Morphogenesis is one of the grand challenges in biology. Our focus has been on the larger scale questions of self-organized cellular and tissue shape, in asking how we should describe it, how we may predict it, and finally, how we may control it. Shape arises because cells change in number, size, shape, and movement; understanding how this actually happens and how it is controlled is a natural goal.

To correctly quantify the geometry of tissue morphogenesis, we have developed geometrical approaches to tissue tectonics using invariants based on velocity gradients to characterize and compare dynamic fate maps of multi-cellular tissues, borrowing ideas from hydrodynamics. This approach allows one to distinguish the change in tissue shape because of the relative magnitudes of cell shape changes and cell movements, with many experimental implications for robustness in developing tissues (e.g. in convergent extension, etc.). More recently, we have used ideas from nonlinear dynamics to characterize properly invariant measures of deformation that describe the dynamic morphoskeletons – the organizers of large scale cell movements in multicellular tissues (***).  A recent outcome has been to complement this with direct measurements of the forces associated with morphing tissues.

Moving towards a predictive theory for how genes link to geometry, we have explored the physical basis of morphogenesis in plants and animals in a variety of examples – from fungi to flowers, from the vertebrate gut to the mammalian brain. For example, inspired by observations from our experimental colleagues, we provided a theoretical basis for the evolution of multicellular tissue organization in choanoflagellates, likely the closest eukaryotic relatives of metazoans using simple, testable geometrical and biophysical principles. We have also worked on understanding the form of plant organs, such as leaves and flowers, treated as mathematical problems of how to embed non-Euclidean surface metrics in three-dimensional space – in such contexts as the shape of a leaf (117), the blooming of a flower and the elongation of Columbine petal spurs using cell shape change- and understanding the recent evolutionary radiation of this species.  We have also shown how slow movements of water can lead to morphological changes in such instances as the opening and closing of a pine cone or the coiling and overwinding of tendrils, explaining observations going back to Darwin’s work on the power of movement in plants.

In the study of morphogenesis in animals, using differential growth as a guiding principle, we have studied the growth and form of the coiled vertebrate gut (139) with scaling ideas, results corroborated using biophysical experiments at multiple time points in a growing organism and across species. The same principle of differential growth suffices to explain how villi, the protrusions responsible for nutrient absorption, in the intestine form (186), and our study linking experiments and computations explains the process in a growing organism and across species. To link these to genetic causes, we have studied the geometrical and physical basis for differential gene expression during villi formation.

When combined with our study of sulcification (130),  the principle of differential growth allowed us to explain the gyrification of the mammalian brain that has a cortex that grows but is constrained by the underlying white matter, and our experiments using a humble swelling gel along with a computational model showed that we can recapitulate the basic folding patterns in the human fetal brain (***), and also construct a morphospace for the gyrification patterns seen across species in terms of two geometric parameters that correspond to the relative thickness of the cortex and the relative growth of the cortex.

Differential growth is just one of many principles evoked to understand patterning in living systems. We have also explored the role of differential motility and contractility from a theoretical perspective (***), and are currently working to see how these principles might also be relevant to understanding the elongation of the body and the patterning processing associated with somitogenesis. Linking evolution and development, we have explored the form, morphogenesis, and function of avian egg shape, the statistical geometry of insect wings, and that of mammalian brains.

Our work linking theory and experiment quantifies the ideas espoused by the early pioneers of developmental biology such as Roux, His, and D’Arcy Thompson more than a century ago and has led to better integration of the role of physics in organogenesis.