{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Primer on differential geometry" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this work, we introduced **dynamical systems theory** and **differential geometry** analysis to single-cell genomics. A dynamical system describes the time dependence of points in a geometrical space, e.g., planetary motion or cell fate transitions, whereas differential geometry uses the techniques of differential/integral calculus and linear/multilinear algebra to study problems in geometry, e.g., the topology or geometric features along a streamline in vector field of the gene expression space. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A vector field function $\\mathbf{f}$, a fundamental topic of dynamical systems theories, takes spatial coordinate input $\\mathbf{x}$ (e.g., single-cell expression in gene state space) in a high-dimensional space (each gene corresponds to a dimension) as input and outputs a vector $\\mathbf v$ (e.g., corresponds to gene expression velocity vector from a single cell) in the same space, i.e. $\\mathbf v = \\mathbf f(\\mathbf x)$. In this study, we specifically discuss velocity vector fields that can be used to derive acceleration and curvature vector fields (see **below**). With analytical velocity vector field functions, including the ones that we learned directly from data, we can move beyond velocity to high-order quantities, including the Jacobian, divergence, acceleration, curvature, curl, etc., using theories developed in differential geometry. The discussion of the velocity vector field in this study focuses on transcriptomic space; vector fields, however, can be generally applicable to other spaces, such as morphological, proteomic, or metabolic space." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Because $\\mathbf f$ is a vector-valued multivariate function, a $d\\times d$ matrix encoding its derivatives, called the *Jacobian*, plays a fundamental role in differential geometry analysis of vector fields:\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\\begin{align*}\n", " \\mathbf J &= \\begin{bmatrix}\n", " &\\dfrac{\\partial f_1}{\\partial x_1} & \\dfrac{\\partial f_1}{\\partial x_2} & \\cdots & \\dfrac{\\partial f_1}{\\partial x_d} &\\ \\\\[3ex]\n", " &\\dfrac{\\partial f_2}{\\partial x_1} & \\dfrac{\\partial f_2}{\\partial x_2} & \\cdots & \\dfrac{\\partial f_2}{\\partial x_d} &\\ \\\\[1.5ex]\n", " &\\vdots & \\vdots & \\ddots & \\vdots &\\ \\\\[1.5ex]\n", " &\\dfrac{\\partial f_d}{\\partial x_1} & \\dfrac{\\partial f_d}{\\partial x_2} & \\cdots & \\dfrac{\\partial f_d}{\\partial x_d} &\\ \n", " \\end{bmatrix} \\ .\n", "\\end{align*}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A Jacobian element $\\partial f_i/\\partial x_j$ reflects how the velocity of $x_i$ is impacted by changes in $x_j$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "