Vector and Representations�The Dance of the Eigenworm
Quantitative Biology Bootcamp
Fall 2026
Vectors and Representations
Newton and the Representation of Functions by Polynomials or Infinite Series: Replacing Functions by a Few Numbers!
We Are Accustomed to Viewing the Earth as a Mercator Projection, but It Is a Representation
https://www.worldatlas.com/geography/world-map-mercator-projection.html
Greenland:
2.2 x 106 km2
Continental US:
8 x 106 km2
A Representation Based on the Earth’s Oceans: The Spilhaus-Adams Projection
https://blog.map-projections.net/spilhaus-projections-a-quintet-of-world-ocean-maps
Though Different Choices of Our Coordinates Are
“Equivalent” in Their Answers, Not in Their Psychology
We Also Often Use Fourier Series as a Representation of a Function
We Can Represent the Deformation of a Material Using the Strain Tensor
Representing the Discrete by the Continuous - One of the Greatest Intellectual Achievements of Mathematics and Science!
Representing Networks by Graphs
Representing Functions on a Computer Using
Extremely Simple Shape or Basis Functions
Represent function as weighted sum of basis functions
We Have Many Ways of Representing Ourselves! I Find It Strange How Often People Accept a Scalar Order Parameter
Generalizing Our Concept of Vector:
Often, Our Representations Are Vectors
The Math of Vectors
My Claim: We Need to Find a Way to Understand Our Vectors
(Berman et al.)
Images as Vectors
`
(Berman et al.)
Transcriptome as a Vector
`
(Berman et al.)
Shape as a Vector
Allele Frequencies as Vectors
Ceres Is Between Mars and Jupiter
(Berman et al.)
These two views of Ceres were acquired on Feb. 12, 2015 from a distance of roughly 80,000 kilometers. The two images capture the dwarf planet ten hours apart, making it possible to see different aspects due to its rotation. Images: NASA/JPL–Caltech/UCLA/MPS/DLR/IDA.
Legendre on an Astronomical Problem Too - The Method of Least Squares
(Berman et al.)
Gauss’ Invents Least Squares Too
(Berman et al.)
Least Squares as an Estimation of a Two-Dimensional Vector
(Berman et al.)
Finding the Right Representation
Key Idea: Finding the “Right” Coordinates
Theoretical Models Can Successfully Predict Bacterial Transcriptional Regulation
Garcia and Phillips, PNAS (2011);
Brewster et al., Cell (2014); Weinert et al., PRL (2014)
Rob
Phillips
Querying the Transcriptome
Querying the Transcriptome at the Single Cell Level
The Single-Cell Sequencing Revolution
The Tabula muris project
Transcriptomes Really Force Us to Come to Terms With the Question of Natural Coordinates
(Berman et al.)
Estimate: The Cost of the Dataset
(Berman et al.)
The Cost of the Dataset
(Berman et al.)
Cellular Decisions Are Often Driven by a Handful of Genes
Our Toy Model: 2D Synthetic Transcriptome
A Two Gene Transcriptome!
Identifying Cell Types in a 2D Synthetic Transcriptome
Finding the Right Coordinate System to Describe Our Data
Finding the Natural Coordinates of the 2D Synthetic Transcriptome
Using PCA for Dimensional Reduction - Building a New “Reduced” Theory
(Berman et al.)
The Eigenvalues Report on the Spread of the Data Along Each Dimension
The Error in a Reduced Description of Our System
The 3D Synthetic Transcriptome
Finding the Natural Coordinates of Our Synthetic Transcriptome
Dimensionality Reduction�Most of the Relevant Information Lives on a Plane
The Error in a Reduced Description of Our System
Homework�Adding Downstream Genes
Homework�Adding Downstream Genes
Homework�Adding Downstream Genes
Homework�Adding Noisy, Uncorrelated Genes
A More Common Definition of PCA
A Simple Synthetic Transcriptome
Finding Cell Types in the Transcriptome Using k-means Clustering
Finding the Right Coordinates in a “Mystery” Data Set
An Interlude on Summary Statistics: Every Distribution Has Its Moments
Examining the Covariance by Hand as a Function of the Orientation of Coordinates
Eigenvectors and Eigenvalues
The Eigenvectors of the Covariance Matrix Define the Normal Modes (or Principal Components)
The “Right” Coordinates Reveal the Natural Variables of the System
A Toy Model From Mechanics of the Key Idea: Finding the “Right” Coordinates for Two Coupled Oscillators
Solving the Coupled Oscillators in a Bad Coordinate System
Finding the “Right” Coordinates
(Berman et al.)
The Question of Shape and the Coordinates of Shape
(Berman et al.)
One of the Main Messages of This Course - Simple Questions From Looking
(Berman et al.)
One of the Main Messages of This Course - Simple Questions From Looking
(Berman et al.)
Visiting Darwin in His Garden
(Berman et al.)
Who Is C. elegans?
(Berman et al.)
Who Is C. elegans?
(Berman et al.)
The Dance of the Worm
(Berman et al.)
This is the “data” - how do we make something of it?
A Truly Beautiful Paper
Getting the Kinematics Right
Quantifying C. elegans Movement and Shape�Turning Images Into a New Kind of “Data”
A Homework Exercise
The Eigenworm!
These Waters Run Even Deeper
(Caltech)
(Caltech)
(Caltech)
These Waters Run Even Deeper
(Berman et al.)
(Caltech)
These Waters Run Even Deeper
(Berman et al.)
Trying to Find the Coordinates of Cell Motion
(Berman et al.)
https://www.facebook.com/mag2artcom/videos/fish-keratocyte-movie/1236883493110342/
Trying to Find the Coordinates of Cell Motion
(Berman et al.)
Using PCA to Find Natural Coordinates?
(Berman et al.)