This document discusses using persistent homology to analyze the topological structure of proteins and relate it to protein compressibility. It summarizes that researchers modeled protein molecules as alpha filtrations to obtain multi-scale insight into their tunnel and cavity structures. The persistence diagrams of the alpha filtrations capture the sizes and robustness of these features in a compact way. The researchers found a clear linear correlation between their topological measure and experimentally determined protein compressibility values.
WiDS Alexandria, Egypt workshop in topological data analysis (Python and R code available on request), covering persistent homology, the Mapper algorithm, and discrete Ricci curvature. Examples include text data and social network data.
Topological Data Analysis: visual presentation of multidimensional data setsDataRefiner
Topology data analysis (TDA) is an unsupervised approach which may revolutionise the way data can be mined and eventually drive the new generation of analytical tools. The idea behind TDA is an attempt to "measure" shape of data and find compressed combinatorial representation of the shape. In ordinary topology, the combinatorial representations serve the purpose of providing the compressed representation of high dimensional data sets which retains information about the geometric relationships between data points. TDA can also be used as a very powerful clustering technique. Edward will present the comparison between TDA and other dimension reduction algorithms like PCA, LLE, Isomap, MDS, and Spectral Embedding.
WiDS Alexandria, Egypt workshop in topological data analysis (Python and R code available on request), covering persistent homology, the Mapper algorithm, and discrete Ricci curvature. Examples include text data and social network data.
Topological Data Analysis: visual presentation of multidimensional data setsDataRefiner
Topology data analysis (TDA) is an unsupervised approach which may revolutionise the way data can be mined and eventually drive the new generation of analytical tools. The idea behind TDA is an attempt to "measure" shape of data and find compressed combinatorial representation of the shape. In ordinary topology, the combinatorial representations serve the purpose of providing the compressed representation of high dimensional data sets which retains information about the geometric relationships between data points. TDA can also be used as a very powerful clustering technique. Edward will present the comparison between TDA and other dimension reduction algorithms like PCA, LLE, Isomap, MDS, and Spectral Embedding.
UMAP is a technique for dimensionality reduction that was proposed 2 years ago that quickly gained widespread usage for dimensionality reduction.
In this presentation I will try to demistyfy UMAP by comparing it to tSNE. I also sketch its theoretical background in topology and fuzzy sets.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
Tensor Train (TT) decomposition [3] is a generalization of SVD decomposition from matrices to tensors (=multidimensional arrays).
It represents a tensor compactly in terms of factors and allows to work with the tensor via its factors without materializing the tensor itself.
For example, we can find the elementwise product of two TT-tensors of size 2^100 and get the result in the TT-format as well.
In the talk, we will show how Tensor Train decomposition can be used to represent parameters of neural networks [1] and polynomial models [2].
This parametrization allows exponentially many 'virtual' parameters while working only with small factors of the TT-format.
To train the model, i.e. optimize the objective subject to the constraint that the parameters are in the TT-format, [2] uses stochastic Riemannian optimization.
[1] Novikov, A., Podoprikhin, D., Osokin, A., & Vetrov, D. P. (2015). Tensorizing neural networks. In Advances in Neural Information Processing Systems.
[2] Novikov, A., Trofimov, M., & Oseledets, I. (2016). Tensor Train polynomial models via Riemannian optimization. arXiv:1605.03795.
[3] Oseledets, I. (2011). Tensor-train decomposition. SIAM Journal on Scientific Computing.
High Dimensional Data Visualization using t-SNEKai-Wen Zhao
Review of the t-SNE algorithm which helps visualizing the high dimensional data on manifold by projecting them onto 2D or 3D space with metric preserving.
Rough sets and fuzzy rough sets in Decision MakingDrATAMILARASIMCA
Rough sets, Fuzzy rough sets, lower approximation, upper approximation, positive region and reduct, Equivalence relation, dependency coefficient, Information system for road accident system
Discrete mathematics counting and logic relationSelf-Employed
In the mathematics world logic is the main fundamental key wi=ord to deal with the problems .If any problems are tried to solve by using certain logics and information provided by the statement then it became quite easier to get the required result effectively and efficiently without any lag of time.
017_20160826 Thermodynamics Of Stochastic Turing MachinesHa Phuong
Show how to construct stochastic models which mimic the
behavior of a general-purpose computer (a Turing machine).
Discrete state systems obeying a Markovian master equation,
which are logically reversible and have a well-defined and
consistent thermodynamic interpretation
UMAP is a technique for dimensionality reduction that was proposed 2 years ago that quickly gained widespread usage for dimensionality reduction.
In this presentation I will try to demistyfy UMAP by comparing it to tSNE. I also sketch its theoretical background in topology and fuzzy sets.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
Tensor Train (TT) decomposition [3] is a generalization of SVD decomposition from matrices to tensors (=multidimensional arrays).
It represents a tensor compactly in terms of factors and allows to work with the tensor via its factors without materializing the tensor itself.
For example, we can find the elementwise product of two TT-tensors of size 2^100 and get the result in the TT-format as well.
In the talk, we will show how Tensor Train decomposition can be used to represent parameters of neural networks [1] and polynomial models [2].
This parametrization allows exponentially many 'virtual' parameters while working only with small factors of the TT-format.
To train the model, i.e. optimize the objective subject to the constraint that the parameters are in the TT-format, [2] uses stochastic Riemannian optimization.
[1] Novikov, A., Podoprikhin, D., Osokin, A., & Vetrov, D. P. (2015). Tensorizing neural networks. In Advances in Neural Information Processing Systems.
[2] Novikov, A., Trofimov, M., & Oseledets, I. (2016). Tensor Train polynomial models via Riemannian optimization. arXiv:1605.03795.
[3] Oseledets, I. (2011). Tensor-train decomposition. SIAM Journal on Scientific Computing.
High Dimensional Data Visualization using t-SNEKai-Wen Zhao
Review of the t-SNE algorithm which helps visualizing the high dimensional data on manifold by projecting them onto 2D or 3D space with metric preserving.
Rough sets and fuzzy rough sets in Decision MakingDrATAMILARASIMCA
Rough sets, Fuzzy rough sets, lower approximation, upper approximation, positive region and reduct, Equivalence relation, dependency coefficient, Information system for road accident system
Discrete mathematics counting and logic relationSelf-Employed
In the mathematics world logic is the main fundamental key wi=ord to deal with the problems .If any problems are tried to solve by using certain logics and information provided by the statement then it became quite easier to get the required result effectively and efficiently without any lag of time.
017_20160826 Thermodynamics Of Stochastic Turing MachinesHa Phuong
Show how to construct stochastic models which mimic the
behavior of a general-purpose computer (a Turing machine).
Discrete state systems obeying a Markovian master equation,
which are logically reversible and have a well-defined and
consistent thermodynamic interpretation
018 20160902 Machine Learning Framework for Analysis of Transport through Com...Ha Phuong
• Propose a data-driven framework to study the relationship
between fluid flow at the macro scale and the internal pore
structure, across the micro and mesoscales, in porous, granular media.
Quantifies a hypothesized link between high permeability and
efficient shortest paths that thread through relatively large
pore bodies connected to each other by high conductance pore throats, embodying connectivity and pore structure.
The variational Gaussian process (VGP), a Bayesian nonparametric model which adapts its shape to match com- plex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity.
Using Topological Data Analysis on your BigDataAnalyticsWeek
Synopsis:
Topological Data Analysis (TDA) is a framework for data analysis and machine learning and represents a breakthrough in how to effectively use geometric and topological information to solve 'Big Data' problems. TDA provides meaningful summaries (in a technical sense to be described) and insights into complex data problems. In this talk, Anthony will begin with an overview of TDA and describe the core algorithm that is utilized. This talk will include both the theory and real world problems that have been solved using TDA. After this talk, attendees will understand how the underlying TDA algorithm works and how it improves on existing “classical” data analysis techniques as well as how it provides a framework for many machine learning algorithms and tasks.
Speaker:
Anthony Bak, Senior Data Scientist, Ayasdi
Prior to coming to Ayasdi, Anthony was at Stanford University where he did a postdoc with Ayasdi co-founder Gunnar Carlsson, working on new methods and applications of Topological Data Analysis. He completed his Ph.D. work in algebraic geometry with applications to string theory at the University of Pennsylvania and ,along the way, he worked at the Max Planck Institute in Germany, Mount Holyoke College in Germany, and the American Institute of Mathematics in California.
Existence results for fractional q-differential equations with integral and m...IJRTEMJOURNAL
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
Talk presented on this workshop "Workshop: Imaging With Uncertainty Quantification (IUQ), September 2022",
https://people.compute.dtu.dk/pcha/CUQI/IUQworkshop.html
We consider a weakly supervised classification problem. It
is a classification problem where the target variable can be unknown
or uncertain for some subset of samples. This problem appears when
the labeling is impossible, time-consuming, or expensive. Noisy measurements
and lack of data may prevent accurate labeling. Our task
is to build an optimal classification function. For this, we construct and
minimize a specific objective function, which includes the fitting error on
labeled data and a smoothness term. Next, we use covariance and radial AQ1
basis functions to define the degree of similarity between points. The further
process involves the repeated solution of an extensive linear system
with the graph Laplacian operator. To speed up this solution process,
we introduce low-rank approximation techniques. We call the resulting
algorithm WSC-LR. Then we use the WSC-LR algorithm for analysis
CT brain scans to recognize ischemic stroke disease. We also compare
WSC-LR with other well-known machine learning algorithms.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Information-theoretic clustering with applicationsFrank Nielsen
Information-theoretic clustering with applications
Abstract: Clustering is a fundamental and key primitive to discover structural groups of homogeneous data in data sets, called clusters. The most famous clustering technique is the celebrated k-means clustering that seeks to minimize the sum of intra-cluster variances. k-Means is NP-hard as soon as the dimension and the number of clusters are both greater than 1. In the first part of the talk, we first present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means but also other kinds of clustering algorithms like the k-medoids, the k-medians, the k-centers, etc.
We extend the method to incorporate cluster size constraints and show how to choose the appropriate number of clusters using model selection. We then illustrate and refine the method on two case studies: 1D Bregman clustering and univariate statistical mixture learning maximizing the complete likelihood. In the second part of the talk, we introduce a generalization of k-means to cluster sets of histograms that has become an important ingredient of modern information processing due to the success of the bag-of-word modelling paradigm.
Clustering histograms can be performed using the celebrated k-means centroid-based algorithm. We consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We prove that the Jeffreys centroid can be expressed analytically using the Lambert W function for positive histograms. We then show how to obtain a fast guaranteed approximation when dealing with frequency histograms and conclude with some remarks on the k-means histogram clustering.
References: - Optimal interval clustering: Application to Bregman clustering and statistical mixture learning IEEE ISIT 2014 (recent result poster) http://arxiv.org/abs/1403.2485
- Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms.
IEEE Signal Process. Lett. 20(7): 657-660 (2013) http://arxiv.org/abs/1303.7286
http://www.i.kyoto-u.ac.jp/informatics-seminar/
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
Similar to 013_20160328_Topological_Measurement_Of_Protein_Compressibility (20)
Universal Approximation Property via Quantum Feature Maps
----
The quantum Hilbert space can be used as a quantum-enhanced feature space in machine learning (ML) via the quantum feature map to encode classical data into quantum states. We prove the ability to approximate any continuous function with optimal approximation rate via quantum ML models in typical quantum feature maps.
---
Contributed talk at Quantum Techniques in Machine Learning 2021, Tokyo, November 8-12 2021.
By Quoc Hoan Tran, Takahiro Goto and Kohei Nakajima
CCS2019-opological time-series analysis with delay-variant embeddingHa Phuong
Q. H. Tran and Y. Hasegawa, Topological time-series analysis with delay-variant embedding, Oral Presentation at Conference on Complex Systems, Singapore, Singapore, Oct. 2019.
SIAM-AG21-Topological Persistence Machine of Phase TransitionHa Phuong
Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021.
Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
1. A topological measurement
of protein compressibility
Tran Quoc Hoan
@k09hthaduonght.wordpress.com/
28 March 2016, Paper Alert, Hasegawa lab., Tokyo
The University of Tokyo
Marcio Gameiro et. al. (Japan J. Indust. Appl. Math (2015) 32:1-17)
2. Abstract
Topological Measurement of Protein Compressibility 2
…we partially clarify the relation between the compressibility of
a protein and its molecular geometric structure. To identify and
understand the relevant topological features within a given
protein, we model its molecule as an alpha filtration and hence
obtain multi-scale insight into the structure of its tunnels and
cavities. The persistence diagrams of this alpha filtration
capture the sizes and robustness of such tunnels and cavities in a
compact and meaningful manner…
Our main result establishes a clear linear correlation between
the topological measure and the experimentally-determined
compressibility of most proteins for which both PDB information
and experimental compressibility data are available…..
3. Tutorial of
Topological Data Analysis
Tran Quoc Hoan
@k09hthaduonght.wordpress.com/
Hasegawa lab., Tokyo
The University of Tokyo
Part I - Basic Concepts
4. Outline
TDA - Basic Concepts 4
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
5. Outline
TDA - Basic Concepts 5
1. Topology and holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
3. Definition of holes
6. Topology
I - Topology and Holes 6
The properties of space that are preserved under continuous
deformations, such as stretching and bending, but not tearing or
gluing
⇠= ⇠= ⇠=
⇠= ⇠= ⇠=
⇠=
7. Invariant
7
Question: what are invariant things in topology?
⇠= ⇠= ⇠=
⇠= ⇠=
⇠=
⇠=
Connected
Component Ring Cavity
1 0 0
2 0 0
1 1 0
1 10
Number of
I - Topology and Holes
8. Holes and dimension
8
Topology: consider the continuous deformation under the
same dimensional hole
✤ Concern to forming of shape: connected component, ring, cavity
• 0-dimensional “hole” = connected component
• 1-dimensional “hole” = ring
• 2-dimensional “hole” = cavity
How to define “hole”?
Use “algebraic” Homology group
I - Topology and Holes
9. Homology group
9
✤ For geometric object X, homology Hl satisfied:
k0 : number of connected components
k1 : number of rings
k2 : number of cavities
kq : number of q-dimensional holes
Betti-numbers
I - Topology and Holes
Image source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
10. Outline
TDA - Basic Concepts 10
1. Topology and holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
3. Definition of holes
11. Simplicial complexes
11
Simplicial complex:
A set of vertexes, edges, triangles, tetrahedrons, … that are closed
under taking faces and that have no improper intersections
vertex
(0-dimension)
edge
(1-dimension)
triangle
(2-dimension)
tetrahedron
(3-dimension)
simplicial
complex
not simplicial
complex
2 - Simplicial complexes
k-simplex
12. Simplicial
12
n-simplex:
The “smallest” convex hull of n+1 affinity independent points
vertex
(0-dimension)
edge
(1-dimension)
triangle
(2-dimension)
tetrahedron
(3-dimension)
n-simplex
= |v0v1...vn| = { 0v0 + 1v1 + ... + nvn| 0 + ... + n = 1, i 0}
A m-face of σ is the convex hull τ = |vi0…vim| of a non-empty subset
of {v0, v1, …, vn} (and it is proper if the subset is not the entire set)
⌧
2 - Simplicial complexes
14. Simplicial complex
14
Definition:
A simplicial complex is a finite collection of simplifies K such that
(1) If 2 K and for all face ⌧ then ⌧ 2 K
(2) If , ⌧ 2 K and ⌧ 6= ? then ⌧ and ⌧ ⌧
The maximum dimension of simplex in K is the dimension of K
K2 = {|v0v1v2|, |v0v1|, |v0v2|, |v1v2|, |v0|, |v1|, |v2|}
K = K2 [ {|v3v4|, |v3|, |v4|}
NOT YES
2 - Simplicial complexes
16. ✤ Let be a covering of
Nerve
16
= {Bi|i = 1, ..., m} X = [m
i=1Bi
✤ The nerve of is a simplicial complex N( ) = (V, ⌃)
2 - Simplicial complexes
17. Nerve theorem
17
✤ If is covered by a collection of convex closed
sets then X and are
homotopy equivalent
X ⊂ RN
= {Bi|i = 1, ..., m} N( )
2 - Simplicial complexes
18. Cech complex
18
P = {xi 2 RN
|i = 1, ..., m}
Br(xi) = {x 2 RN
| ||x xi|| r}
✤ The Cech complex C(P, r) is the nerve of
✤
= {Br(xi)| xi 2 P}
✤ From nerve theorem: C(P, r)
Xr = [m
i=1Br(xi) ' C(P, r)
✤ Filtration
ball with radius r
2 - Simplicial complexes
19. Cech complex
19
✤ The weighted Cech complex C(P, R) is the nerve of
✤ Computations to check the intersections of balls are not easy
ball with different radius= {Bri
(xi)| xi 2 P}
Alpha complex
2 - Simplicial complexes
21. General position
21
✤ is in a general position, if there is no
✤ If all combination of N+2 points in P is in a general
position, then P is in a general position
x1, ..., xN+2 2 RN
x 2 RN
s.t.||x x1|| = ... = ||x xN+2||
✤ If P is in a general position then
The dimensions of Delaunay simplexes <= N
Geometric representation of D(P) can be
embedded in RN
2 - Simplicial complexes
22. Alpha complex
22
✤
✤
✤ The alpha complex is the nerve of
↵(P, r) = N( )
✤ From Nerve theorem:
Xr ' ↵(P, r)
2 - Simplicial complexes
23. Alpha complex
23
✤
✤
✤ The weighted alpha complex is defined
with different radius
if P is in a general position
filtration of alpha complexes
2 - Simplicial complexes
24. Alpha complex
24
✤ Computations are much easier than Cech complexes
✤ Software: CGAL
• Construct alpha complexes of points clouds data in RN with
N <= 3
Filtration of alpha complex
Image source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
2 - Simplicial complexes
25. Outline
TDA - Basic Concepts 25
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
27. What is hole?
27
✤ 1-dimensional hole: ring
not ring have ring
boundary
without
ring
without
boundary
Ring =
1-dimensional graph without boundary?
However, NOT
1-dimensional graph without
boundary but is 2-dimensional graph
’s boundary
Ring = 1-dimensional graph without boundary and is not boundary
of 2-dimensional graph
3 - Definition of Holes
28. What is hole?
28
✤ 2-dimensional hole: cavity
not cavity have cavity
boundary
without
cavity
without
boundary
However, NOT
2-dimensional graph without
boundary but is 3-dimensional graph
’s boundary
Cavity = 2-dimensional graph without boundary and is not boundary
of 3-dimensional graph
Cavity =
2-dimensional graph without boundary?
3 - Definition of Holes
29. Hole and boundary
29
q-dimensional hole
q-dimensional graph without boundary and
is not boundary of (q+1)-dimensional graph=
We try to make it clear by “Algebraic” language
3 - Definition of Holes
30. Chain complexes
30
Let K be a simplicial complex with dimension n. The group of q-
chains is defined as below:
The element of Cq(K) is called q chain.
Definition:
Cq(K) := {
X
↵i
⌦
vi0
...viq
↵
|↵i 2 R,
⌦
vi0
...viq
↵
: q simplicial in K}
0 q nif
Cq(K) := 0, if q < 0 or q > n
3 - Definition of Holes
31. Boundary
31
Boundary of a q-simplex is the sum of its (q-1)-dimensional faces.
Definition:
vil is omitted
@|v0v1v2| := |v0v1| + |v1v2| + |v0v2|
3 - Definition of Holes
32. Boundary
32
Fundamental lemma
@q 1 @q = 0
@2 @1
For q = 2
In general
• For a q - simplex τ, the boundary ∂qτ, consists of all (q-1) faces of τ.
• Every (q-2)-face of τ belongs to exactly two (q-1)-faces, with different direction
@q 1@q⌧ = 0
3 - Definition of Holes
33. Hole and boundary
33
q-dimensional hole
q-dimensional graph without boundary and is
not boundary of (q+1)-dimensional graph
(1)
(2)
(1)
(2)
:= ker @q
:= im@q+1
(cycles group)
(boundary group)
Bq(K) ⇢ Zq(K) ⇢ Cq(K)
@q @q+1 = 0
3 - Definition of Holes
34. Hole and boundary
34
q-dimensional hole
q-dimensional graph without boundary and is
not boundary of (q+1)-dimensional graph
(1)
(2)
Elements in Zq(K) remain after make Bq(K) become zero
This operator is defined as Q
=
:= ker @q := im@q+1
Q(z0
) = Q(z) + Q(b) = Q(z)
(z and z’ are equivalent in
with respect to )
q-dimensional hole = an equivalence
class of vectors
ker @q
im @q+1
For z0
= z + b, z, z0
2 ker @q, b 2 im @q+1
3 - Definition of Holes
35. Homology group
35
Homology groups
The qth
Homology Group Hq is defined as Hq = Ker@q/Im@q+1
= {z + Im@q+1 | z 2 Ker@q } = {[z]|z 2 Ker@q}
Divided in groups with operator [z] + [z’] = [z + z’]
Betti Numbers
The qth
Betti Number is defined as the dimension of Hq
bq = dim(Hq)
H0(K): connected component H1(K): ring H2(K): cavity
3 - Definition of Holes
36. Computing Homology
36
v0
v1 v2
v3
All vectors in the column space of Ker@0 are equivalent with respect to Im@1
b0 = dim(H0) = 1
Im@2 has only the zero vector
b1 = dim(H1) = 1
H1 = { (|v0v1| + |v1v2| + |v2v3| + |v3v0|)}
3 - Definition of Holes
37. Computing Homology
37
v0
v1 v2
v3
H1 = { (hv0v1i + hv1v2i + hv2v3i hv0v3i)}
All vectors in the column space of Ker@0 are equivalent with respect to Im@1
b0 = dim(H0) = 1
Im@2 has only the zero vector
b1 = dim(H1) = 1
3 - Definition of Holes
38. Outline
TDA - Basic Concepts 38
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
39. Persistent Homology
Persistent homology 39
✤ Consider filtration of finite type
K : K0
⇢ K1
⇢ ... ⇢ Kt
⇢ ...
9 ⇥ s.t. Kj
= K⇥
, 8j ⇥
✤ : total simplicial complexK = [t 0Kt
Kk
Kt
k
T( ) = t 2 Kt
Kt 1
: all k-simplexes in K
: all k-simplexes in K at time t
: birth time of the simplex
time
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
40. Persistent Homology
40
✤ Z2 - vector space
✤ Z2[x] - graded module
✤ Inclusion map
✤ is a free Z2[x] module with the baseCk(K)
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
41. Persistent Homology
41
✤ Boundary map
✤ From the graded structure
✤ Persistent homology
(graded homomorphism)
face of σ
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
42. Persistent Homology
42
✤ From the structure theorem of Z2[x] (PID)
✤ Persistent interval
✤ Persistent diagram
Ii(b): inf of Ii, Ii(d): sup of Ii
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
43. Persistent Homology
43
birth time
death time
✤ “Hole” appears close to the
diagonal may be the “noise”
✤ “Hole” appears far to the
diagonal may be the “noise”
✤ Detect the “structure hole”
Persistent homology Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
44. Outline
TDA - Basic Concepts 44
1. Topology and holes
3. Definition of holes
5. Some of applications
2. Simplicial complexes
4. Persistent homology
see more at part2 of tutorial
45. Applications
5 - Some of applications 45
• Persistence to Protein compressibility
Marcio Gameiro et. al. (Japan J. Indust. Appl. Math (2015) 32:1-17)
46. Protein Structure
Persistence to protein compressibility 46
amino acid 1 amino acid 2
3-dim structure of hemoglobin
1-dim structure of protein
folding
peptide bond
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
47. Protein Structure
Persistence to protein compressibility 47
✤ Van der Waals radius of an atom
H: 1.2, C: 1.7, N: 1.55 (A0)
O: 1.52, S: 1.8, P: 1.8 (A0)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
Van der Waals ball model of hemoglobin
48. Alpha Complex for Protein Modeling
Persistence to protein compressibility 48
✤
✤
✤
: position of atoms
: radius of i-th atom
: weighted Voronoi Decomposition
: power distance
: ball with radius ri
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
49. Alpha Complex for Protein Modeling
Persistence to protein compressibility 49
✤
✤
✤
Alpha complex nerve
k - simplex
Nerve lemma
Changing radius
to form a filtration (by w)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
50. Topology of Ovalbumin
Persistence to protein compressibility 50
birth time
deathtime
birth time
deathtime
1st betti
plot
2nd betti
plot
PD1 PD2
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
51. Compressibility
Persistence to protein compressibility 51
3-dim structureFunctionality
Softness
Compressibility
Experiments Quantification
Persistence diagrams
(Difficult)
…..…..
Select generators and fitting parameters
with experimental compressibility
holes
52. Denoising
Persistence to protein compressibility 52
birth time
deathtime
✤ Topological noise
✤ Non-robust topological features depend on a status of
fluctuations
✤ The quantification should not be dependent on a
status of fluctuations
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
53. Holes with Sparse or Dense Boundary
Persistence to protein compressibility 53
✤ A sparse hole structure is deformable to a much larger
extent than the dense hole → greater compressibility
✤ Effective sparse holes
: van der Waals ball
: enlarged ball
birth time
deathtime
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
54. # of generators v.s. compressibility
Persistence to protein compressibility 54
# of generators v.s. compressibility
Topological Measurement Cp
Compressibility
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
56. Protein Phylogenetic Tree
Persistence to Phylogenetic Trees 56
✤ Phylogenetic tree is defined by a distance matrix for a
set of species (human, dog, frog, fish,…)
✤ The distance matrix is calculated by a score function
based on similarity of amino acid sequences
amino acid sequences
fish hemoglobin
frog hemoglobin
human hemoglobin
distance matrix of
hemoglobin
fish
frog
human
dog
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
57. Persistence Distance and Classification of Proteins
Persistence to Phylogenetic Trees 57
✤ The score function based on amnio acid sequences does not
contain information of 3-dim structure of proteins
✤ Wasserstein distance (of degree p)
Cohen-Steiner, Edelsbrunner, Harer, and Mileyko, FCM, 2010
on persistence diagrams reflects similarity of persistence
diagram (3-dim structures) of proteins
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
58. Persistence Distance and Classification of Proteins
Persistence to Phylogenetic Trees 58
birth time
deathtime
birth time
birth time
deathtime
deathtimeWasserstein distance
Bijection
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
59. Distance between persistence diagrams
Persistence to Phylogenetic Trees 59
Persistence of sub level sets
Stability Theorem (Cohen-Steiner et al., 2010)
birth time
deathtime
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
60. Phylogenetic Tree by Persistence
Persistence to Phylogenetic Trees 60
✤ Apply the distance on persistence diagrams to classify
proteins
Persistence diagram used the noise band same as
in the computations of compressibility
3DHT
3D1A
1QPW
3LQD
1FAW
1C40
2FZB
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
61. Future work
TDA - Basic Concepts 61
✤ Principle to de-noise fluctuations in persistence diagrams (NMR
experiments)
✤ Finding minimum generators to identify specific regions in a
protein (e.g., a region inducing high compressibility, hereditarily
important regions)
✤ Zigzag persistence for robust topological features among a
specific group of proteins (quiver representation)
✤ Multi-dimensional persistence (PID → Grobner basic)
Slide source: http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
62. Applications more in part … of tutorials
5 - Some of applications 62
✤ Robotics
✤ Computer Visions
✤ Sensor network
✤ Concurrency & database
✤ Visualization
Prof. Robert Ghrist
Department of Mathematics
University of Pennsylvania
One of pioneers in applications
Michael Farber Edelsbrunner
Mischaikow Gaucher Bubenik
Zomorodian
Carlsson
63. Software
TDA - Basic Concepts 63
• Alpha complex by CGAL
http://www.cgal.org/
• Persistence diagrams by Perseus (coded by Vidit Nanda)
http://www.sas.upenn.edu/~vnanda/perseus/index.html
http://chomp.rutgers.edu/Project.html
• CHomP project
64. Reference link
Topological Measurement of Protein Compressibility 64
✤ Original paper
✤ Author slides
http://www2.math.kyushu-u.ac.jp/~hiraoka/protein_homology.pdf
http://www.sas.upenn.edu/~vnanda/source/compressibility-final.pdf
✤ Books (very good)
- (Japaneses) タンパク質構造とトポロジー パーシステントホモロジー群入
門 平岡 裕章
- (English) Computational Topology - An Introduction, Herbert Edelsbrunner, John
L. Harer