A dynamical systems treatment of transcriptomic trajectories in hematopoiesis

Abstract

Inspired by Waddington's illustration of an epigenetic landscape, cell-fate transitions have been envisioned as bifurcating dynamical systems, wherein the dynamics of an exogenous signal couples to a cell's enormously complex signaling and transcriptional machinery, eliciting a qualitative transition in the collective state of a cell -- its fate. Single-cell RNA sequencing (scRNA-seq) measures the distributions of possible transcriptional states in large populations of differentiating cells, making it possible to interrogate cell fate-transitions at whole-genome scales with molecular-scale precision. However, it remains unclear how to bridge the disparate scales of the dynamics of whole transcriptomes to the molecules that define the collective fate-transitions in Waddington's geometric vision. We bridge these scales by showing that bifurcations in transcriptional states can be analytically pinpointed and their genetic bases revealed, directly from data. We demonstrate the power of our conceptual framework and analytical scheme in the context of a recent scRNA-seq based investigation of a classic case-study of sequential fate decisions -- the transition of hematopoietic stem cells to neutrophils. Our work provides a rigorous and model-independent mathematical framework for detecting and categorizing transitions in cell-fate directly from sequencing data, aiding in gene network inference and determination of the salient properties of cellular differentiation, such as when cell fate transitions are reversible and how these transitions generate a diversity of cell types.

Characterizing transition cells in developmental processes from scRNA-seq data

“…We assume that most of cells captured by scRNA-seq are approximately equilibrium according to Boltzmann distribution. Taking the derivative of the covariance matrix, we arrive at equation (4), the continuous-time Lyapunov equation: According to Simon et al (Freedman et al, 2022), one of the conclusions that can be drawn from equation (4) is when a cell is in a transition state, its gene pair-wise Pearson’s correlation coefficients are more likely to be close to ±1. Briefly, the C matrix can be diagonalize into P Λ P -1 , and equation (4) can be written as , where .…”

“…According to Simon et al (Freedman et al, 2022), one of the conclusions that can be drawn from equation ( 4) is when a cell is in a transition state, its gene pair-wise Pearson's correlation coefficients are more likely to be close to ±1. Briefly, the C matrix can be diagonalize into 𝑷𝚲𝑷 −𝟏 , and equation ( 4) can be written as 𝝀 𝚺 ̃+ 𝚺 ̃𝝀𝑯 + 𝑫 ̃= 0, where 𝚺 ̃= 𝑷 −𝟏 𝚺(𝑷 𝑯 ) −𝟏 , 𝑫 ̃= 𝑷 −𝟏 𝝈𝝈 𝑻 (𝑷 𝑯 ) −𝟏 .…”

Dynamical landscapes of cell fate decisions

“…The notion that bifurcations are involved in cellular decisions is relatively well accepted [ 16 – 23 ]. They have been postulated in a range of decision-making systems including the triggering of human promyelocytic HL60 cells to neutrophil differentiation [ 17 ], differentiation of progenitor FDCP-mix cells into either the erythroid/megakaryocyte or the myelomonocyte lineage [ 18 ], early mouse embryonic development [ 12 , 20 , 24 ], differentiation of a primitive streak-like cell population into mesodermal and endodermal lineages [ 21 ], somitogenesis [ 22 ] and the transition of haematopoietic stem cells to neutrophils [ 23 ]. However, current discussions are largely restricted to local bifurcations where saddles and attractors collide.…”

“…Signature gene expression patterns and clustering algorithms have been used to identify attractors and end-states corresponding to specific cell types [ 11 , 12 , 20 , 21 , 35 , 46 – 51 ]. In addition, much interest has focused on identifying cells at decision points and understanding the detailed structure of transitions at or near a bifurcation [ 23 , 35 , 52 ]. The aim of these methods is to determine the critical points where cells change state and provide criteria that identify these from experimental data.…”

“…Others have examined the effect on gene–gene correlations resulting from the critical slowing down that occurs as fold or pitchfork bifurcations are approached [ 23 ]. Further work will be necessary to define clearly transition states and decision points.…”