About

Pseudodynamics+ solves PDE with time and state dependent parameters

pseudodynamics+ is a computational framework for reconstructing population-aware cell state dynamics from time-resolved single-cell data. It is designed to bridge the gap between static single-cell snapshots and tissue-scale population dynamics by jointly modelling cell state transitions and changes in total population size.

At its core, pseudodynamics+ formulates cellular differentiation as a continuous density evolution process governed by a partial differential equation (PDE) that integrates three fundamental dynamic behaviours: proliferation (growth), directed state transitions (differentiation), and stochastic dispersion (diffusion).

\[ \frac{\partial }{{\partial t}}u\left( { \mathbf{s},t} \right) = \underbrace{g\left( { \mathbf{s},t} \right)u\left( { \mathbf{s},t} \right)}_{\textrm{growth}} - \underbrace{\nabla_{\mathbf{s}} \left( {v\left( { \mathbf{s},t} \right)u( \mathbf{s},t)} \right)}_{\textrm{drift}} + \underbrace{\nabla_{\mathbf{s}} \left( {D\left( { \mathbf{s},t} \right)\nabla_{\mathbf{s}} u( \mathbf{s},t)} \right)}_{\textrm{diffusion}} \]

To solve this high-dimensional PDE without discretising the cell state space, pseudodynamics+ leverages physics-informed neural networks (PINNs), enabling scalable and mesh-free inference directly in low-dimensional representations of single-cell landscapes.

Downstream analyses

By integrating single-cell transcriptomic time series with independent measurements of population size, pseudodynamics+ infers state- and time-resolved dynamic parameters at single-cell resolution. These parameters provide a quantitative description of how cells proliferate, differentiate, and redistribute across complex, branching developmental landscapes.

pseudodynamics+ enables a range of downstream analyses, including:

• Estimation of growth, differentiation velocity fields, and diffusion across cell states

• Reconstruction and interpolation of continuous cell density landscapes at unobserved time points

• Simulation of cell state trajectories and future population dynamics

• Quantification of continuous density transport between progenitor and descendant states

• Identification of genes associated with dynamic changes in differentiation behaviour

Overall, pseudodynamics+ provides a population-aware, mechanistic framework for interpreting time-series single-cell data, offering insights into tissue development, homeostasis, and lineage bias that are not accessible from cell state information alone. It is applicable to complex multi-lineage systems and is particularly suited for studies where both molecular resolution and quantitative population dynamics are essential.