Dynamics

Dynamics combines parameters estimation and velocity calculation, establishing the foundation for vector field reconstruction and subsequent analyses. This paragraph provides a concise overview of the diverse approaches we have implemented to cater to different experiment types. More details can be found in the “Method” section in our paper.

Spliced data:

When the dataset only contains spliced and unspliced RNA data, the following methods can be applied.

  • Conventional: The kinetics of RNA is transcription, splicing, and degradation is represented as follows:

    \[\begin{split}\dot{u} = \alpha - \beta u \\ \dot{s} = \beta u - \gamma s\end{split}\]

    where \(u\) and \(s\) are the copies of unspliced and spliced RNA for a particular gene in a cell, \(\alpha\), \(\beta\) and \(\gamma\) are the rate constants for transcription, splicing, and degradation rate. Transcription rate is a function of the cell state and other variables while splicing and degradation rate are often regarded as constants for specific cell types in most cases. Assuming pseudo-steady state for cells with extreme high unspliced and spliced RNA expressions, we can perform linear regression between the spliced and unspliced RNA to estimate the parameters (\(\beta u = \gamma s\)). Then the velocity can be determined by applying the corresponding ordinary differential equations (ODEs) in conjunction with the obtained parameter values. In the original paper the velocity is defined as \(v = u - \tilde{\gamma} s\), where \(\tilde{\gamma} = \frac{\gamma}{\beta}\).

    The parameters configuration for this method is as follows: experiment_type -> conventional, assumption_mRNA -> ss, model -> deterministic. Additionally, we offer multiple options for the est_method parameter, including ols, rlm, and ransac. For a comprehensive understanding of the est_method options and their respective functionalities, please refer to the detailed documentation provided within the code’s docstring.

  • stochastic splicing: The stochastic splicing method, denoted as GMM (Generalized method of moments) in the code, enhances the estimation of kinetic parameters by incorporating the first, second, and mixed moments of unspliced and spliced RNAs for each gene across multiple cells. In contrast to the original velocity method, which only relies on the first moment, this approach takes advantage of the additional moments, thereby augmenting the robustness of parameter estimation. The parameters configuration for this method is as follows: experiment_type -> conventional, model -> stochastic, assumption_mRNA -> ss, est_method -> gmm.

  • Negative binomial: The negative binomial is an alternative procedure based on the observation that in most cases total RNA counts at steady state follow the NB distribution. It reformulates the moments condition with the NB-distribution variable and solves the formulated equation with a nonlinear least squares optimizer. The configuration for this method is as follows: experiment_type -> conventional, model -> stochastic, assumption_mRNA -> ss, est_method -> negbin.

Labeled data:

The metabolic labeling dataset measures the synthesis or degradation of labeled RNA within a known period of time in an experimentally programmable manner. This dataset offers a more direct assessment of the kinetics underlying gene expression. Consequently, leveraging the metabolic labeling dataset presents an opportunity to address certain challenges encountered in velocity estimation. Here we introduce multiple dynamics models tailored to different types of experiments.

  • One shot: In “one-shot” experiments, there is only one labeling time point, and the splicing process is not explicitly considered. In this method, we substitute the spliced and unspliced RNA variables from the original velocity method with labeled and total RNAs, then we can incorporate the second moments using the negative binomial method to get a more accurate parameter. The configuration for this method is as follows: experiment_type -> one_shot/one-shot, assumption_mRNA -> ss. For one shot experiment, we can tune the model and est_method just like conventional methods.

  • Kinetic: Kinetic experiments represent a time series of 4sU or other nucleotide analog treatment to observe the accumulation of metabolically labeled RNA over time. Our implementation includes two distinct estimation methods: the two-step approach and the direct approach. The two-step approach is a generalization of one-shot method on multiple time points, which relies on two consecutive linear regressions to estimate the degradation rate. Furthermore, the two-step approach can be further generalized to the kinetic assumption. On the other hand, the direct approach directly employs the kinetic model to estimate the rate parameters. In terms of velocity calculation, the two-step approach generally outperforms the direct approach. The configuration for this method is as follows: experiment_type -> kin, assumption_mRNA -> ss/kin, model -> deterministic/stochastic, est_method -> two-step/direct.

  • Degradation: In degradation experiments, samples are chased after an extended 4sU (or other nucleotide analog) labeling period and the wash-out to observe the decay of the abundance of the (labeled) unspliced and spliced RNA decay over time. Splicing and degradation rate can be estimated from this process using the nonlinear least squares. The configuration for this method is as follows: experiment_type -> deg, assumption_mRNA -> kinetic, model -> deterministic/stochastic/mixture.

  • Mix: Furthermore, our framework offers support for a range of mixture experiments like mixed steady-state and stimulation labeling experiment (mix_std_stm) and mixed kinetic and degradation experiment (mix_kin_deg/mix_pulse_chase). More details can be found in the code.