Quantifying cell transitions in C. elegans with data-fitted landscape models

Approximate Bayesian computation for inferring Waddington landscapes from single-cell data

Single-cell technologies allow us to gain insights into cellular processes at unprecedented resolution. In stem cell and developmental biology snapshot data allow us to characterize how the transcriptional states of cells change between successive cell types. Here, we show how approximate Bayesian computation (ABC) can be employed to calibrate mathematical models against single-cell data. In our simulation study, we demonstrate the pivotal role of the adequate choice of distance measures appropriate for single-cell data. We show that for good distance measures, notably optimal transport with the Sinkhorn divergence, we can infer parameters for mathematical models from simulated single-cell data. We show that the ABC posteriors can be used (i) to characterize parameter sensitivity and identify dependencies between different parameters and (ii) to construct representations of the Waddington or epigenetic landscape, which forms a popular and interpretable representation of the developmental dynamics. In summary, these results pave the way for fitting mechanistic models of stem cell differentiation to single-cell data.

Combinatorial interpretation of BMP and WNT controls the decision between primitive streak and extraembryonic fates

BMP signaling is essential for mammalian gastrulation, as it initiates a cascade of signals that control self-organized patterning. As development is highly dynamic, it is crucial to understand how time-dependent combinatorial signaling affects cellular differentiation. Here, we show that BMP signaling duration is a crucial control parameter that determines cell fates upon the exit from pluripotency through its interplay with the induced secondary signal WNT. BMP signaling directly converts cells from pluripotent to extraembryonic fates while simultaneously upregulating Wnt signaling, which promotes primitive streak and mesodermal specification. Using live-cell imaging of signaling and cell fate reporters together with a simple mathematical model, we show that this circuit produces a temporal morphogen effect where, once BMP signal duration is above a threshold for differentiation, intermediate and long pulses of BMP signaling produce specification of mesoderm and extraembryonic fates, respectively. Our results provide a systems-level picture of how these signaling pathways control the landscape of early human development.

Open problems in mathematical biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles

One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors.

Dynamical landscapes of cell fate decisions

The generation of cellular diversity during development involves differentiating cells transitioning between discrete cell states. In the 1940s, the developmental biologist Conrad Waddington introduced a landscape metaphor to describe this process. The developmental path of a cell was pictured as a ball rolling through a terrain of branching valleys with cell fate decisions represented by the branch points at which the ball decides between one of two available valleys. Here we discuss progress in constructing quantitative dynamical models inspired by this view of cellular differentiation. We describe a framework based on catastrophe theory and dynamical systems methods that provides the foundations for quantitative geometric models of cellular differentiation. These models can be fit to experimental data and used to make quantitative predictions about cellular differentiation. The theory indicates that cell fate decisions can be described by a small number of decision structures, such that there are only two distinct ways in which cells make a binary choice between one of two fates. We discuss the biological relevance of these mechanisms and suggest the approach is broadly applicable for the quantitative analysis of differentiation dynamics and for determining principles of developmental decisions.

Statistically derived geometrical landscapes capture principles of decision-making dynamics during cell fate transitions

Fate decisions in developing tissues involve cells transitioning between discrete cell states, each defined by distinct gene expression profiles. The Waddington landscape, in which the development of a cell is viewed as a ball rolling through a valley filled terrain, is an appealing way to describe differentiation. To construct and validate accurate landscapes, quantitative methods based on experimental data are necessary. We combined principled statistical methods with a framework based on catastrophe theory and approximate Bayesian computation to formulate a quantitative dynamical landscape that accurately predicts cell fate outcomes of pluripotent stem cells exposed to different combinations of signaling factors. Analysis of the landscape revealed two distinct ways in which cells make a binary choice between one of two fates. We suggest that these represent archetypal designs for developmental decisions. The approach is broadly applicable for the quantitative analysis of differentiation and for determining the logic of developmental decisions.

Cell state transitions: definitions and challenges

A fundamental challenge when studying biological systems is the description of cell state dynamics. During transitions between cell states, a multitude of parameters may change - from the promoters that are active, to the RNAs and proteins that are expressed and modified. Cells can also adopt different shapes, alter their motility and change their reliance on cell-cell junctions or adhesion. These parameters are integral to how a cell behaves and collectively define the state a cell is in. Yet, technical challenges prevent us from measuring all of these parameters simultaneously and dynamically. How, then, can we comprehend cell state transitions using finite descriptions? The recent virtual workshop organised by The Company of Biologists entitled 'Cell State Transitions: Approaches, Experimental Systems and Models' attempted to address this question. Here, we summarise some of the main points that emerged during the workshop's themed discussions. We also present examples of cell state transitions and describe models and systems that are pushing forward our understanding of how cells rewire their state.

Geometry of gene regulatory dynamics

Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types that elaborate this complexity results from the activity of gene networks and was likened by Waddington to a flow through a landscape in which valleys represent alternative fates. Geometric methods allow the formal representation of such landscapes and codify the types of behaviors that result from systems of differential equations. Results from Smale and coworkers imply that systems encompassing gene network models can be represented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all three-way cellular decisions realizable by tuning at most two parameters, which can be generalized to include spatial coordinates in a tissue. All diagrams of cell states vs. model parameters are thereby enumerated. We unify a number of standard models for spatial pattern formation by expressing them in potential form (i.e., as topographic elevation). Turing systems appear nonpotential, yet in suitable variables the dynamics are low dimensional and potential. A time-independent embedding recovers the original variables. Lateral inhibition is described by a saddle point with many unstable directions. A model for the patterning of the Drosophila eye appears as relaxation in a bistable potential. Geometric reasoning provides intuitive dynamic models for development that are well adapted to fit time-lapse data.