David Rand

Warwick Mathematics Institute & Zeeman Institute


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Developmental Biology, Dynamics & Landscapes

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Bifurcation set for the compact elliptic umbilic that has been used to describe the landscape underlying a key part of the developmental process creating the worm vulva (Corson & Siggia). Thanks to Meritxell Saez.

Developmental biology is about the dynamic processes that play out at the cellular and tissue level to create an organism. Its three fundamental aspects are morphogenesis (what causes an organism to develop its shape), the control of cell growth, and cellular differentiation. Key to these are the mechanisms that underlie a cell's decision-making to determine its fate and specification. Recent developments in single-cell transcriptomics, proteomics and imaging have opened up exciting opportunities to probe these decision-making mechanisms in much deeper ways than previously possible.

Dynamical systems & Waddington landscapes

Developmental dynamics admits an intuitive topographical representation that can be formalised mathematically (C H Waddington, The Strategy of the Gene, London, 1957). The developmental path of a cell is conceived as a downhill flow in a landscape with height function that is itself shifting because of changes in the signals the cell receives. In this picture, cell fates are determined by which of the valleys the downhill trajectory descends to. Changing signals can cause the attractor in that valley to disappear (bifurcate) allowing a dynamically trapped cell to transition to a new fate.

Part of my work to to make this intuitive picture rigorous using the modern theory of dynamical systems. The dynamical system we are concerned with are those which describe relevant gene regulatory networks comprising transcription factors and intercellular signals. Under general conditions they have a gradient-like description which allows the link with landscapes.

Cell differentiation whereby cells change their state and develop specialised functions is at the heart of developmental biology. In our landscape picture it occurs when there is a bifurcation which destroys an attracting state allowing the cells that were trapped by that attractor to escape and transition to a new attractor which corresponds to a new fate. The developmental transitions produced by these bifurcations are caused by changes in the signals that the cells see and typically there is a relatively small number of signals. A key aim of our theory is a classification of such bifurcations as part of a classification of compact landscapes. Already, with the classification of landscapes in dimension 2 we find interesting examples highly relevant to stem cell differentiation.

Landscape dynamics of verterbrate trunk development

The formation of the vertebrate trunk provides an important example of how cell fate decisions in developing tissues are made by signal controlled gene regulatory networks. Successively more posterior neural and paraxial presomitic mesodermal (PSM) cells of the trunk are generated from a bipotential population of cells termed neuromesodermal progenitors (NMPs) at the posterior end of the embryo.Several signals including members of the Fgf and Wnt families are known to be involved in the induction of mesodermal and neural tissue. In the mouse embryo, an anterior-posterior gradient of Fgf and Wnt signalling is created and the corresponding signals induce expression of the genes Cdx and Bra and this in turn stimulates Wnt expression. Upregulation of Cdx factors induces posterior Hox gene expression which establishes the posterior identity of NMP derived neural and mesodermal progenitors (Fig. 1).

We are studying the landscape dynamics of this process in cell lines using ideas from landscape dynamics.

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Development from an information-theoretic point of view

Multidimensional data coming from contemporary expression analysis, FACS and similar technologies enables calculation of the correlation structures of the corresponding multidimensional concentration distributions, opening up great opportunities for the application of mathematical analysis that have not been investigated systematically.

The strongly anthropomorphic phrase "decision-making" has been widely adopted by biologists to describe the processes whereby, in response to signals, cells are switched into specific states as discussed above. This boils down to the cell somehow being able to use its highly stochastic response to the noisy signals it sees to decide which genes to turn on that will cause the appropriate changes. The multidimensional data coming from contemporary expression analysis, FACS and similar technologies enables calculation of the correlation structures of the corresponding multidimensional concentration distributions, opening up great opportunities for the application of information geometry and information theory to this situation. This has not been investigated systematically. These information based theories address such questions but the biological problem opens up new challenges and opportunities and is pretty unique in having the really interesting and appropriate data to use.

Rather than just focus on either mutual information or ad hoc methods, there is a critical need to take advantage of information geometry (IG)\footnote{ Amari, S.-i. 2016 Information geometry and its applications, Springer. } and related statistical areas to develop a more systematic and rational approach. At the same time the biologically motivated problems about information transduction on networks will open up new directions for IG. IG will bring in the key ideas around the various f-divergences such as KL, Bregman, $\chi$-squared, Chernoff and Bhattacharyya divergences) and geometric ideas developed as part of its dual Riemannian geometry structure. The main parts of this theory are about the broad class of exponential distributions.

Making reliable systems from unreliable components

As mentioned above recent developments in single-cell transcriptomics, proteomics and imaging have enabled much greater understanding of the levels of variation and heterogeneity in expression levels within cells. We are now able not only to see the levels of variability in mRNAs and protein but also the extent to which multiple mRNAs or proteins have the correlation they should if they are under the control of regulatory networks. Moreover, we know from the study of individual components such as genes that there are input-output systems with very low mutual information between the input and output. This and the high levels of stochasticity that are found raise a number of important questions but the outstanding two are:
1. How can one evolve highly effective decision-making networks using low reliability components?
2. How can a population of cells which are so highly stochastic interact through signalls so as to produce an almost deterministic developmental pattern?


Eric Siggia (Rockefeller), James Briscoe (Crick), Robert Blassberg (Crick), Elena Camacho Aguilar (Houston), Meritxell Saez (Warwick)

© David Rand
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