Recent Publications and Preprints
Bistable boundary conditions implying codimension 2 bifurcations
Sáez, M & Rand, D.
Preprint
We consider generic families Xθ of smooth dynamical systems depending on parameters θ ∈ P where P is a 2-dimensional simply connected domain and assume that each Xθ only has a finite number of restpoints and periodic orbits. We prove that if over the boundary of P there is a S or Z shaped bifurcation graph containing two opposing fold bifurcation points while over the rest of the boundary there are no other bifurcation points, then, if there is no fold-Hopf bifurcation in P then there is at least one cusp in the interior of P .
TimeTeller: A tool to probe the circadian clock as a multigene dynamical system.
Vlachou D, Veretennikova M, Usselmann L, Vasilyev V, Ott S, Bjarnason GA, Levi F, & Rand, D.
PLoS Comput Biol 20(2): e1011779. (2024) https://doi.org/10.1371/journal.pcbi.1011779
Recent studies have established that the circadian clock influences onset, progression and therapeutic outcomes in a number of diseases including cancer and heart diseases. Therefore, there is a need for tools to measure the functional state of the molecular circadian clock and its downstream targets in patients. Moreover, the clock is a multi-dimensional stochastic oscillator and there are few tools for analysing it as a noisy multigene dynamical system. In this paper we consider the methodology behind TimeTeller, a machine learning tool that analyses the clock as a noisy multigene dynamical system and aims to estimate circadian clock function from a single transcriptome by modelling the multi-dimensional state of the clock. We demonstrate its potential for clock systems assessment by applying it to mouse, baboon and human microarray and RNA-seq data and show how to visualise and quantify the global structure of the clock, quantitatively stratify individual transcriptomic samples by clock dysfunction and globally compare clocks across individuals, conditions and tissues thus highlighting its potential relevance for advancing circadian medicine.
Sir Erik Christopher Zeeman.
D A Rand
Biographical Memoirs of Fellows of the Royal Society. August, 2022. https://doi.org/10.1098/rsbm.2022.0012
Sir Erik Christopher Zeeman, formerly Foundation Professor of Mathematics at the University of Warwick, Principal of Hertford College, Oxford, and Vice-President of the Royal Society, was for over 40 years a leading figure in British mathematical life. A brilliant mathematician, exceptional lecturer, prodigious polymath and deep-thinking leader and administrator, Christopher Zeeman had a remarkable influence on British mathematics. He made major research contributions to topology, dynamical systems, catastrophe and singularity theory, and applications of mathematics in the physical, biological and social sciences. Particularly notable among his many other achievements were his foundation of the Warwick Mathematics Institute and his contributions to the public understanding of science.
Dynamical landscapes of cell fate decisions
M. Sáez, J. Briscoe and D. A. Rand
Interface Focus 12: 20220002. https://doi.org/10.1098/rsfs.2022.0002
The generation of cellular diversity during development involves differen- tiating cells transitioning between discrete cell states. In the 1940s, the developmental biologist Conrad Waddington introduced a landscape meta- phor 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.
A quantitative landscape of cell fate transitions identifies principles of cellular decision-making.
M. Sáez, R. Blassberg, E. Camacho-Aguilar, E. D. Siggia, D. A. Rand, J. Briscoe
Cell Systems 12:1–17 https://doi.org/10.1016/j.cels.2021.08.013
Fate decisions in developing tissues involve cells transitioning between a set of discrete cell states, each defined by a distinct protein expression profile. Geometric models, often referred to as Waddington landscapes, in which developmental paths are given by the gradient and cell states by the minima of the model, are an appealing way to describe differentiation dynamics and developmental decisions. To construct and validate accurate dynamical landscapes, quantitative methods based on experimental data are necessary. To this end we took advantage of the differentiation of neural and mesodermal cells from pluripotent mouse embryonic stem cells exposed to different combinations and durations of signalling factors. We developed a principled statistical approach using flow cytometry data to quantify differentiating cell states. Then, using a framework based on Catastrophe Theory and approximate Bayesian computation, we constructed the corresponding dynamical landscape. The result was a quantitative model that accurately predicted the proportions of neural and mesodermal cells differentiating in response to specific signalling regimes. Analysis of the data revealed two distinct ways in which cells make a binary choice between one of two fates which correspond to two distinct landscape geometries. We discuss the biological relevance of these mechanisms and suggest that they represent general archetypal designs for developmental decisions. Taken together, the approach we describe is broadly applicable for the quantitative analysis of differentiation dynamics and for determining the logic of developmental cell fate decisions.
Geometry of Gene Regulatory Dynamics.
David A. Rand, Archishman Raju, Meritxell Sáez, Francis Corson, and Eric D. Siggia.
Proceedings of the National Academy of Sciences of the USA. (2021) 118:38:e2109729118 https://doi.org/10.1073/pnas.2109729118
Embryonic development leads to the reproducible and ordered appearance of complexity from egg to adult. The successive differentiation of different cell types, that elaborates this complexity, result 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 rep- resented as potential gradients with a Riemann metric, justifying the Waddington metaphor. Here, we extend this representation to include parameter dependence and enumerate all 3-way cellular decisions realisable 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 non-potential yet in suitable variables the dynamics are low dimensional, potential, and a time independent embedding recovers the biological 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
Quantifying cell transitions in C. elegans with data-fitted landscape models.
Elena Camacho-Aguilar , Aryeh Warmflash, David A. Rand
PLoS Computational Biology, 17(6), p.e1009034
Standard models of cell differentiation focus on creating and analyzing gene regulatory networks (GRNs), which can be used to predict the evolution of a gene expression profile and determine stable states that correspond to cell fates. These models require a large number of parameters variables, and the difficulty in constraining the parameters can reduce their predictive value. Further, model complexity often limits the ability to offer mechanistic insight. Recently there has been an increased interest in simplified models that mathematically formalize Waddington’s landscape metaphor, focusing on cell fates and the transitions between them at the phenotypic level without reference to the underlying GRN. However, to date there is no general, systematic method to develop and fit landscape models to new biological problems. Here we present a methodology to formulate landscape models and fit them to quantitative data, and apply it to model the well-studied process of vulval development in the C. elegans worm. This model represents a qualitatively novel way of thinking about this process, reproduces a large quantity of existing data, and makes novel predictions. Moreover, we provide all necessary mathematical background and implementation details as well as software that can serve as a resource for the broader community to apply in other contexts.