Aldo Guzmán, IBM: Topology in Computational Genomics

Aldo Guzman sence from IBM introduces the concept of topology in computation and genomics. He discusses the intersection of algebraic topology with computational genomics, which involves studying genetic properties across various types of data known as 'omics'. Guzman highlights the successful application of algebraic topology tools such as persistent homology in this field. He also provides a brief historical context of how mathematics has been applied in biology and ecology, starting with Fibonacci's observation on rabbit population growth, which has its roots in Indian mathematics as early as 200 AD. The narrative continues with the transition from macroscopic observations to microscopic levels, leading to the development of population studies and the understanding of life expectancy, as exemplified by Daniel Bernoulli's study on smallpox inoculation.

The paragraph delves into the historical developments in the application of mathematical models to biological studies. It starts with the debate between Daniel Bernoulli and John Baptist Lonber, which underscores the societal impact of mathematical models. The discussion then shifts to Gregor Mendel's work with plant experiments that led to the formulation of genetic laws, which are fundamental to understanding how information is passed across generations. The paragraph also touches on the advancements in microscopy, mathematical and statistical developments in the 20th century, and the discovery of DNA. It emphasizes the transition from studying populations to focusing on individual levels, such as the bacteria in our gut, which can be seen as both individual and population entities. The paragraph concludes with an explanation of the terms 'genotype' and 'phenotype', which are central to genomics.

This paragraph explains the concept of gene expression and how it is measured. Gene expression levels indicate how much a gene is expressed, which can be determined by measuring the amount of protein present in a cell or by measuring the messenger RNA itself. The process is simplified to illustrate the mechanism of gene expression: DNA in the nucleus is unzipped by enzymes, copied to create messenger RNA, which then leaves the nucleus and is used by ribosomes and transfer RNA to assemble proteins. The paragraph emphasizes the importance of gene expression levels in understanding cell functions and how they can be used to create a table of data with rows representing subjects and columns representing gene expression levels, which can be used for further analysis.

The focus of this paragraph is on using gene expression levels to predict certain phenotypes, such as Parkinson's disease. It describes the process of creating a table of data points and using algebraic topology, specifically persistent homology, to analyze these data points. The paragraph explains the construction of filtered simplicial complexes from the data and the computation of homology. It also includes an animation to illustrate the process of recovering the homotopy type of a simplicial complex using neighborhoods around points. The concept of barcodes is introduced as a way to represent the persistence of holes in the data over time. The application of this method to Parkinson's disease is mentioned, highlighting the potential of using topological signatures for patients in machine learning settings.

This paragraph discusses the application of persistent homology in machine learning, specifically for predicting phenotypes from gene expression levels. It describes the process of enriching a point cloud with weights based on sample values and using this to construct a weighted VOR complex. The resulting persistent landscapes, which are vector spaces, are then used as inputs for machine learning models. The paragraph emphasizes the use of convolutional neural networks (CNNs) for their ability to exploit spatial coherence in the representation. It also mentions the performance of the model in predicting Parkinson's disease, noting its low false positive rate. The discussion concludes with the idea of mapping features back to the topological space and original data to understand which features were important in the machine learning model.

The paragraph introduces persistent harmonic homology, a generalization of harmonic chains to the persistent setting, which allows for the selection of representative cycles in homology. This method provides a way to map features back to the topological space and then to the original data, assigning weights based on the harmonic representatives. The process involves constructing a barcode and then propagating the information to lower orders, which can be used to directly influence the feature space. The paragraph discusses how this method can be applied to different problems and how it has been implemented to get actual results on real data.

This paragraph discusses the application of harmonic persistent homology in distinguishing subtypes of lung cancer and other cancers using the TCGA dataset. It explains how traditional tools like PCA do not separate subtypes well, but supervised learning and the use of harmonic weights can improve performance. The paragraph also mentions the use of harmonic weights in breast cancer subtyping and chronic lymphocytic leukemia studies. The results show that genes with higher harmonic weights are known to be associated with lung cancer, confirming the method's effectiveness. The paragraph highlights how this approach can help overcome challenges in studies with limited data points and discover new patterns in an unsupervised manner.

The final paragraph discusses the future of genomic data analysis, emphasizing the increasing amount of data available, such as single-cell sequencing data. It highlights the importance of advanced mathematical and statistical methods, as well as AI and ML techniques, in handling this data and gaining insights into biological processes. The paragraph concludes by emphasizing the importance of conferences in fostering the development of these advanced methods and the potential for these tools to move research forward.

The speaker concludes the talk by reiterating the importance of Topological Data Analysis (TDA) in the context of the larger picture of connecting various tools to gain insights into biological structures. The speaker leaves the audience with a sense of the potential of these methods to move research forward and invites questions from the audience.