We highlight dynamo’s power to overcome fundamental limitations of conventional splicing-based RNA velocity analyses to enable accurate velocity estimations on a metabolically-labeled human hematopoiesis scRNA-seq dataset. Furthermore, differential geometry analyses reveal mechanisms driving early megakaryocyte appearance and elucidate asymmetrical regulation within the PU.1–GATA1 circuit. Leveraging the Least-Action-Path method, dynamo accurately predicts drivers of numerous hematopoietic transitions. Finally, in silico perturbations predict cell-fate diversions induced by gene perturbations. Dynamo thus represents an important step in advancing quantitative and predictive theories of cell-state transitions.Single-cell (sc)-RNA-seq, together with RNA-velocity and metabolic labeling, reveals cellular states and transitions at unprecedented resolution. Fully exploiting these data, however, requires kinetic models capable of unveiling governing regulatory functions. Here, we introduce an analytical framework  A hallmark of metazoans is the ability of a single zygote to differentiate into a multitude of cell types while maintaining the same genome. To illustrate this process, Waddington introduced the epigenetic landscape, a metaphor in which differentiation proceeds like a ball sliding downhill into various valleys . This mex) in a multi-dimensional expression space in which the elements are the instantaneous concentrations of molecules. Neglecting stochasticity, the time derivative of the cell state, or its velocity f is a vector field function of the instantaneous cell states (x(t)). Although efforts have been made to perform whole-cell simulations of bacteria  and dynamo, available at https://github.com/aristoteleo/dynamo-release.This framework represents a notable advance from the metaphor of epigenetic landscape to a quantitative and predictive theory of the time evolution of single cell transcriptomics, applicable to many biological systems and at genome-wide scale. We have made the associated computational framework as an open-source software, A1, each propagating along a trajectory (streamline) defined by the vector field function to settle at the same attractor state A1  and moves toward a stable attractor state , its velocity tends to increase before it slows down in the attractor state , the conventional RNA velocity as defined in the original study  and calculate absolute unspliced, spliced, new, or total RNA velocity  by first estimating the degradation rates from labeling data and then the scaled degradation rate constant Estimating the parameters and RNA velocities with labeling data involves some technical subtleties, which we took into account when developing the corresponding algorithms. Overall, we estimate absolute splicing and degradation constants  of just 14 minutes, 81 times faster than Ank2 (t1/2 of 18.6 hours)  . Still, onstant) . For exa6 hours) . Houseke6 hours) .In the scEU-seq cell-cycle data , genes wHMGB2 is greater than its degradation rate, across cells its unspliced RNA is less abundant than its spliced RNA  yielded velocity flows that closely recapitulate the established knowledge of hematopoiesis  activation are explored (Tet 1/2/3 triple knockout (TetTKO) on the scNT-seq mESC dataset  to the basophil master regulators GATA2 and RUNX1, suggesting that Bas lineage can arise via two potential trajectories, from either GMP or MEP cells, consistent with  decreases as the level of GATA1 increases (rectangle B of SPI1 (∂fSPI1/∂xSPI1) decreases with increased GATA1 level (third subpanel of fSPI1/∂xGATA1 and ∂fGATA1/∂xGATA1). These behaviors are in marked contrast to SPI1 and GATA1 antagonize each other through a “competitive inhibition”-type mechanism, as reported in  values even at low levels of GATA1, reflecting the absence of a threshold for GATA1 inhibition/self-activation  is a principled method  that hasThe ability of the LAP method to estimate the dominant transition path and associated gene expression dynamics broadly yields non-trivial predictions regarding these transitions –I. As anOnce the LAP is calculated in the PCA space, we can project it back to the original gene expression space to predict the full transcriptomic kinetics along the path. Exploring the dynamics of TFs along the LAP from HSC to the Bas lineage revealed three distinct waves of TFs activation . We nextThese analyses reveal the potential of the LAP approach to predict the optimal path and TF cocktails of cell fate transitions with high accuracy, paving the road for à la carte reprogramming between any cell types of interest for applications in regenerative medicine .in silico perturbation predictions of expression response for each gene in each cell    to direcurements , therebyin silico tracing the transcriptomic dynamics of cell ensembles over time .Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Jonathan Weissman  is implemented as a Python package and is available through GitHub (https://github.com/aristoteleo/dynamo-release). Notebooks, tutorials for reproducing all figures in this study, and tutorials of dynamo usage cases are also available through GitHub .6 cell/mL. On the first day of the differentiation experiment, cells were seeded at 200,000 cell/mL in 12 well plates (unless stated otherwise) and treated with 1 μM ATRA  to differentiate into either the neutrophil-like cells. Cell differentiation status was confirmed by flow cytometry analysis of CD14  and CD11b .HL60 cells (ATCC® CCL-240™) were grown in RPMI 1640 medium (Gibco), with 20% FBS + 5% Penicillin-Streptomycin) at 37°C under 5% CO2, supplemented with 10% fetal bovine serum (Sigma) and 1% Penicillin/Streptomycin (HyClone). Cells were maintained below a density of 10For the human hematopoiesis dataset, we cultured primary human CD34+ hematopoietic stem and progenitor cells obtained from the Fred Hutchinson Cancer Research Center. Cells were thawed and cultured in StemSpan SFEM II human hematopoietic stem cell expansion media  supplemented with StemSpan CC100  and 50 ng/ml TPO . Cells were allowed to differentiate over the course of 1 week.HL60 differentiations were initialized on different days so that all samples could be harvested in a single scRNA-seq reaction to minimize batch effects. Cells were treated with 1 μM ATRA and differentiated for 0 (no ATRA treatment), 1, 2, 3, 4, or 5 days, with all differentiations performed in biological replicates. Samples were tagged or “cell hashed”  with disμL of 10x PBS and 4.4μL of alkylation mix (20 mM IAA in 100% DMSO) was added for a final concentration of 10 mM IAA, 50% DMSO. Alkylation was stopped by addition of 1.3 μL of 100 mM DTT and incubating for 5 minutes at room temperature. Alkylated RNA was purified with 1.1 volume of Ampure XP beads and two washes with fresh 80% ethanol, and eluted into an RNA elution buffer , 3.125 μΜ Oligo-dT30VN , 0.5 U/μL Recombinant RNase Inhibitor, 1:24million ERCC RNA spike-in mix ). cDNA and the remaining library preparation was performed according to a modified version of the protocol for Smart-seq2  were introduced using a lentiviral transduction strategy . Given tOur scNT-seq protocol was adapted from . For theUpon completion of the labeling phase, cells were washed twice with 0.01% bovine serum albumin  in Dulbecco’s Phosphate-Buffered Saline (DPBS) and filtered through a 40 μm strainer . Subsequently, the samples were diluted to a concentration of 120 cells/μL in DPBS-0.01% BSA, and loaded into a 10-mL Luer lock syringe  containing a magnet , and stirred gently. 10mL of lysis buffer containing 4 mL of water, 3 mL of 20% Ficoll PM 400 , 100 μL of N-lauroylsarcosine sodium salt solution 20% , 400μL of 0.5M EDTA , 2 mL of 1M Tris-HCl, pH 7.5, and 500 μL of 1 M DTT  was used to resuspend barcoded oligo-dT primer on beads for Drop-seq  at a concentration of 130 beads/μL in a 10-mL Luer lock syringe containing a magnet. Droplet generation oil  was dispensed using a 30-mL Luer lock syringe . Cells , oil  and beads  were transferred into uFluidix Drop-seq chips with hydrophobic coating using KD Scientific Syringe Pumps  and micromedical tubing , and visualized using a Photron Fastcam SA5 camera. Droplets were collected in 50-mL conicals.Droplet breakage was performed by removing oil and adding 30 mL of 6X SSC  and 1 mL of 1H,1H,2H,2H-perfluoro-1-octanol 97% , followed by vigorous shaking. The supernatant was removed following a 1-minute centrifugation at 1,000 g, and 20 mL of 6X SSC was added twice to resuspend the beads, which were then transferred to new tubes and centrifuged at 1,000 g for 2 minutes. Beads were resuspended in 6X SSC, transferred to 2-mL Lobind tubes, and washed twice with 1 mL of 6X SSC. This and all subsequent washing steps were performed by centrifuging the samples in a spinning bucket centrifuge at 1,000 g for 1 minute.4sU was converted to cytosine analogs using TimeLapse-seq chemistry, which gives rise to apparent U-to-C mutations following reverse transcription . BrieflyBeads were then washed once with 1mL 10mM Tris-HCl pH 8.0, and treated with a mix of 10μL of exonuclease I , 20μL 10X exonuclease I buffer  and 170μL of water at 37°C for 45 minutes, and washed with 1 mL of TE-SDS and twice with 1 mL of TE-TW. To prepare beads for second strand synthesis, beads were resuspended in 500μL fresh 0.1N NaOH, incubated at room temperature with rotation for 5 minutes, and neutralized using 500μL 0.2M Tris-HCl (pH 7.5). A wash with TE-TW and one with 10mM Tris-HCl (pH 8) were performed. Subsequently, the beads were resuspended in a mix of 20μL 10X Blue buffer , 40μL 20% Ficoll PM-400, 20μL 10mM dNTPs, 10μL 100μM TSO-N9 primer (/5SpC3/AAGCAGTGGTATCAACGCAGAGTGAAT(N1:25252525)(N1)(N1)(N1)(N1)(N1)(N1)(N1)(N1)), 5μL 100μM TSO-GAATG primer (/5SpC3/AAGCAGTGGTATCAACGCAGAGTGAATG), 10μL Klenow exo-  and 95μL of water, and incubated at room temperature for 10 minutes and at 37°C for 60 minutes with rotation. Finally, beads were washed with 1 mL of TE-SDS and twice with 1 mL of TE-TW.We next determined the optimal number of cycles required for whole-transcriptome amplification by performing qPCR on an aliquot of 6,000 beads. Following two washes in water, beads were resuspended in a mixture of 25 μL KAPA HiFi HS ReadyMix , 0.4 μL of 100 μM TSO-PCR primer (AAGCAGTGGTATCAACGCAGAGT), and 24.6 μL of water. A first PCR was run using the following parameters: 95°C for 3 minutes; 4 cycles of ; 9 cycles of ; 72°C for 5 minutes; and hold at 4°C. The PCR product was purified using one round of 0.7X AMPURE XP beads . One μL of purified cDNA was added to 4.5 μL of KAPA HiFi HS ReadyMix spiked with SYBR Green Dye , 0.07 μL of 25 μM TSO-PCR primer and 3.53 μL of water. qPCR was run using the following parameters: 95°C for 3 minutes; 25 cycles of 95°C for 15 seconds, 63°C for 30 seconds, and 72°C for 30 seconds). The extra number of cycles to add to the last stage of the first PCR was three-fourths of the cycle number coinciding with the exponential amplification stage of the qPCR.Large-scale PCR amplification was performed on the rest of beads with the same parameters as the first PCR above, plus the additional number of cycles in the last stage determined by qPCR. To ensure high diversity in our libraries, multiple tagmentations were performed for the cDNA from each of the timepoints using the Nextera XT DNA Library Prep Kit , and amplified in a second round of PCR  and unlabeled reads of each gene g for the scSLAM-seq data that we produced. The probability of y T-to-C mutations in a read that contains n possible mutation sites can be defined with the following equation:pe is the background T-to-C mutation rate that is independent of the mutations introduced by metabolic labeling, and pc is the T-to-C mutation rate introduced by metabolic labeling. B is the binomial probability mass function. Estimation of pc, pe, and πg is performed using the pipeline from (πTrue) and estimated (πg) fraction of labeled reads as the labeling correction coefficient, denoted as ρ is larger than 1 and vice versa.We use the binomial mixture model first described in the GRAM-SLAM study  to estimate the fraction of labeled (ine from  with a fdynast (https://github.com/aristoteleo/dynast-release), an inclusive and efficient command-line toolkit for preprocessing data from metabolic labeling–based scRNA-seq experiments and quantifying the following four types of mRNA species : unspliced, spliced, labeled (or new), and total RNAs (details will be reported elsewhere). Samples from days 4 and 7 were each subjected to standard dynast runs, using human genome hg38 as the alignment reference. The resultant objects were first filtered to keep only high quality cells and then merged to obtain in total about 2,000 high-quality cells. This merged object was then used for all downstream analyses.We developed l and total RNA r is proportional to the labeling correction coefficient ρ:For one-shot experiments, the slope of the linear relationship between labeled RNA ρ amounts to a high NTR  at steady state. For one-shot experiment, we assume that the labeling data have been statistically well corrected with the mixture binomial model, and thus the ρ is effectively close to 1. Then the slope is approximately k = 1 − eγt−, allowing us to obtain the degradation rate constant γ from the NTR slope. We can evaluate the error between the estimated γ under this assumption, and the true degradation rate constant γtrue:Therefore, an overestimated ρ < 1, γ < γtrue, γ is underestimated. Consequently, the velocity of total RNA, considering only its magnitude, differs from the true velocity by,When n/k − r determines the sign of the velocity, i.e., n/k − r > 0 amounts to a positive velocity, and vice versa. Therefore, under-correction of labeled RNA leads to underestimation of the velocity. It is also apparent that a ρ higher than one leads to the overestimation of both the degradation rate constant γ and the velocity of total RNA. The labeling correction coefficient ρ, which is assumed to be constant across all time points, has minimal impacts on curve fitting methods for degradation rate constants because the time scale of a first order degradation is independent of initial concentrations. However, for kinetics experiments, ρ affects the curve fitting of both the synthesis of labeled RNAs, and the degradation of unlabeled RNAs. The transcription rate α is under-estimated when ρ < 1, and overestimated when ρ > 1. The kinetics of unlabeled RNA are not merely a degradation process when ρ < 1, as there are artificial increases of new RNA due to underestimation of labeled RNA, and the degradation appears slower than the true rate. By contrast, when ρ > 1, the degradation of the unlabeled RNA is unaffected for similar reasons as in degradation experiments. As a result, the cluster-wise velocity for kinetic experiments is underestimated when ρ < 1 and overestimated when ρ > 1. In the extreme cases, an underestimated labeling RNA fraction can lead to a sign change in the velocity. On the other hand, because the cell-wise velocity is:Note that α, and an inaccurately estimated γ alters its magnitude but not the sign.It is unaffected by the inaccurate estimation of dynamo, consists of four integral stages: 1) estimation of genome-wide kinetic rate constants and velocity vectors, 2) reconstruction of single-cell vector field functions with the resultant cell state and velocity samples, 3) in-depth analyses leveraging various differential geometry analyses, and 4) predictions of optimal paths and cell fate diversions induced by genetic perturbations.Our analytical framework, a and activation rate b, between an active state  and an inactive state  (πg) fraction of labeled reads for gene g as the labeling correction coefficient. Our model further incorporates RNA splicing dynamics with the splicing rate constant β. The degradation of the spliced RNA is captured by the degradation rate constant γs. The protein translation rate constant η and degradation rate constant γp are also modeled in dynamo for possible datasets from single-cell transcriptomic–proteomic coassays. For the purpose of simplicity, this work mainly focuses on RNA transcription, splicing, degradation, and metabolic labeling. We analyze various types of scRNA-seq data with and without metabolic labeling. For the former, we consider four possible experimental scenarios   Next, wecenarios ; for eaccenarios  to descru and s are the copies of unspliced and spliced RNA for a particular gene in a cell, respectively; α, β, and γ are the rate constants for transcription, splicing, and degradation , respectively. In this study, we classify such a model system as Model 1. If we can estimate the kinetic parameters , together with u and s measured by scRNA-seq, we can derive a measure of “RNA velocity” of unspliced α is not constant, but rather a function of the cell state and other variables , it is difficult to obtain the unspliced RNA velocity. On the other hand, splicing and degradation rate constants (β) can in most cases be approximated as constants for certain cell types. The question, then, is how to estimate those kinetic parameters. Assuming pseudo-steady state Most existing pseudotime ordering methods merely reveal the central trend of a population of cells. By contrast, RNA velocity , an impoLet A linear regression of cells at steady states can be performed to obtain v is equal to β, which is in general gene-specific as revealed in Note that Because the intron reads are generated through mis-priming on polyA- or polyT- enriched intronic regions of nascent pre-RNA, it can be difficult to apply conventional RNA velocity to most transcription factors, which are typically expressed at low levels, and genes with no polyA/T-enriched intron regions;αon for the induction phase, αoff for the repression phase, Although many biological systems, including hematopoiesis, involve rapid but coordinated changes of RNA transcription rates for a considerable number of genes , existinThe linear regression methods used by conventional RNA velocity ignores the distribution of unspliced and spliced RNA, which can be used to improve the estimation of kinetic parameters;For systems far away from the pseudo-steady state, using cells with extreme RNA expression levels for linear regression may lead to inaccurate velocity calculations for most cells;The time scale for the degradation rate constant in conventional RNA velocity Conventional RNA velocity only estimates velocities for observed cells. Thus, it is a discrete, sparse, and local measure of cell dynamics and often merely used as a descriptive rather than a predictive tool.Although conventional RNA velocity has been successfully applied to a variety of studies, it has several limitations:A great deal of efforts have been devoted to the improvement of conventional RNA velocity estimation  in regark-nearest neighbor graph of cells, instead of the raw expression:dynamo) is the set of k-nearest neighbors of each individual cell, often constructed in the space of the top PCs  , reduced from the original gene expression space of highly variable genes. These can be considered as estimators of the first moments of the distribution of unspliced and spliced RNAs. RNA velocity calculations performed on the first moments lead to a cleaner phase plane and therefore smoother velocity vectors , in lieu of the linear regression on mean expressions as used in the original RNA velocity method. Instead of directly fitting the distribution, GMM seeks to solve the following equations of moments for θ, also known as moment conditions:g1, g2, g3, … are functions of the random variables X (e.g. the copies of spliced and unspliced RNA across cells) and parameters θ. The optimal θ can be found by minimizing the Euclidean norm of the above expectations:g is a vector-valued function consisting of the moment conditions, and W is a positive definite weighting matrix, defined as the inverse of the covariance matrix:ε is the error term for the moment conditions when applied to actual data.Second moments (uncentered variances and covariances) provide information in addition to first moments on the shape of the underlying distribution. It is thus desirable to also take advantage of the second moments to improve the estimation robustness and accuracy of the kinetic parameters, and thus that of the RNA velocity measurements. The second moments of unspliced and spliced RNA, as well as their mixed moments, also rely on the Model 1  distribution method, based on an observation that in most cases total RNA counts at steady state follow the NB distribution  and the mean μ satisfy the following relationship:ϕ is the reciprocal of the dispersion parameter of NB distribution. Assuming that the number of spliced RNA s is an NB-distributed variable, the variance of spliced RNA satisfies:ϕ and is computed from:We also developed an alternative procedure, the ribution . With thPut all together, these give the moment conditions for the first and second moments:A nonlinear least squares optimizer can then be used to solve for A and I are the active and inactive states of the promoter, respectively, and r the number of mRNAs. The first two lines describe the spontaneous switching of the promoter, and the last two lines correspond to the synthesis and degradation of the total mRNA. At steady state, the distribution of r follows a negative binomial distribution, where:φ is the reciprocal dispersion parameter. The burst frequency (BF) is defined as the rate constant for promoter activation, kon, whose reciprocal characterizes the expected time required for the promoter to switch from the inactive state I to the active state A : state A . FollowiModel 2, which explicitly considers RNA labeling but not splicing, and Model 3, which considers both labeling and splicing (Because metabolic labeling–based scRNA-seq (time-resolved RNA-seq or tscRNA-seq) measures the synthesis or degradation of labeled RNA within a known period of time in an experimentally programmable manner, it offers a more direct measurement of the kinetics of gene expression than cscRNA-seq. Thus, in principle, it also provides an opportunity to overcome some of the challenges of the cscRNA-seq in RNA velocity estimation. However, it is nontrivial to properly estimate kinetic parameters and compute RNA velocity for tscRNA-seq data with various metabolic labeling approaches, including three general labeling strategies given in splicing . In the Model 2, we take into account labeling (with a labeling correction coefficient ρ) but not splicing. The total RNA has a synthesis rate constant α and a degradation rate constant γ. The labeled RNA has a reduced synthesis rate constant ρα but the same degradation rate constant. The ODEs for describing the dynamics of labeled (l) and total (r = l + o) are,In r over time t is:r0 is the initial concentration of the total RNA r. For the labeled RNA, the solution is:The general solution for the total RNA l) is well corrected with the binomial mixture model /(1 − eγt−)  . TogetheModel 3, we consider both the labeling and the splicing processes. The solutions for labeled, unspliced RNA (ul) and labeled, spliced RNA (sl) are equivalent to those for unspliced and spliced RNA in Model 1, with an additional ρ modifying the effective transcription rate of the labeled RNA:In β=γ, the solution for sl is instead:When dynamo for the sake of completeness and robustness for kinetic parameter estimations.We will omit this special scenario for simplicity in the following sections, although it is included in Below, we detail the respective estimation procedures of the four labeling scenarios given in Now we will introduce the respective estimation procedures and the corresponding models for each of the three general labeling strategies given in Model 2 is:t is the labeling time and we denote k = ρ(1 − eγt). When the dynamics of total RNA is at steady state In “one-shot” experiments, there is only one labeling time point, and the splicing process is not explicitly considered. The solution for new RNA in k can be obtained through a simple linear regression with zero intercept of the first moments of labeled and total RNAs , for cells with extreme high expressions of both l and r (top right corner of the phase plane). This approach effectively replaces u and s in the original RNA velocity method with l and r, and was previously reported as the “NTR”  velocity method , and at steady state the total RNA follows the negative binomial distribution, one can easily incorporate second moments using the negative binomial method:k, and can be used to compute the velocity of total RNA.Because in one-shot experiments the labeled and total RNAs are linearly correlated with a slope of The “two-step” approach :With data collected at multiple labeling time points in a kinetics (pulse) experiment, on the phase plane of labeled and total RNA, we find that cells from the same labeling period are distributed on a line whose slope increases as the labeling period increases. We realize that this phenomenon can be explained by the fact that the slope γ based on Model 2 (k for the labeled (l) and total (r) RNA for different labeling time t, based on the linear relationship (see the “one-shot” method):We then take advantage of this discovery and develop the “two-step” approach, which relies on two consecutive linear regressions to estimate the degradation rate constant  Model 2 , and thek(t) = 1 − eγt−, it is apparent that the slope increases with longer labeling time and asymptotically approaches one. Rearranging this equation, we have:When labeling correction coefficient is close to one, from t and the quantity −ln(1 − k). In the second step, we then estimate the parameter γ using a simple linear regression of t. The total RNA velocity is again:A linear relationship exists between the labeling time k. We note that not every single gene in the dataset may follow this kinetics, and in general we use R-square of the “two-step” model fitting to select genes with confident fittings for downstream analysis.Note that the “two-step” approach can be regarded as a generalization of the above “one-shot” method for one-shot labeling experiments to kinetics experiments with multiple labeling time points. The negative binomial method can also be applied here in the first step to achieve a more robust estimation of the slope 2). Curve fitting methods  for each gene using nonlinear least-squares methods. In general, given m experimental data points, y(1),y(2),…,ym),t(2),…, tm) is the solution of the ODEs at the time point t, given parameters θ. When there are multiple species  quantified from the experiment, we cast the ODEs into a matrix form while the composite loss function is the summation for loss function of all species, and weights can be added to the loss function to adjust the importance of each species. For example, a higher weight is assigned to the labeled than the unlabeled species (2:1 by default) for the kinetics experiment, because the unlabeled species does not strictly follow the degradation kinetics due to imperfect labeling:When single-cell kinetics (pulse) or degradation (chase) data using RNA metabolic labeling, e.g., scEU-seq or scNT-seq , at multThis general procedure is applied to all following curve-fitting methods; the key is to find solutions of each species for various RNA labeling strategies.θ in a predetermined range (see Estimation of Parameter Ranges for Curve Fitting Methods) as the initial guesses for the parameters θ required by the nonlinear least squares optimizer.We used Latin hypercube sampling to randomly initialize a set of values of l0 = l(0) = 0, ul,0 = ul(0) = 0 and sl,0 = sl(0) = 0). During the labeling process, because we assume that the labeling period is much shorter than the time scale of the biological process of interest, transcriptional rates are treated as constant in all cells. Therefore, based on the solutions of Model 3, the abundance of labeled, unspliced labeled and spliced labeled RNA increase over time:In kinetics experiments, the samples are collected after a short period of 4sU  labeling. At the beginning of the experiment, the concentrations for labeled RNA, unspliced labeled and spliced labeled RNA, are zero , and the solutions are more sensitive to β and γ than those of the labeled species:With sufficient sampling of the labeling time points (at least three), all three kinetic parameters can be estimated in theory. Because cells at different states may have different transcription rates, clustering can be performed first and the fitting is done for each cluster to derive cluster or cell-type specific kinetic rates . The aboThe spliced RNA velocity can be computed as before:ul above also allows us to compute the velocity for unspliced RNA in individual cells:The solution for Model 2 can be used:If no splicing data are available, the solution for αc denotes the transcription rate constant of cluster c:The total RNA velocity can be computed either for each cluster, where The velocity for new RNA can be computed in a similar way:Model 1 for data obtained from the kinetics experiments. Because the current labeling time of a tscRNA-seq kinetics experiment typically requires at least 1 hour (because of the low sensitivity of single- cell methods), which is much longer than the time scale of RNA splicing , the labeling kinetics do not have sufficient time resolution for reliable estimation of the splicing rate constant β. We can circumvent this by first computing u = ul + uu) and spliced RNA (s = sl + su) using the conventional RNA velocity method. Then, we can use either model to estimate the actual degradation rate constant γ, and the splicing rate constant is simply given by:There is, however, a practical issue when using curve-fitting methods with With this, we can then estimate absolute RNA velocities for total, spliced, unspliced, and new RNAs according to the model and data available. Note that a similar procedure can also be applied to relative kinetic parameters estimated with the dynamical method from  that genul and spliced sl RNA decay over time. The process can be formulated as below :In degradation experiments (Case 3 in β and degradation rate constant γ using the nonlinear least squares. The (labeled and unlabeled) spliced RNA velocity is then given by:These two equations can be substituted into the loss function, and we obtain splicing rate constant uu, su) indeed increase over time due to transcription, cell-wise transcription rates α cannot be directly estimated from such experiments because each cell has different transcription activity. However, with a two-state promoter stochastic expression model, we can assume a universal αon and αoff for all cells, similar to the dynamical model  follows the first-order decay kinetics . The methods for obtaining “guesstimations” are different for each parameter:Kinetics experimentslt = n(t), i.e. the number of copies of new RNA at labeling time t. Thus, the guesstimated α is simply the averaged ratio of new RNA and labeling time. The degradation rate constant can be roughly estimated from the old RNA:If the RNA dynamics are far from steady state and degradation is negligible, then the amount of newly synthesized RNA is proportional to the labeling time:Degradation experimentsl0, ul,0, and sl,0, are simply the average abundance of labeled RNAs across all cells belonging to the initial labeling time point. The degradation rate constant is guesstimated with the labeled RNA, using a equation similar to the one for kinetics experiments:The guesstimated values for the initial conditions, including To overcome the local optima of the cost function and speed up parameter estimation, we need to have good guesses of parameters and the valid ranges of those parameters. A set of parameter ranges are used for initial parameter value sampling and providing upper and lower bounds for optimizers to avoid unrealistic results. The “guesstimated” values n cells, the goodness of fit is determined using the standard R-squared:k species, where each xi and yi is a vector of model predictions for m time points, the Gaussian log-likelihood is:σ(x) are the mean and standard deviation of x, respectively. To balance the numerical difference between species, the data and model predictions are normalized by the maximal value of data for each species.For linear regression models, given the data and model predictions A is:v is the reaction rate, k the first-order rate constant, and [A] is the concentration of product A. Because the time scale of a first-order reaction is often characterized by the reciprocal of the rate constant , “rate” and “rate constant” are often used interchangeably in certain contexts , and that of the first-order rate constant is one per unit time (1/T). For a zeroth-order reaction, the rate constant is also the reaction rate, and therefore they share the same dimension.The rate law connects the rate of a reaction and concentrations of involved (bio)chemical species. As an example, the rate law for a first-order reaction that generates a product contexts . They arN/T. Because RNA splicing and degradation are first-order reactions, β and γ are first-order rate constants with dimension 1/T. In the original RNA velocity method , and thus the “velocity” is relative to the splicing rate constant β. Consequently, suppose that one obtains a small relative RNA velocity for a gene, the actual change in the copy number of spliced RNA per unit time can be large if the splicing rate is fast.In the context of RNA velocity, the velocities of unspliced and spliced RNA for a gene are essentially net reaction rates for the production and depletion of unspliced and spliced RNA, with the dimension of y method , the degα is a zeroth-order rate constant with the dimension N/T. Note that RNA transcription is not an elementary reaction in which products are formed in a single step, but instead a complex reaction with multiple steps involving various trans- and cis-elements. The zeroth-order rate constant α is thus an apparent rate constant under a reduced reaction scheme that lumps many intermediate steps, which are in fact regulated by a variety of internal and external signals. As a result, the transcription rate constant α is a function of cell state in the gene expression space. This has also been shown to be the case for splicing and degradation rate constants , it is straightforward to incorporate cell-specific volume information in our parameter estimation procedure.We also want to comment that in practice, we additionally assume that all cells share the same total RNA content. In our preprocessing steps, we scale the total UMI counts in each cell to 10,000 molecules, similar to many other scRNA-seq analysis toolkits. The normalized gene expression in each cell can be regarded as the fraction of total RNA content occupied by each gene. This normalization scheme is believed to help remove library size differences incurred during library construction and sequencing .If the expression of a particular gene in the progenitor is low, it should start to increase as cells differentiate from progenitor to terminal cell states. There should be progenitors that are above the steady-state fitting line in the phase plane. However, if most of the progenitor cells are located below the line, their velocities are negative, leading to reversed vector flow.If the expression of a particular gene in progenitors is high, it should start to decrease as the cells differentiate to terminal cell states. There should be progenitors that are below the steady-state fitting line. However, if most of the progenitors are located above the steady-state line, their velocities are positive, leading to reversed vector flow.Similar reasoning can be applied to the mature cell states.In some scenarios, we may find unexpected wrong velocity backflow from your RNA velocity analysis. To diagnose those cases, we can identify genes showing up in the wrong phase portrait position that may contribute to the wrong flow direction. We can then remove those genes to correct velocity vectors. This requires some prior knowledge about the progenitor and terminal cell types in the system. The underlying rationale boils down to the following scenarios :If the dynamo.We first assess whether, when each progenitor state differentiates into each terminal cell state, a gene is in the induction or repression phase based on the shift of the median gene expression between these two states. If it is in the induction phase, cells should mostly have positive or close to zero velocity  and vice versa. Those thresholds can be provided by the users or inherited from the default values provided by 1 - fraction of cells having velocity passing those thresholds in each state is then used as a measure of velocity confidence.Thus, we design a heuristic algorithm to quantify the confidence of each gene by assessing whether it obeys the above constraints:Note that this heuristic method requires one to provide meaningful progenitor groups and mature cell groups, and the thresholds of velocity. In particular, the progenitor groups should in principle have cells going out , whereas mature groups should end up in a different expression state, and there are intermediate cells going to the dead end cells in each terminal group .dynamo. By default it uses the Jaccard index, which measures how well each velocity vector meets the geometric constraints defined by the local neighborhood structure . The Jaccard index is calculated as the fraction of the number of the intersected set of nearest neighbors from each cell at the current expression state (x) and that from the future expression state (x + v) over the number of the union of these two sets, namely:xi, vi, S(xi) and S(xi + vi) are respectively the current expression state for cell i, the current velocity vector for cell i, the set of nearest neighbor cells for cell i based on the current expression states (x), and the set for nearest neighbor cells for cell i based on the future expression states (x + v).Several confidence metrics for cell-wise velocity vectors are implemented in scVelo ; second, the trajectories of cell transitions in the gene expression space are continuous and differentiable; and third, the dynamics can be described by a set of memoryless equations, i.e., the temporal propagation of the system depends only on the present state, but not those at prior times. Here, we provide some justifications for those assumptions and discuss the limitations of the vector field reconstruction. Moreover, we discuss the sources of noise in RNA velocity data and how their effects are minimized in vector field reconstruction.z = {x, y}, where x represents the measured spliced and unspliced transcripts, and/or labeled and total RNA in the case of labeling-based scRNA-seq experiments, and y represents all other unmeasured species such as the proteome and epigenome. It should be noted that x can be different from the raw RNA counts , as in many cases the domain of the vector field is the size-factor normalized and then logarithm-transformed transcripts . Let us assume that one can describe the dynamics of a cell by a set of stochastic differential equations ,Generically, one can represent the internal state of a cell by the expression levels  of intracellular molecular species, e.g., spliced or unspliced RNAs. Mathematically, one represents the cell state as a vector F and G form a vector field in the full space that describes interactions among intracellular species, influence from extracellular environmental factors (μ) including external stimuli and the extracellular secretome, and direct interactions with neighboring cells. Biologically, we expect that different layers of gene regulation, e.g., the proteome and transcriptome, are coupled. The extracellular factors μ are in general explicitly time dependent. The terms ζx and ζy refer to random noise, and we assume them to be white noise with zero means. Much of the effort in this study focuses on the reconstruction of F, and there are two theoretical issues that must be considered when reconstructing the vector field from single-cell transcriptome data alone. First, in a typical scRNA-seq experiment, only x is measured, and the other variables are hidden. Second, a cell is generally subject to a time-varying extracellular environment.The functions For simplicity, we restrict ourselves to the case that external stimuli are constant and spatially uniform, whereas in a more general situation the vector field is time-dependent. We also treat direct and indirect cell–cell interactions in a mean-field sense instead of treating the many-body cell–cell interaction problem explicitly. With single-cell multi-modality co-assays that are also augmented with spatial and temporal resolution , our fraζx= ζy = 0, cells evolve along a manifold M embedded in the state space of {x, y}  centered at the manifold (represented as the gradient around M in O(x), e.g., the number of unspliced (u) and spliced (s), or labeled (l) and total (r) RNA, should be understood as being projected to the subspace of x, i.e., averaged over the hidden variables,Z is the normalization factor δ is Dirac’s delta function, which sifts out the x among all possible x within an integral. In the case of time scale separation between transcription and other slower processes , one may further assume that x reaches quasi-steady-state for a given set of y, and one can expect that ρ ≈ ρ1(xss(y)) ρ2 also varies slowly in time.The presence of stochasticity loosens the coupling. Instead of moving strictly on the manifold, the population of cells follows an evolving probability distribution k neighboring cells in the state space, weighted with a specific kernel function , and K is often chosen as a fast-decaying function (Gaussian kernel) to |xi − x|, compared to ρ, or a k-nearest-neighbor step function with a sufficiently small neighborhood (30 by default in dynamo), compared to the total number of cells in the dataset. In fact, the RNA velocity is also such an average, for example:In practice, the above average is typically performed by averaging k = 1 − e−γt (see Estimating absolute RNA velocity for metabolic labeling–based scRNA-seq experiments across various labeling strategies). This approximates the following average in the continuous domain:κ is the continuous analogue of K, and:Note that here xi, yi) are supposedly sampled from the distribution, the probability density ρ is implicitly included in the summation. Comparison of the continuous average  around the mean of ρ, ρ:Note that in the discrete version , because average  with the average  makes itx and y are tightly coupled, ρ is a very narrow unimodal distribution  becomes multimodal and the projection is compromised  are almost always measured with errors, in addition to the error introduced by the hidden variables. The stochastic differential equations in ρ:D(z) is the diffusion tensor associated with the white noises. A(z) is the drift, which concatenates F and G :Due to stochasticity in gene expression and technical errors from scRNA-seq experiments, the observable A dictates the evolution of cells in the gene expression space and can be understood as the averaged velocity of both x and the unmeasured y. The velocity vectors obtained using the first moment averaging (F(x)), the projection of F(z) on x. During vector field reconstruction, the sparseVFC algorithm minimizes noise by optimizing the sum-of-squares of the difference between the vector field and the data, as well as detecting outliers based on a Bayesian approach (see Outlier detection in vector field reconstruction). Although this study focuses on the reconstruction of the vector field, which corresponds to the deterministic term (drift) of dynamo, we provide such a possibility by leveraging the sdeint python package.veraging  approximc in Note that the vector field, defined as β for each gene, we explore whether the velocity directionality would be affected by this scaling with relative RNA velocity, especially in the UMAP space. Randomly scaling the velocity vector by a positive value on a few cscRNA-seq data, sampled from the uniform distribution  for each gene, however, does not change the velocity vector directionality in UMAP (data not shown), indicating that the sign of velocity is the most important information for revealing the directionality of RNA velocity, especially when projected to a lower dimension. This result may explain why the conventional RNA velocity method, although relative, still proves useful in revealing the directionality of cell fate transitions. When the RNA velocity estimates are relative, the resultant vector field and differential geometry quantities are also relative. In this study, we demonstrated that importance of absolute vector field analyses with the cell-cycle dataset from  approaches in functional approximation with the interpretability of dynamical systems formulations.In the second and third stages of our f that maps any point (i.e. expression state of a cell) x in a (subset of) d dimensional (gene expression) space to a vector v (e.g. the RNA velocity vectors) in the same space, i.e., v=f(x). Thus, RNA velocity estimates  space or two- or three-dimensional UMAP (Uniform Manifold Approximation and Projection) space . In orden is the number of cells, and d is the dimension  of the cell state space. We assume that the measured single-cell RNA velocity is sampled from a smooth, differentiable vector field that assigns each cell expression state x with an RNA velocity vector v. Normally, single-cell RNA velocity measurements are results of biased, noisy, and sparse sampling of the cell expression state space. Therefore, the goal of velocity vector field reconstruction is to robustly learn a mapping function f, which outputs an RNA velocity vector v, based on the observed data f should recover the true velocity vector field on the entire domain To formally introduce the problem of velocity vector field learning in the context of scRNA-seq, we consider a set of pairs of cell expression states straints . Ideallyor field .f* can be written in a least-squares fashion:pi is a weight deciding the importance of the i-th data point in the loss function. However, it is not a trivial task to minimize the above loss function with respect to a function f. Approximating vector-valued functions in a sparse reproducing kernel Hilbert space (RKHS) has been shown to be effective in learning vector field functions for 2D applications, and can be easily generalized to high dimensional data  algorithm (pi as a likelihood function (see details in Outlier detection in vector field reconstruction). The final loss function has an additional parameter σ accounting for inlier noise:In this representation, the loss function can be optimized with respect to the coefficient vectors lgorithm  improvesC = [c1c2 ··· cm]T, and it can be shown that the solution C* to the following linear equation contains the coefficient vectors for the optimal vector field function f*:U is an m-by-m matrix whose elements are K an m-by-m Gram matrix consisting of P matrix is a diagonal matrix of the weights pi, and V=.Let E-step: calculation of the diagonal matrix P based on the likelihood function for outlier rejection, and 2) an M-step: Solving the above linear system for C, and updating the vector field function evaluations at sample points f(x) with the optimal ci’s. Other parameters, for example σ, are also updated accordingly in this step. The algorithm finishes when the loss function converges, or the number of optimization steps surpasses the designated maximum iterations.The sparseVFC algorithm  consistsv of inliers with a Gaussian distribution, i.e.:z is an indicator variable, such that z = 1 when the cell is an inlier, and z = 0 otherwise. θ contains all parameters, including the variance of the Gaussian distribution σ2, the vector field f, and the prior probability q mentioned below. The probability distribution of outliers is modeled with a uniform distribution:a is the volume of the domain for velocity vectors. Empirically, this is a parameter used for adjusting the aggressiveness of the outlier detection. Denote the fraction of inliers as q:Outlier detection is vital for robust vector field reconstruction from noisy RNA velocity data. The sparseVFC algorithm  models nThen, this is essentially a mixture model where the likelihood is:error in ):P RNA velocity data samples, one can construct a diagonal matrix P = diag, where pi = P. The E-step of the EM algorithm evaluates this matrix, which is used as the weight in the loss function for sparseVFC.For σ and q at the M-step of each EM iteration following standard EM algorithm procedure, the updated parameters are the solutions of the following optimization problem  is a conditional expectation of the complete-data log-likelihood function:To update  problem :θnew=aWith i.i.d. samples, one can show that :Q w.r.t. σ2 and q and equating them to zero, one obtains the solutions for updating the parameters:F = ⏉.By taking derivatives of m, the regularization parameter λ, the inverse bandwidth of the Gaussian kernel w, and the maximal number of iterations Nmax. Their default values and effects of changes in these values on the resultant vector field are summarized in the following table:The sparseVFC algorithm with an isotropic Gaussian kernel has four main parameters: the number of control points w is determined in the following way:The inverse bandwidth k-nearest-neighbors for each cell (by default 20% of the number of cells);Find dm;Compute the mean distance of each cell to its neighbors σ = dm/1.5.The inverse bandwidth SciPy fsolve is used in our case). Because the solver can only find fixed points closest to an initial guess x0, we simply randomize n such initial points in a domain containing all data points. We used Latin hypercube sampling technique : all eigenvalues are negative;Unstable fixed point (repulsor): all eigenvalues are positive;Saddle point: The eigenvalues are a mixture of positive, negative values, or even zeros.If one is interested in fixed points of a specific order , one may use a quasi-Newton conditional root-finding algorithm developed by Wang et al. .f = [fx fy]⏉, the x-nullcline consists of points where:Nullclines are lines (in 2D) or surfaces (in higher dimensions) when at least one component of the vector field is zero. For example, for a 2D vector field x- and y-nullclines, so we compute nullclines using a pseudo-arclength continuation method /|p1 − p0|, and the first tangent vector at the fixed point is a normalized random vector.This guarantees that the solution n points k (k = 4 by default) intervals, each of which contains m points j = 1, 2, …, k. If a portion of the trajectory enters a stable limit cycle and orbits around the corresponding fixed point, the x- and y-coordinates of the points are periodic. We use the fast Fourier transform to obtain the frequency spectra for the x- and y- coordinates of points in the last two intervals:j = k − 1, k. Let Stable limit cycles cause redundant sampling for trajectories integrated using vector fields. In this study, we focus on detecting limit cycles for a 2D vector field, but the method can be easily generalized to higher dimensions. Suppose we have a trajectory of dynamo.We notice that some identified fixed points are far away from regions populated with data points, where the reconstructed vector field may be less reliable. We quantify the confidence of the fixed points based on how far they are from domains populated with cells, and use the filled color of each node (corresponds to the fixed points) to represent the confidence of those fixed points when creating the topography plot in Once the vector field was learned, either in reduced UMAP space, top PCA space or even the original gene expression space, we can use it to predict the historical and future cell expression states over arbitrary time scales given any initial cell state. This can be achieved by integrating the continuous vector field from one or a set of initial cell states forward or backward in time. When the integration was performed for the early cells of a particular clone, the integration paths can be used to calculate the minimal distances from clone cells at later time points to the paths, as well as the fate bias (see below), to validate the accuracy and single cell trajectory predictability of the reconstructed vector field, as demonstrated in the HL60 or the hematopoiesis clone tracing datasets analyses .Fate probability is currently calculated as the percentage of points along the predicted cell fate trajectory whose nearest observed cells belong to a particular cell group, e.g., cell type. The distances to the nearest cells are required to be small, and are determined by the median distance of cells and a distance threshold, see details below:Cell fate trajectories predicted by our vector field method sometimes end up in regions where few or no cells were actually measured (see the above section). A heuristic method is thus used to iteratively move backwards along the integrated trajectory to assign cell fate. We first identify the region with small velocity in the tail of the trajectory, which is determined by a threshold of speed, and then check whether the points in the region are close enough to the observed data points (cells), determined by a distance threshold. If not, we select another set of points upstream along the trajectory by one time step while keeping the same amount of points. This moving-back procedure stops when all the selected points of the trajectory are sufficiently close to the observed cells.In case that the trajectory is close to a few random cells, we find the second nearest cells for selected points on the trajectory to include more cells as the nearest neighbors, especially those from terminal cell types. We then use group information of those observed cells to define the fate probability. Fate probability for a particular cell group is then defined as:Animating cell fate commitments relies on the numerical integration of vector fields, as in the above section. Note that this two-dimensional space can be either UMAP, any two dimensions from PCA, or any two genes of interest, as long as we first reconstruct the vector field on this two-dimensional space. A vector field animation visualizes the movement of a set of cells in gene expression space, and the long-range trajectory predicted by the reconstructed vector field. Thus, it provides intuitive visual understandings of the RNA velocity, curvature, acceleration, and cell fate commitment in action.c are the combination coefficient vectors. Because the vector field is a linear combination of Gaussian kernels, whose derivative is:We derive the analytical formula of Jacobian of the vector field which improves the computational efficiency tremendously compared to numerical approaches. The vector field function obtained from the sparseVFC algorithm has the following form , as they are essentially trajectories of cells moving in the vector field. The acceleration is the time derivative of the velocity:v is the velocity vector. The curvature vector of a curve is defined as the derivative of the unit tangent vector v∥):Many differential geometry quantities are defined on In the context of velocity vector fields and streamlines, the unit tangent vector is the normalized velocity. In 2D, the curvature formula has an equivalent but simpler form:Although acceleration and curvature are mathematically defined on streamlines, the actual calculation, as shown above, can be done pointwise using only the velocity and the Jacobian evaluated at the point of interest. Because the acceleration or the curvature can be calculated for any point in the state space, one obtains the acceleration or curvature vector field.k principal components form a d-by-k matrix Q, where d is the dimension of the original gene expression space, then the gene-specific Jacobian G is:Because the vector field function is often learned in a PCA-reduced space, and to acquire gene-specific information, a transformation of the Jacobian from the PCA space to the original gene expression space is needed. Suppose the first ij-th element of G is the partial derivative of the velocity of gene i with respect to the expression level of gene j. The obtained Jacobian G here is only an approximation of the true gene-specific Jacobian, as only k < d principal components are used.Thus, the n × m matrix Q, where n is the number of cells, and m the number of genes, one can obtain a gene-wise vector of such quantities by averaging over cells:Generally, given some quantity  calculated for each gene in each cell, i.e. a C is the set of cells in cluster c, and nc the number of cells in C. When one is interested in the absolute values of the quantities, the average is calculated with |Q|. Then, genes can be ranked based on qc for each cluster. For the ranking of the Jacobian, since each cell is associated with an m × m Jacobian matrix, the whole data is an m × m × n 3D matrix. The same averaging method is applied to all cells or each cluster:Suppose that cells are divided into several clusters, e.g., distinct cell types, the above average can be calculated for each cluster:m × m matrix, and the ranking can be performed in various ways to identify putative interactions, regulators, and effectors:Top interactions: because each element in the averaged Jacobian indicates the change in the velocity of the effector with respect to the change in the expression of the putative regulator, top elements suggest strong gene–gene interactions in each cell or cell type, as below.Top regulators for each effector: we rank each row of the averaged Jacobian matrix, so that for each effector, one obtains the top genes that potentially regulate the effector.Top effectors for each regulator: we rank each column of the averaged Jacobian matrix, so that for each regulator, one obtains top genes potentially regulated by the regulator.ζ, we sum up its averaged Jacobian elements with respect to all possible regulators:Rζ, which shows the top genes potentially involved in the regulation of others;Top regulators: For effector ζ, a summation is taken across all effectors:Eξ reveals top regulated genes.Top effectors: For regulator Because for each cell or cluster, the Jacobian or average Jacobian is an Because the reconstructed vector field is expressed as a set of implicit basis functions, not explicitly as Hill functions, in the current framework we are not able to directly obtain kinetic parameters such as the Hill coefficient. Nevertheless, the reconstructed vector field encodes such information, and additional computations are applied to extract that information. We demonstrate this possibility on simplistic network motifs such as PU.1–GATA1, by fitting the derivatives of inhibitory or activation Hill equations to the corresponding Jacobian elements. Further efforts will be needed to make such efforts generally applicable to systems with more sophisticated mechanisms.x on the target gene takes the form of an activating Hill function:Formally, we assume that the activation effect of gene For self-interactions, there is also an additional degradation term, so:Taking the derivatives:SciPy curve_fit function  cell-wise analyses; 2) trajectory-wise analyses; and 3) plane-wise analyses. The cell-wise analyses cor- respond to regular analyses across measured cells, whereas the trajectory-wise and plane-wise analyses are unique to generative vector fields learned with ion path  or the pion path . Becauseion path  by varyiK1 = K2 = a1 = a2 = b1 = b2 = 1, and n = 4. In the following two subsections, we will describe how the demonstration of the vector field analysis and the benchmarking of our vector field reconstruction with this two-gene system are performed.We use the simple canonical self-activating and mutual-inhibiting two-gene motif that frequently appears in a variety of cell fate bifurcation systems to introduce key concepts in dynamical systems and differential geometry employed in this study . The vecx1 and x2 to  and plot the velocity values calculated with the above vector field function on a 25-by-25 grid with even spacing in this space. The velocity values on the grid are also used to create the streamline plot. Individual trajectories associated with states 1, 2, 3 are obtained via numerical integration of the vector field function. Fixed points are solved analytically from the vector field function. To obtain the separatrices, we integrate the vector field function backwards in time, starting from initial points that are close to saddle points in both directions of eigenvectors whose eigenvalues are negative. The Jacobian of this system is a 2-by-2 matrix:To make the quiver plot of the two-gene system, we first set the expression range of dynamo using default parameters. Attractor, saddle points, and nullclines are estimated with the reconstructed vector field function and plot with the streamline plot that is also based on the reconstructed vector field function via dynamo. We used the reconstructed vector field function to calculate analytical Jacobian, acceleration, curvature, curl, and divergence using dynamo. Scatterplots from dynamo, including a frontier showing the boundary of all those cells, are used to plot the 5000 sampled cells, colored according to either the four elements of the Jacobian, divergence, or curl at those points. Dynamo is also used to estimate the acceleration and curvature for those sampled cells, and then plot their magnitudes together with the corresponding vector fields (i.e. acceleration or curvature vector field). We calculate the analytical Jacobian, acceleration, curvature, divergence, and curl with the true vector field function at those sampled data points and compare the corresponding values estimated from dynamo with scatterplots /Δt, where Δt is the time step for the cell to move from xk − 1. In addition to the deterministic vector field, we also assume a certain degree of stochasticity in the system:η(t) is a stochastic white noise and σ the size of it. The action S along the discrete path is defined as /2. We have also assumed the diffusion matrix to be a constant D, such that D = σ2/2. It is intuitive that a path whose tangential velocities v align with the vector field has smaller action than paths that do not. The LAP is a path such that:The least action path (LAP) principle has previously been used to predict the optimal transition path of cell fate transition for simple and designed systems , 2014. Bted from , aims tosegments . For eacfined as :S is analytically computed:f/∂xa is the a-th row of the Jacobian of the vector field mentioned in the section “differential geometry analysis of the reconstructed single-cell vector field”. With the analytical Jacobian, the computation efficiency of the LAP optimization improves tremendously, making the LAP calculation feasible to operate in high-dimensional space, such as the top 30 PCs.Minimization of the action by varying the path without moving the starting and end points. The optimal path kNN graph of cells.The LAP is found by iterating between the two steps, and empirically we found that the path converges in two or three iterations. By default, the LAP optimization is initialized with the interpolated shortest path on the A and B are two cell types, C a proportionality constant. Furthermore, the transition time, or more specifically the mean first passage time (MFPT), is related to the transition rate:For rare transitions with ion rate :R(A→B)Therefore, the action of the optimal path predicts both the likelihood and transition time for such rare transitions. Again, most reprogramming experiments take a few weeks or months, depending on the exact initial and terminal cell states T*, we perform an additional linear search with respect to the traversal time T < T*, and find the LAP whose traversal time is at the elbow of the action vs. traversal time curve. To determine the elbow Te, we interpolate the normalized action vs. traversal time curve using the cubic spline. Then we calculate the second derivative of the interpolated curve numerically, and find the traversal time that is closest to T* and exceeds the numerical threshold. The resulting LAP is the fastest LAP (F-LAP).For natural transitions, there are many paths following the vector field streamlines, which have nearly zero actions. The LAP is not unique, and the algorithm often ends up finding one with non-optimal traversal time. Therefore, after the algorithm converges to a LAP with traversal time i along the path:à la carte reprogramming  for every gene gramming  for regeVector field predictions with LAP and in silico perturbations. We used the LAPs calculated in UMAP space to visualize the developmental, dedifferentiation, and transdifferentiation LAPs in We used the LAP method to predict an optimal transition matrix between all stable hematopoietic cell types. First, we identified the cell states in either UMAP or PCA space closest to the identified fixed points, and treated them as the stereotypical states for hematopoietic cell types. We then looped through all possible transitions (5 × 6 = 30) from one stereotypical cell state to another and performed LAP analyses. Optimizations of LAPs for all transitions were done in either UMAP or PCA space with default parameters and settings, except that when searching for developmental LAPs (LAPs starting from HSCs state to other mature cell types), we explicitly used the F-LAPs, as recommended in R) of each TF for each transition, we then defined a priority score S:sklearn (y-axis) and that (x-axis) of false positives (TFs not on the compiled list) across all transitions, eventually creating the universal ROC curve across all cell transitions. We also calculate the AUC (area under the curve) of the ROC curve. The ROC analysis and AUC of LAP TF prioritization are presented in Based on the ranking  w.r.t. to small perturbations in the expression of all genes in the network (encoded by the Jacobian matrix J), exact differential:We leverage the analytical Jacobian of the reconstructed vector field function to make In vectorized form:The matrix on the right hand side is the Jacobian of the vector field. Replacing infinitesimal changes with finite perturbations, the above equation becomes:c is often added to the perturbation Δx to amplify the response Δf. Furthermore, because vector fields are often learned in the PCA space, the perturbations in the d-dimensional gene space are first transformed to the k-dimensional PCA space by:Q is the d-by-k PCA loading matrix, and μ is the mean of the PCA-transformed data. The response Δf can be transformed back to the PCA space:In practice, a proportionality constant f, a gene by cell matrix, to identify the strongest positive or negative responders of the genetic perturbation across cells , and were retrieved via the GEO access ID GSE128365. Because scEU-seq manually separated labeled and unlabeled RNA, there is no need for a statistical estimation. However, the manual separation of labeled and unlabeled RNA may introduce potential cross-contamination, and the preparation of two libraries may lead to batch effects. Correction of those possible cross-contamination and batch effects represents an interesting future direction. Data for the sci-fate (The data deposited by the scEU-seq  study prsci-fate  and scNTTet1/2/3 triple-knockout (TetTKO) datasets for studying the bidirectional transition between mESC pluripotent and totipotent state from  were used in this study. The wild-type experiment used the degradation metabolic labeling scheme, whereas the TetTKO experiment used the one-shot metabolic labeling scheme. From both experiments, we obtained unspliced, spliced, labeled, and total RNA data for each gene in each cell. To estimate the absolute degradation rates for each gene in the wild-type dataset, we used the labeled and total RNA data and apply a curve fitting estimation approach (see Estimating absolute RNA velocity for metabolic labeling–based scRNA-seq experiments across various labeling strategies) that builds on Model 2 . On the other hand, the spliced and unspliced RNA were used to estimate the relative degradation rate constants. Combining the relative and absolute degradation rate constants, we obtained the absolute splicing rate constant, which allows us to compute the absolute spliced RNA velocity. The absolute total RNA velocity or spliced RNA velocity was then projected to the total RNA-based or spliced RNA–based 2D UMAP and used for vector field reconstructions (in the top 30 PC space), differential geometry analyses , etc.dynamo with default parameters . The streamline plots in dynamo, as noted. The splicing/labeling kinetics–based phase plots of example gene Fos in dynamo.To benchmark the performance of labeling vs. splicing based RNA velocity analyses for the neuronal activ- ity dataset, we provided scVelo with the unspliced and spliced RNA counts of 97 neuronal activity genes, as features from the original study. Similar to the hematopoietic tscRNA-seq dataset analyses, we closely followed scVelo’s tutorials to perform spliced RNA velocity estimation with either the deterministic, stochastic, or dynamical method. We reproduced what we have reported previously on the labeling RNA velocity analysis using uu, ul, su, sl) for each gene in each cell from all experiments which then gave us unspliced (u), spliced (s), labeled (l) and total (r) RNA data . We mainly focused on analyzing the kinetics and degradation labeling experiments, while demonstrating the generalizability of our estimation framework and revealing the high transcription rates for mitochondrial genes with the mixture labeling experiment. For the kinetics experiment, we used the labeled and total RNA data and the “two-step” method to estimate the absolute transcription  and degradation rate constants. We estimated absolute splicing rate constants similar to the previous section. With the absolute transcription, splicing, and degradation rate constants, we can obtain absolute unspliced, spliced, labeled (or new), and total RNA velocities. The absolute total RNA velocity or spliced RNA velocity was then projected to the total RNA–based or spliced RNA–based 2D UMAP, and are used for vector field reconstructions (in the top 30 PC space), differential geometry analyses , etc. For the mixture experiment, which had a fixed time period that includes a variable initial kinetics experiment and later accompanying degradation experiment  is the Iverson bracket, which outputs 1 if the statement P is true and otherwise 0. The matrix L is used in the same way as the Jacobian matrix to perform the “interaction ranking” with absolute values but to identify the top toggle-switch gene pairs . Intuitively, the preprocessing described here ensures that the gene pair with strongest mutual inhibitions  will be ranked the highest. Note that the two Iverson brackets are used to filter out positive interactions. Also note that because L is a symmetric matrix, only the interactions above the main diagonal are needed for ranking.We relied on the Jacobian matrix dyn.vf.perturbation function to perform in silico perturbation and visualize the cell fate diversion with streamlines projected from the perturbation effect vectors (In silico perturbation to predict gene-wise perturbation effects and cell fate diversions). We suppressed SPI1 (setting expression to −100), GATA1 (setting expression to −100), and both SPI1  and GATA1 (setting expression to −100) and visualized the resultant streamline plots based on the perturbation effect vectors (KLF1 (setting expression to 100), suppression of HLF1 (setting expression to −100), and triple activation of GATA1, KLF1, and TAL1 , as shown in the streamline plots of  dynamics along the indicated trajectories in 1) Gene expression quickly decreases, 2) while velocity rapidly approaches 0 over time. Taking the derivative of the expression or velocity with respect to time along the indicated trajectory gives velocity 1) or acceleration 2), respectively, represented by red arrows. 3) Increasing the gene expression linearly decreases the velocity of the other genes. 4) The velocity of gene x1 positively correlates with that of x2, but with different strengths across the three trajectories.A. Gene expression (both f1/∂x2 (left), ∂f2/∂x2 (right) along the horizontal dashed line indicated in f2/∂x1 (left), ∂f1/∂x1, are shown in B. The Jacobian of ∂f = ∂f2/∂x1 − ∂f1/∂x2 and divergence (∇·f = ∂f1/∂x1 + ∂f2/∂x2) landscapes in the phase space of the two-gene system.C. Heatmaps of the curl  and tscRNA-seq data that do not incorporate splicing (Model 2) or do (Model 3).A. Three main models of cscRNA-seq data  degradation rates γ and the splicing rate middle. The murine splicing rate constant (β) calculated based on scNT-seq data is generally higher than that for humans calculated based on scEU-seq data (right).B. Estimating RNA degradation and splicing rates with data from degradation or kinetics labeling tscRNA-seq experiments. Scatterplot of 1) degradation rates, eq study  on the lAnk2 (slow degradation) and Slc25a32 (fast degradation) chase data, using the ESC experiment data from the scNT-seq study (C. Deterministic first-order decay model fitting of eq study .β) are in general much larger than the degradation rate constants (γ) in both the scNT-seq (left) and scEU-seq (right) dataset analysis based on the density plot.D. Splicing rate constants (left) but slower degradation (right) than other genes based on the cumulative distribution plot.E. Housekeeping genes tend to have faster splicing (left) or degradation (right) are enriched in transcription and cell cycle–related pathways.F. The top 10% genes from the scEU-seq dataset with highest splicing  lineage. CD34, CD33, and CD41a are markers for HSPCs, committed myeloid cells, and the Meg lineage, respectively.A. FACS plots showing human HSPCs (hematopoietic stem and progenitor cells) exiting from CD34PLEK, HBB, LMO4, MPO, and LYZ, markers of the Meg, Ery (Erythrocyte), Bas (Basophil), Mon (Monocyte), and Neu (Neutrophil) lineages, respectively, on the UMAP space.B. Gene expression distribution of dynamo under different models/methods.C. RNA velocity results of the splicing data from the hematopoietic tscRNA-seq experiment, obtained using scVelo and D. Corrected splicing RNA velocity flow via identification of reliable expression dynamics by scoring the velocity confidence of each gene in the phase plane when provided with a prior lineage relationship. See more discussion in E. Distribution of velocity confidence among all dynamical genes, identified via the dynamical model from scVelo. The dashed vertical line indicates the threshold (0.8) used by the correction algorithm to select confident velocity genesF. Box plot of the number of non-confident vs. confident velocity genes.G. Spliced RNA velocity flow after the correction.H. RNA velocity flow on the splicing data from the neuronal activity scNT-seq dataset  using thFos.I. Unsuccessful capture of introns and the constant transcription rate assumption causes most cells to have negative velocities, as evidenced by the fact that they are mostly under the estimated steady state line on the example gene, J. Unbiased capture of nascent RNA via metabolic labeling and the gene/cell dependent RNA transcription rate modeling strategy give rise to correct RNA velocity flow of neuronal activity under KCl polarization.left), cycle–related genes (middle), and a combination of GR and cell cycle–related genes (right) for the sci-fate dataset (K. Streamline plots with only GR-related genes ( dataset  on UMAP Tet 1/2/3 (TetTKO) are biased to differentiate into 2C-like cells, based on the splicing RNA velocity streamline plot produced with dynamo.L. Cells with triple KO of dynamo. Both L and M are the same as the middle and bottom panels from M. Cells with TetTKO are biased to differentiate into 2C-like cells, based on the labeling RNA velocity streamline plot produced with Fig SI4right) along the trajectory at points xa and xb have single-peaked distributions along X as well as the unmeasured Y , whereas the interval at xc has a two-peaked distribution for unmeasured Y . Note the conditional distribution for Y at point xa (P(y|xa)) is much wider than that at point xb (P(y|xb)). Vector field reconstruction is expected to perform well when hidden variables from the system correlate with the observed variables (at xa), but less well when they are loosely coupled (xb and xc). K(|x − xa|) is a fast-decay kernel function  of single cells.p ≤ 10−4.D. Boxplot of the expression distance distribution (in the PCA space) of sister and cousin cell pairs, as well as that of random cell pairs. Mann–Whitney–Wilcoxon test (two-sided) was used to calculate the p-value between groups. ****: R2) is shown for each panel.E. Scatterplots of the distance of spliced RNA expression states of single cells vs. the distance of unspliced RNA expression states, and vs. that of RNA velocity vectors of single cells show strong correlations. Distances were calculated for any pairs of cells in PCA space. R-squared value (p ≤ 0.01.F. Distances between first-nearest neighbor cells show no difference among cells from the same or different linkages. Mann–Whitney–Wilcoxon test (two-sided) was used to calculate the p-value between groups. **: Left) and the identified outlier cells on the gene expression space of x1 and x2 (right). Accuracy of estimated Jacobian deteriorates in boundary regions of sampled cells due to insufficient and biased sampling.G. Pairwise scatterplots of estimated and analytical Jacobian elements (indicated by the notation for each column) from corresponding cells (numdifftools).H. Analytical differential geometric analyses enable nearly 1000x faster computation than state-of-the-art numeric algorithms . X-axis is in log-scale. Each row is associated with the indicated benchmarks annotated above the corresponding horizontal line.I. Cosine correlation or RMSE between estimated vector field quantities  under different cell downsampling depth, noise level, and values of various parameters  and is used to validate the vector field prediction .A. Clonity of cells, which were sequentially sampled at different time points, is inferred based on the static barcodes (top) and plate-based metabolic labeling scRNA-seq (scSLAM-seq or NASC-seq) coupled with sequential clonal cell tracking via lentivirus lineage barcodes (bottom) for neutrophil fate commitment of HL60 cells under ATRA treatment.B. Experimental schemes of conventional 10x Chromium–based scRNA-seq (CD38) and neutrophil marker (CD11b or ITGAM) expression, respectively.C. RNA velocity streamline plot of UMAP embedding based on labeling data from the scRNA-seq experiment reveals neutrophil-lineage commitment. From left to right, the cells on the streamline plot are colored with experimental time, progenitor marker  was used to calculate the p-value between groups. ****: in vivo and in vitro systems. Cells are colored by the cell type identity.G. RNA velocity streamline plot of SPRING embedding (embedding is from ) revealsH. A majority of cells from the same clone are biased towards a specific lineage or remain in the undifferentiated cell state. In the heatmap, row, column, and color correspond to a particular clone of cells, a particular cell lineage, and the probability that cells eventually commit to a cell lineage (or maintain the undifferentiated cell state).I. Comparing vector field–based lineage fate predictions with other state-of-the-art methods. Smoothed clone fate (red dot) is the prediction based on clone barcodes. PBA, WOT, and fateID (black dots) are predictions based on other state-of-art algorithms that do not use velocity information. Those first four methods are from the original study . PredictFig SI6A. The reconstructed vector field, and identified fixed points of hematopoiesis from the hematopoietic tscRNA-seq dataset. Cells are colored based on vector field–based pseudotime, which is calculated based on the RNA velocity transition matrix.B. Megakaryocytes have the largest RNA speed (velocity magnitude) among all cell types.Mt, same as below, see more at FLI1 (Meg lineage master regulator) relative to KLF1 (Ery lineage master regulator) in progenitors.C. Earlier expression  analyses of the PU.1/SPI1–GATA1 network motif across all cells. i) Transcription rates of SPI1 and GATA1 in the SPI1 and GATA1 expression space. ii) Similar to i) (each column of the i-th subpanel matches with that of the ii-th subpanel), but replaced with a response heatmap  or self-activation (second two subpanels), respectively. White dashed line corresponds to the zero Jacobian value. The blue stars at each H. For more details, see I. The velocity kinetic curves over gene expression changes of the corresponding fitted Hill equations of panel SPI1 and GATA1 along the integration path. iii) Repression of GATA1 by SPI1 (∂fGATA1/∂xSPI1), repression of SPI1 by GATA1 (∂fSPI1/∂xGATA1), and self-activation of SPI1 (∂fSPI1/∂xSPI1) and GATA1 (∂fGATA1/∂xGATA1) along the integration paths reveal linear or sigmoid-like, time-dependent interactions.J. Trajectory-wise analyses of the SPI1-GATA1 network motif along a trajectory from HSCs to the Mon lineage. From left to right and top to bottom: i) The trajectory from HSCs to the Mon lineage in the first two PCA components. Cells are colored by cell-type identities, whereas trajectories are colored by integration time . ii) The switch kinetics of SPI1, repression of SPI1 by GATA1 and of GATA1 by SPI1, and self-activation of GATA1 as a function of SPI1 (x-axis) and GATA1 (y-axis) expression.K. Plane-wise analyses of the SPI1–GATA1 network motif in a GMP-like state. Self-activation of Fig SI7A. Predicted optimal dedifferentiation path  from each of the terminal cell types to HSC in the UMAP embedding. Color of the node along the paths indicates the LAP transition time.B. Same as above but for transdifferentiation paths (a.k.a transdifferentiation LAPs).C. Transcriptomic kinetics along the LAP of the basophil lineage back to HSC.D. The LAP time between any hematopoietic cell states. Megakaryocyte developmental LAP has the overall shortest time among all developmental LAPs. Developmental time is significantly smaller than that of reprogramming.E. The cell fate transition matrix between stable hematopoietic cell types. The expression kinetics of the top three genes for each transition (from the cell type in that row to the cell type in that column) along the LAP are plotted as a function of the LAP transition time (Unit: hour).F. Majority of TFs for known hematopoietic transitions are accurately prioritized by LAP predictions.Supplementary Table 1Supplementary Table 1, related to Top toggle switch gene pairs identified for each hematopoietic cell fate bifurcation of the tscRNA-seq dataset.Supplementary Table 2Supplementary Table 2, related to Key transcription factors involved in known hematopoietic cell fate transitions