Abstracts (HeKKSaGOn Summer School 2014)

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

Abstracts of Lectures

Pattern formation and applications to materials science, I--III

Abstract: My talk consists of two main parts: one is an application of persistence homology to materials science and the other is on the dynamics of localized patterns in dissipative systems and its applications. If time allows, I will talk about more recent results on self-recovery dynamics for a class of network system.
1. Topological approach in materials science
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. Dynamics of localized patterns in heterogeneous media
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. Behaviors of amoeba (Physarum plasmodium) in heterogeneous environments
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.

Non-Euclidean Statistics with Applications in Biology, I--III

Abstract: This set of lectures is motivated by the question how to deal statistically with non-Euclidean data arising in many applications of biology, biometry, imaging and medicine. Here, due to restrictions, or when equivalence classes are considered, originally vector-valued data often become non-Euclidean.
(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).

Abstract: Bacteria form a great variety of colonies. The spatio-temporal population structures have a major impact on many scientific fields, and so it becomes more desirable to predict the development of colonies; nevertheless the prediction still remains difficult because the growth kinetics and its dependence on the environmental conditions have not yet been fully understood. Here we show morphological changes of colonies grown on nutrient agar media due to pH and nutrient-richness (peptone concentration) variations for pre-cultured, motile cells of bacterial species Bacillus subtilis. Then we perform a reproduction analysis consisting of a series of culture and measurement experiments, modelling and simulations on the basis of theoretical considerations. The resulting model enables us to predict the development of colonies of motile bacteria for diverse environmental settings. Furthermore, our findings provide insights into bacterial survival strategies and the role of each bacterial activity.
This is a joint work with Madoka Nakayama (Sendai National College of Technology) and Wataru Shoji (Tohoku University).

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.

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.

HeKKSaGOn Summer School @ Göttingen

Inference on Pattern Formation:
Applications to Biology and Materials Science

September 11 -- September 20, 2014

Abstracts of Lectures

Time Table

HeKKSaGOn Summer School @ Göttingen

Inference on Pattern Formation: Applications to Biology and Materials Science

September 11 -- September 20, 2014

Abstracts of Lectures

Time Table

Inference on Pattern Formation:
Applications to Biology and Materials Science