Learning noisy tissue dynamics across time scales

TL;DR

The paper tackles predicting noisy multicellular tissue dynamics across multiple time scales by modeling tissues as neural stochastic differential equations on a dynamic cell graph, augmented with a dual signaling graph to capture cascades. It jointly deploys graph neural networks, normalizing flows, and WaveNet-inspired causal dilations to address discreteness, stochasticity, and non-Markovianity, enabling both accurate predictions and sampling of single-cell distributions from limited data. Key contributions include a dual-graph architecture for signaling, a Graph WaveNet for memory, and a graph-conditioned normalizing flow for joint noise modeling, demonstrated on MDCK epithelial monolayers, fly wing development, and ERK wave signaling, highlighting mechanochemical coupling and multiscale dynamics. The approach offers a data-efficient pathway toward digital twins in bioengineering and clinical contexts by enabling probabilistic, high-fidelity reconstructions of tissue dynamics from imaging data.

Abstract

Tissue dynamics play a crucial role in biological processes ranging from inflammation to morphogenesis. However, these noisy multicellular dynamics are notoriously hard to predict. Here, we introduce a biomimetic machine learning framework capable of inferring noisy multicellular dynamics directly from experimental movies. This generative model combines graph neural networks, normalizing flows and WaveNet algorithms to represent tissues as neural stochastic differential equations where cells are edges of an evolving graph. Cell interactions are encoded in a dual signaling graph capable of handling signaling cascades. The dual graph architecture of our neural networks reflects the architecture of the underlying biological tissues, substantially reducing the amount of data needed for training, compared to convolutional or fully-connected neural networks. Taking epithelial tissue experiments as a case study, we show that our model not only captures stochastic cell motion but also predicts the evolution of cell states in their division cycle. Finally, we demonstrate that our method can accurately generate the experimental dynamics of developmental systems, such as the fly wing, and cell signaling processes mediated by stochastic ERK waves, paving the way for its use as a digital twin in bioengineering and clinical contexts.

Figures (6)

• Figure 1: Deep learning for noisy multicellular dynamics.A. Cells in a tissue undergo three characteristic dynamical behaviors, motion, cell state transitions, and cell signaling (bottom row). These cellular dynamics are often subject to intrinsic stochasticity due to the complex physical and chemical processes occurring inside the cells. As the basic structural and functional unit of living organisms, cells also exhibit strong discreteness in many aspects, ranging from their granularity in space to cell division or state change in time. Moreover, through stress response and chemical signaling, cells experience complex interactions from their neighbors, which could even in turn regulate the internal states of the cells. The cell network in a tissue is naturally represented as a cell graph where the nodes are the nuclei of the cells and the edges denote their interconnectivity through touching membranes. In practice, it is also convenient to construct a signaling graph with structure dual to the cell graph, which is used in the graph neural networks to describe interactions (see Methods for details). In the dual graphs, the vertices and edges are interchanged, so the information on signaling is contained on the dual nodes. This enables the GNN to capture directional signaling cascades that would be difficult to handle through the convolutional structure of a GNN based on the cell graph only. B. The main challenges encountered in learning the dynamics of cells in tissues are discreteness, stochasticity, and non-Markovianity (i.e. history dependence). These challenges originate from the processes at play in the multicellular dynamics of a biological tissue (panel A). C. These are addressed by blending three machine learning techniques: graph neural networks (neural networks working on data stored on the vertices and edges of graphs), normalizing flows (generative models that represent complex probability distributions by a sequence of change of variables), and WaveNet (autoregressive models using causal dilated convolutions).
• Figure 2: Learning cell dynamics.A. We aim at predicting the velocity $\bm{v}_i$ of all individual cells from their positions $\bm{r}_i$ and cell shapes and areas ($c_i$) in tissues. This requires handling discrete and stochastic data, but no non-Markovian effect. B. The ML pipeline allows us to predict the evolution of both average features (red and blue lines) and noise (manifested through observed variability; red and blue regions). C-F. We apply the technique to a monolayer of epithelial MDCK cells, which undergo stochastic migration and division in a confined environment (panel C). Cells become more rounded as they divide and gradually transit from a fluid state into a glassy state. Their motility, measured by the spatially averaged cell speed, decreases over time. Our ML model not only captures the same decreasing trend (panel B), but also accurately predicts the spatial variation (shaded region). Panel D shows the velocity field at single-cell resolution. Even though an experimental movie only provides a particular realization of the stochastic dynamics, the probabilistic design enables us to infer the underlying statistics through maximum likelihood optimization (panel E). This allows us to further extract the ensemble properties of individual cells (panel F). G-J. We applied our ML algorithm to study the growth of a fly wing Etournay2015Etournay2016. It automatically recognizes the deterministic nature of such a developmental system. The predicted probability distribution of cell migration velocity is nearly a $\delta$-function (panel J). Note that all the ML predictions shown in this paper are produced on the test dataset, which has never been seen by the ML model during training. L. We aim at predicting the state of individual cells within the cell division cycle from the individual positions in the tissue. In the division process, the cells go through three states, $G_1$, $S$, and $G_2$ and then undergo mitosis ($M$). It is a stochastic process discrete in time meanwhile subject to regulations from neighboring cells through mechanical stresses, but here we do not consider non-Markovian effects. M. In a monolayer of MDCK cells, we can identify the division states of individual cells at any given time. N-O. Comparison between experimental ground truth (panel N) and ML predictions (panel O). After training, our model can predict the cell division state with accuracy over 80%.
• Figure 3: Learning dynamic cell signaling.A. By sending out ERK signals, the cells can coordinate their movements (such as lateral expansion and contraction), leading to collective migration of the cells through a feedback loop between ERK activity and cell strain. Here we perform a ML study of the propagation of ERK waves where we aim at predicting the future of the system from its past (the state of the system is represented by the positions $\bm{r}(t)$ of the cells as well as their internal states $\bm{s}(t)$). In addition to the challenges of discreteness and stochasticity already present in the other examples, here we need to take non-Markovian effects into account. Given this long-term correlation in ERK waves, here we integrate our graph neural network with an advanced sequence model called WaveNet. These two networks work seamlessly in our architecture: the WaveNet encodes the environmental changes of individual cells over time, whereas the graph neural networks collect and redistribute such sequential information among each cell and its neighbors. B-C Comparison between experimental ground truth (panel B) and ML prediction (panel C) of ERK signals. D-F. Our ML model accurately predicts the overall decreasing trend of the spatially averaged ERK intensity $s$ (panel D) as well as its spatial variations (shaded region). Furthermore, our model also captures the right statistics of the ERK waves: the predicted time and spatial correlation functions (panels E and F) match well with the experimental ground truth. In panels D-F, the ML prediction is shown in red while the experimental ground truth is shown in blue.
• Figure 4: Graph neural networks.a. Dual graph representation. We represent the spatial arrangement of the agents as an agent graph $\mathcal{G}^\text{a}$, where nodes $n^\text{c}_i$ correspond to cell nuclei and edges $e^\text{c}_{ij}$ indicate the connectivity between two adjacent cells sharing a common membrane. To allow complex cell-cell interactions like signaling, we further employ a message-passing graph $\mathcal{G}^\text{m}$, which form a dual structure in relation to the cell graph: $n^\text{m}_\mu \leftrightarrow e^\text{c}_{ij}$ and $e^\text{m}_{\mu\nu} \leftrightarrow \left(e^\text{c}_{ij}, e^\text{c}_{jk}\right)$. A signal cascade causes statistical dependency of the $\mathcal{G}_\text{c}$ edges in a sequential manner, e.g. $e_{ij} \to e_{jk} \to e_{kl}$, which is converted into the dependency of the $\mathcal{G}_\text{m}$ nodes $n_{\mu} \to n_{\nu} \to n_{\xi}$. The latter can then be explicitly encoded by graph convolution. b. Graph convolution. Information of individual cells and any explicit relations with its neighbors are passed onto the nodes and edges of the cell graph $\mathcal{G}^\text{c}$, respectively. Then we duplicate the same information on the message-passing graph $\mathcal{G}^\text{m}$ and encode how each cell-cell relation depends on local environment using a sequence of graph convolutions. The obtained embeddings are returned back to the corresponding $\mathcal{G}_\text{c}$ edges, capturing the relation between any given cell and its neighbors in a directional fashion. Together with the information of the cell, they provide a complete spatial embedding on the local environment, which can then be used in other ML modules.
• Figure 5: Graph WaveNet. The standard WaveNet uses a stack of 1D dilated convolutions to process sequential data. The receptive field of WaveNet grows exponentially with the depth of the network (see the tree of blue nodes), allowing us to handle long-term memory effects. Here we integrate graph neural networks into each dilated convolution (see green arrows), to further incorporate spatial relation between agents. The resulting model can therefore provide the spatiotemporal embedding of each individual agents, which encode how its local environment varies in the past.