Mathematical modeling with single-cell sequencing data

Abstract

Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two approaches for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; by solving partial differential equations on a graph representing discrete cell state relationships, and by solving the equations on a continuous cell state-space. We demonstrate how to calibrate model parameters from single or multiple time-point single-cell sequencing data, and examine the effects of data processing algorithms on the model calibration and predictions. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell types during the pathogenesis of acute myeloid leukemia. The mathematical modeling framework we present is general and can be applied to study cell state-transitions in any single-cell genome sequencing dataset. Author summaryHere we compare and contrast graph-and continuum-based approaches for constructing mathematical models of cell state-transitions using single-cell RNA-sequencing data. Using two publicly available datasets, we demonstrate how to calibrate mathematical models of hematopoeisis and how to use the models to predict dynamics of acute myeloid leukemia pathogenesis by mathematically perturbing the process of cellular proliferation and differentiation. We apply these modeling approaches to study the effects of perturbing individual or sets of genes in subsets of cells, or by modeling the dynamics of cell state-transitions directly in a reduced dimensional space. We examine the effects of different graph abstraction and trajectory inference algorithms on calibrating the models and the subsequent model predictions. We conclude that both the graph-and continuum-based modeling approaches can be equally well calibrated to data and discuss situations in which one method may be preferable over the other. This work presents a general mathematical modeling framework, applicable to any single-cell sequencing dataset where cell state-transitions are of interest.July 21, 2019 1/32 47 impact of various graph construction and trajectory inference methods on the geometry 48 of the cell state space, and solve the model on these geometries. We then use the model 49 the study the effects of perturbing 1) the graph structure 2) expression of select subsets 50 of genes 3) and cell state transition dynamics by perturbing flow on the graph or by 51 modifying the dynamics in the continuous space. We predict novel dynamics of leukemia 52July 21, 2019 2/32 65 embedding. While different techniques provide different shapes and differentiation 66 sp...

Mapping Transcriptomic Vector Fields of Single Cells

“…Although this work establishes a general and powerful framework, important directions remain for further development. First, our vector field learning approach focuses on the deterministic part of the dynamics, i.e., the convection part of a convection–diffusion process (Cho and Rockne 2019) in transcriptome space. Biological systems are, however, intrinsically stochastic; accordingly, future work should seek to reconstruct the (stochastic) diffusion part of the model as well.…”

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“…into unordered pairs, where s is the number of sets. The raw moments, m i 1 i 2 ...i N (t) can then be solved from the expressions for the central moments obtained from equation (12). For a practical example of the Gaussian closure, see [104].…”

“…We highlight the computational cost of introducing complexity into the moment equations through the closure methods. The pair-wise closure, equation (11), which introduces a quotient, and the Gaussian closure, equation (12), which introduces a cubic, take significantly longer using DAISY to assess than the mean-field closure, equation (10), yet give the same result. However, unlike MCMC, we note that structural identifiability results are deterministic, and independent of user choices such as prior, number of particles, and generated or real synthetic data.…”

“…Stochastic mathematical models are rapidly becoming an essential tool for interpreting biological phenomena [1][2][3][4][5][6][7]. These models are necessitated, in part, by increasing experimental interest in capturing finer-scale, time-series observations [8][9][10][11][12] as well as spatial information [13][14][15][16][17][18] rather than coarse-scale deterministic trends (figure 1). As computational inference techniques for stochastic models have improved [19][20][21][22][23], a fundamental question that often remains overlooked is whether or not model parameters can be confidently estimated from the available data.…”

Identifiability analysis for stochastic differential equation models in systems biology

“…Stochastic mathematical models are rapidly becoming an essential tool for interpreting biological phenomena [1][2][3][4][5][6][7]. These models are necessitated, in part, by increasing experimental interest in capturing finer-scale time-series observations [8][9][10][11][12] as well as spatial information [13][14][15][16][17][18] rather than coarse-scale deterministic trends (figure 1). As computational inference techniques for stochastic models have improved [22][23][24][25][26], a fundamental question that often remains overlooked is whether or not model parameters can be confidently estimated from the available data.…”