influences over the dynamics of waves. Here the geometry means slope, height, size, curvature and so on. Localized waves are sensitive to those factors and in fact present a variety of dynamics including rebound, pinning, splitting, and traveling motion around the defect. A reduction method to finite-dimensional system is presented, which clarifies the mathematical structure for those dynamics. In the reference below we mainly focus on a class of one-dimensional traveling pulses the associated parameters of which are close to drift and/or saddle-node bifurcations. The great advantage to study the dynamics in such a class is two-fold: firstly it gives us a perfect microcosm for the variety of outputs in general setting when pulses encounter heterogeneities. Secondly it allows us to reduce the original PDE dynamics to tractable finite dimensional system. Such pulses are sensitive when they run into the heterogeneities and show rich responses such as annihilation, pinning, splitting, rebound as well as penetration. The reduced ODEs explain all these dynamics and the underlying bifurcational structure controlling the transitions among different dynamic regimes. It turns out that there are hidden ordered patterns associated with the critical points of ODEs which play a pivotal role to understand the responses of the pulse. We mainly focus on a bump and periodic types of heterogeneity, however our approach is also applicable to general case. It should be noted that there appears spatio-temporal chaos for periodic type of heterogeneity when its period becomes comparable with the size of the pulse.

3.

We present a mathematical model that describes adaptive behaviors of an amoeboid organism, the Physarum plasmodium of true slime mold. The plasmodium migrating in a narrow lane stops moving for a period of time (several hours but the duration differs for each plasmodium) when it encounters the presence of a chemical repellent, quinine. After stopping period, the organism suddenly begins to move again in one of three different ways as the concentration of repellent increases:

going through the repulsive place (penetration), splitting into two fronts of going throught it and turning (splitting) and turning back (rebound). In relation to the physiological mechanism for tip migration in the plasmodium, we found that the frontal tip is capable of moving further although the tip is divided from a main body of organism. This means that a motive force of front locomotion is produced by a local process at the tip. Based on this finding, a mathematical model for front locomotion is considered in order to understand the dynamics for both the long period of stopping and three kinds of behavior. A model based on reaction-diffusion equations succeeds to reproduce the experimental observation. The origin of long-time stopping and three different outputs may be reduced to the hidden instabilities of internal dynamics of the pulse, which may be a skeleton structure extracted from much more complex dynamics imbedded in the Physarum plasmodium.

- master equation, chemotaxis, cross diffusion
- Shigesada model - relative entropy
- interactions of multi-species
- Lotka-Volterra system - weak scaling limit

(a) Typical examples will be given for such data on circles, spheres, tori, real and complex projective spaces, orbit spaces (shape spaces), SPD matrices (diffusion tensors) and phylogenetic tree spaces.

For statistical analysis, we will address key challenges as the development of

(b) appropriate data descriptors (generalized Frechet means),

(c) their employment for dimension reduction methods inspired by PCA,

(d) their asymptotics (consistency and central limit theorems) leading to

(e) inferential non-Euclidean statistics.

The lectures will conclude with a discussion of

(f) open issues (e.g. uniqueness of Frechet means) and

(g) a few known results about the effects of topology and curvature on asymptotics (stickiness and smeariness).

**Tutorials**

The second tutorial is about the standard schemes for discretization of reaction-diffusion equations with emphasis on dos and don'ts in programming.

**Lectures**

This is a joint work with S.Tsujimoto and A.Zuk.

This is a joint work with Madoka Nakayama (Sendai National College of Technology) and Wataru Shoji (Tohoku University).

**Short Communications**

Benjamin EITZNER (Göttingen) *Reliable extraction of filament data from microscopic live cell
images*

Hiroko YAMAMOTO (Sendai) * Location of concentration points in solutions of a reaction-diffusion equation with variable coefficients
*

**Poster Presentations**

Zhijun GAO (Sendai)
*Dynamics of traveling spots with oscillatory tails for the generalized three-component FitzHugh-Nagumo equations*

**Abstract: ** The research is concerned with the dynamics of traveling spots with oscillatory tails arising in a three-component reaction diusion systems called the generalized three-component FitzHugh-Nagumo equations. As is known by Yves Couder's oil droplet experiments on a liquid surface, particles (droplets) associated with waves (surface waves caused by bouncing of particles) show a macroscopic quantum-like behaviors similar to Young's slit experiments and tunnel's phenomena. The main aim is to show numerically that such traveling spots also display a similar wave-particle duality, and moreover by using center manifold reduction theory, we are able to show that the dynamics of single spot dynamics can be reduced to a 4D system of ODEs. Via the analysis of such ODEs, we can show at least partially the origin of such quantum-like behaviors that is never observed for monotone-tail spots. This is a joint work with Prof. Yasumasa Nishiura.

Steffen HAERTING (Heidelberg) *A receptor-based model with coexistence of Diffusion-Driven Instability and hysteresis*

**Abstract:** The aim of this poster is to contribute to the understanding of stability of steady states of systems of ordinary differential equations coupled to reaction-diffusion equations. Such systems arise naturally from the modeling of biological phenomena such as cell-receptor-ligand binding, so called receptor- based models. We present conditions for stability of irregular, discontinuous steady states of a system of one ordinary differential equation coupled to one reaction-diffusion equation. Moreover, we present a model exhibiting Diffusion-Driven Instability and stable discontinuous irregular steady states, i.e. de-novo formation of discontinuous, non-periodic stable steady states. The poster is based on joint research with I. Takagi (Tohoku University) and A. Marciniak-Czochra (University of Heidelberg).

Kurumi HIRUKO (Sendai) *A dynamical aspect of hybrid system describing intermittent androgen suppression therapy of prostate cancer*

**Abstract: **Since a prostate tumor is influenced by androgen, continuous androgen
suppression (CAS) therapy is the most famous therapy of prostate
cancer in Japan. However the relapse of tumors often occurs in spite
of under the CAS therapy. Recently, clinical studies suggested that
intermittent androgen suppression (IAS) therapy may delay or prevent
the relapse. In the IAS therapy, medication is stopped when the size
of tumor decreases less than a lower threshold, and resumed when the
size exceeds an upper threshold.

We deal with a hybrid system describing IAS therapy and prove mathematically that the size of tumor remains in some bounded interval for any time under the IAS therapy.

We deal with a hybrid system describing IAS therapy and prove mathematically that the size of tumor remains in some bounded interval for any time under the IAS therapy.

Dhisa MINERVA (Osaka) *Pathway network analysis of the MMP2 activation model in the early stage of cancer cell invasion*

**Abstract: ** Cancer is the leading causes of death in the world, particularly in developing countries. Metastasis is believed as a factor that would make the situation worse. Cancer cell has the ability to invade and metastasize to a distant part of body. Sato, et al [2] has revealed the mechanism of cancer cell invasion. The mechanism itself involved three molecules found attached on cancer cell membrane, MT1-MMP, and in the extracellular matrix cell, TIMP2 and MMP2, that the interaction between these molecules leads to the MMP2 activation. We believe the MMP2 activation is the initial process causes the cancer cell metastasis.

We study the MMP2 activation model based on mass conservation and action laws. We propose a method based on conservation law and attachment and detachment rules of basic molecules to analyze and solve the ordinary differential equation system of pathway network associated with the MMP2 activation. The idea is to simplify the original model without affecting its dynamic. The method leads us to derive the explicit solution of such system with some analysis about its equilibrium state. The method is expected to be extended into general case for any system of the pathway network.

**References ** [1] Hoshino, D., et al. 2012. Establishment and validation of computational model for MT1-MMP dependent ECM degradation and intervention strategies. PLoS Computational Bio, vol8:e1002479.

[2] Sato, H., et al. 1994. A matrix metalloproteinase expressed on the surface of invasive tumor cells. Nature, 370: 61-65.

We study the MMP2 activation model based on mass conservation and action laws. We propose a method based on conservation law and attachment and detachment rules of basic molecules to analyze and solve the ordinary differential equation system of pathway network associated with the MMP2 activation. The idea is to simplify the original model without affecting its dynamic. The method leads us to derive the explicit solution of such system with some analysis about its equilibrium state. The method is expected to be extended into general case for any system of the pathway network.

[2] Sato, H., et al. 1994. A matrix metalloproteinase expressed on the surface of invasive tumor cells. Nature, 370: 61-65.

Max SOMMERFELD (Göttingen) *Mathematical analysis and modelling of stress fiber skeletons*

**Abstract: ** The seminal study of Engler et al. (2006) demonstrated that
the mechanical properties (Young痴 modulus E) of the micro-environment
alone can direct stem cell differentiation. Here we are concerned with
the statistical analysis of these effects based on flourescence imaging
of the acto-myosin filament cell skeleton structures of human
mesenchymal stem cells under various matrix elasticities. In order to
reliably extract filament structures we have designed the Filament
Sensor. For an analysis of cumulative histograms (total filament length
over orientations) we have generalized the SiZer of Chaudhuri and Marron
(1999, 2000) for the detection of shape parameters of densities on the
real line to the case of circular data giving an exhaustive circular
scale space theory not available previously. Future challenges of this
research lie in including more parameters (beyond cumulative histograms)
of filament processes, analyses of time series of filament processes and
in a 3D analysis of the filament structures.

Jan-Eric STECHER (Heidelberg) *Mass concentration in a nonlocal model of
clonal selection*

**Abstract: **Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of
self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing the dynamics of a continuum of cell
clones structured by the
self-renewal potential. This model consists of a system of
integro-differential equations, which is coupled by nonlinear,
nonlocal term. The integral term acts as a feedback loop in cell
differentiation and proliferation process. The solution of this model
exhibits a mass concentration, which is located in the maximum of the
self-renewal potential. In a second step we incorporate the idea that
stem cells have the ability to mutate, which is modeled via a
diffusive term in the first equation. We give an existence and
uniqueness result for this system of partial integro-differential
equations, as well as numerical simulations to illustrate the
long-term behaviour of the solutions.

Fabian TELSCHOW (Göttingen) *Removing unwanted experimental side-effects in gait analysis for
discrimination*

**Abstract: ** Recording paths from the motion of the human knee joint in practice
involves marker placements on volunteers' thighs and shanks. Although
done by experts, realistic markers have varying loci resulting in
varying coordinate frames for the femur and tibia, respectively. In
order to discriminate gait of volunteers more effectively, we remove
these unwanted experimental side-effect. This allows, moreover, to
detect possible changes in gait patterns after biomechanical interventions.

Tomoyuki TERADA (Sendai) *Characterization of the nonautonomous system and autonomous system using the topological invariants *

**Abstract: ** In this poster, we will introduce the classical characterization of nonautonomous system and autonomous system using the topologial invariants for linear differential operator on one dimensional spaces. In addition, we will present several topics via dynamical system.

Hoang Duy THAI (Göttingen) *Fingerprint image segmentation via total variation*

**Abstract: ** Segmentation is an important step separating foreground from background in images because the wrong segmentation will have impacted on the performance of the recognition system. For fingerprints, foreground regions contain structure-texture information. Based on ideas from Aujol et al. [1, 3], we have developed a technique for fingerprint segmentation via total variation to decompose the original
image into the parts of cartoon, texture, and noise; then, the region of interest can be extracted by texture information via image processing steps.

Haruki UMAKOSHI (Osaka) *Global solution to quadratic systems of reaction-diffusion*

**Abstract: ** We study the global-in-time behavior of quadratic reaction-diffusion systems. The nonlinearity of this systems satisfies ”entropy inequality” which plays an important role to prove the existence of global weak solution. Furthermore, we can prove
the existence of the global solution provided that the space dimension is less than or equal to 2.

Kazuyoshi WATANABE (Kyoto) *Continuous closed 1-form and the related numbers*

**Abstract: ** We can define a continuous closed 1-form on topological space. Similariy as smooth closed 1-form, this 1-form represent a singular cohomology class,and any cohomology class may be realized by a continuous 1-form. From any cohomology class, the novikov betti number and the category number with respect to the class are defined, and category
number is related to a homoclinic cycle and a number of zero-point. I
show the examples of any relation of their numbers.

**Monday, September 15, 2014**

10:00--10:50 Lecture A1:** Y. Nishiura ** *Pattern formation and applications to materials science, I*

11:05--11:55 Lecture A2: ** Y. Nishiura ***Pattern formation and applications to materials science, II*

12:00--13:30 lunch

13:30--14:20 Lecture M1 (T. Kato) *Spectral coincidence of transition operators, automata groups and BBS in tropical geometry*

14:20--14:50 coffee

14:50--15:40 Tutorial M1 (**I. Takagi, S. Tasaki**) *Structure-preserving discretization*

15:50--16:40 Tutorial M2 (**I. Takagi, S. Tasaki**) *Practical introduction to numerical methods for diffusion equations*

18:00--20:00 Welcome Party

**Tuesday, September 16, 2014**

10:00--10:50 Lecture A3: ** Y. Nishiura ***Pattern formation and applications to materials science, III*

11:00--11:50 Special Lecture (TBA)

12:00--13:30 lunch

13:30--14:20 Short communications by poster presenters

14:30--15:20 Short communications by poster presenters

15:20--15:50 coffee

15:50--17:00 Poster Session

**Wednesday, September 17, 2014**

10:00--10:50 Lecture B1:** T. Suzuki ***Space-time patterns in biological model: thermodynamics of multi-species, I*

11:00--11:50 Lecture B2: ** T. Suzuki ***Space-time patterns in biological model: thermodynamics of multi-species, II*

12:00--13:30 lunch

13:30--14:20 Lecture W1: **S. Tasaki** *Prediction of motile bacterial colonies*

14:20--14:50 coffee

14:50--15:40 Tutorial W1 (**A. Krajina**) *Some aspects of extreme value theory*

15:50--16:40 Tutorial W2 (**T. Aspelmeier**) *Stochastic nano-scale imaging in biology*

16:45--17:15 Round Table

**Thursday, September 18, 2014**

10:00--10:50 Lecture B3:** T. Suzuki ***Space-time patterns in biological model: thermodynamics of multi-species, III*

11:00--11:50 Lecture C1:** S. Huckemann ***Non-Euclidean Statistics with Applications in Biology, I*

12:00--13:30 lunch

13:30--14:20 Lecture Th1: **R. Tanaka** *Phase transitions of random infinite sums*

14:30--15:20 Lecture Th2: **A. Krajina** *An application of extreme value theory in non-Euclidean statistics*

15:20--15:50 coffee

16:00--17:30 Göttingen observatory tour

**Friday, September 19, 2014**

10:00--10:50 Lecture C2:** S. Huckemann ***Non-Euclidean Statistics with Applications in Biology, II*

11:00--11:50 Lecture C3:** S. Huckemann ***Non-Euclidean Statistics with Applications in Biology, III*

12:00--13:30 lunch

13:30--14:50 Lecture F1: ** T. Aspelmeier** * Statistical methods for superresolution fluorescence microscopy*

14:40--15:40 Short communications

15:40--16:00 coffee

16:00--16:50 Lecture F2: **I. Takagi** *Control of patterns by spatial heterogeneity*

18:00--20:00 Farewell Party

created on 23/jul/14