A Geometric Tension Dynamics Model of Epithelial Convergent Extension.

Epithelial tissue elongation by convergent extension is a key motif of animal morphogenesis. On a coarse scale, cell motion resembles laminar fluid flow; yet in contrast to a fluid, epithelial cells adhere to each other and maintain the tissue layer under actively generated internal tension. To resolve this apparent paradox, we formulate a model in which tissue flow occurs through adiabatic remodelling of the cellular force balance causing local cell rearrangement. We propose that the gradual shifting of the force balance is caused by positive feedback on myosin-generated cytoskeletal tension. Shifting force balance within a tension network causes active T1s oriented by the global anisotropy of tension. Rigidity of cells against shape changes converts the oriented internal rearrangements into net tissue deformation. Strikingly, we find that the total amount of tissue extension depends on the initial magnitude of anisotropy and on cellular packing order. T1s degrade this order so that tissue flow is self-limiting. We explain these findings by showing that coordination of T1s depends on coherence in local tension configurations, quantified by a certain order parameter in tension space. Our model reproduces the salient tissue- and cell-scale features of germ band elongation during Drosophila gastrulation, in particular the slowdown of tissue flow after approximately twofold extension concomitant with a loss of order in tension configurations. This suggests local cell geometry contains morphogenetic information and yields predictions testable in future experiments. Furthermore, our focus on defining biologically controlled active tension dynamics on the manifold of force-balanced states may provide a general approach to the description of morphogenetic flow.

The Geometric Basis of Epithelial Convergent Extension.

Shape changes of epithelia during animal development, such as convergent extension, are achieved through concerted mechanical activity of individual cells. While much is known about the corresponding large scale tissue flow and its genetic drivers, key open questions regard the cell-scale mechanics, e.g. internal vs external driving forces, and coordination, e.g. bottom-up self-organization vs top-down genetic instruction. To address these questions, we develop a quantitative, model-based analysis framework to relate cell geometry to local tension in recently obtained timelapse imaging data of gastrulating Drosophila embryos. This analysis provides a systematic decomposition of cell shape changes and T1-rearrangements into internally driven, active, and externally driven, passive, contributions. Specifically, we find evidence that germ band extension is driven by active T1 processes that self-organize through positive feedback acting on tensions. More generally, our findings suggest that epithelial convergent extension results from controlled transformation of internal force balance geometry which we quantify with a novel quantification tool for local tension configurations.

Visceral organ morphogenesis via calcium-patterned muscle constrictions.

Organ architecture is often composed of multiple laminar tissues arranged in concentric layers. During morphogenesis, the initial geometry of visceral organs undergoes a sequence of folding, adopting a complex shape that is vital for function. Genetic signals are known to impact form, yet the dynamic and mechanical interplay of tissue layers giving rise to organs' complex shapes remains elusive. Here, we trace the dynamics and mechanical interactions of a developing visceral organ across tissue layers, from subcellular to organ scale in vivo. Combining deep tissue light-sheet microscopy for in toto live visualization with a novel computational framework for multilayer analysis of evolving complex shapes, we find a dynamic mechanism for organ folding using the embryonic midgut of Drosophila as a model visceral organ. Hox genes, known regulators of organ shape, control the emergence of high-frequency calcium pulses. Spatiotemporally patterned calcium pulses trigger muscle contractions via myosin light chain kinase. Muscle contractions, in turn, induce cell shape change in the adjacent tissue layer. This cell shape change collectively drives a convergent extension pattern. Through tissue incompressibility and initial organ geometry, this in-plane shape change is linked to out-of-plane organ folding. Our analysis follows tissue dynamics during organ shape change in vivo, tracing organ-scale folding to a high-frequency molecular mechanism. These findings offer a mechanical route for gene expression to induce organ shape change: genetic patterning in one layer triggers a physical process in the adjacent layer - revealing post-translational mechanisms that govern shape change.

Sector search strategies for odor trail tracking.

Ants, mice, and dogs often use surface-bound scent trails to establish navigation routes or to find food and mates, yet their tracking strategies remain poorly understood. Chemotaxis-based strategies cannot explain casting, a characteristic sequence of wide oscillations with increasing amplitude performed upon sustained loss of contact with the trail. We propose that tracking animals have an intrinsic, geometric notion of continuity, allowing them to exploit past contacts with the trail to form an estimate of where it is headed. This estimate and its uncertainty form an angular sector, and the emergent search patterns resemble a "sector search." Reinforcement learning agents trained to execute a sector search recapitulate the various phases of experimentally observed tracking behavior. We use ideas from polymer physics to formulate a statistical description of trails and show that search geometry imposes basic limits on how quickly animals can track trails. By formulating trail tracking as a Bellman-type sequential optimization problem, we quantify the geometric elements of optimal sector search strategy, effectively explaining why and when casting is necessary. We propose a set of experiments to infer how tracking animals acquire, integrate, and respond to past information on the tracked trail. More generally, we define navigational strategies relevant for animals and biomimetic robots and formulate trail tracking as a behavioral paradigm for learning, memory, and planning.

Human neural tube morphogenesis in vitro by geometric constraints.

Understanding human organ formation is a scientific challenge with far-reaching medical implications1,2. Three-dimensional stem-cell cultures have provided insights into human cell differentiation3,4. However, current approaches use scaffold-free stem-cell aggregates, which develop non-reproducible tissue shapes and variable cell-fate patterns. This limits their capacity to recapitulate organ formation. Here we present a chip-based culture system that enables self-organization of micropatterned stem cells into precise three-dimensional cell-fate patterns and organ shapes. We use this system to recreate neural tube folding from human stem cells in a dish. Upon neural induction5,6, neural ectoderm folds into a millimetre-long neural tube covered with non-neural ectoderm. Folding occurs at 90% fidelity, and anatomically resembles the developing human neural tube. We find that neural and non-neural ectoderm are necessary and sufficient for folding morphogenesis. We identify two mechanisms drive folding: (1) apical contraction of neural ectoderm, and (2) basal adhesion mediated via extracellular matrix synthesis by non-neural ectoderm. Targeting these two mechanisms using drugs leads to morphological defects similar to neural tube defects. Finally, we show that neural tissue width determines neural tube shape, suggesting that morphology along the anterior-posterior axis depends on neural ectoderm geometry in addition to molecular gradients7. Our approach provides a new route to the study of human organ morphogenesis in health and disease.

Fluctuations can induce local nematic order and extensile stress in monolayers of motile cells.

Recent experiments in various cell types have shown that two-dimensional tissues often display local nematic order, with evidence of extensile stresses manifest in the dynamics of topological defects. Using a mesoscopic model where tissue flow is generated by fluctuating traction forces coupled to the nematic order parameter, we show that the resulting tissue dynamics can spontaneously produce local nematic order and an extensile internal stress. A key element of the model is the assumption that in the presence of local nematic alignment, cells preferentially crawl along the nematic axis, resulting in anisotropy of fluctuations. Our work shows that activity can drive either extensile or contractile stresses in tissue, depending on the relative strength of the contractility of the cortical cytoskeleton and tractions by cells on the extracellular matrix.

A variational method for image-based inference of internal stress in epithelial tissues.

Cellular mechanics drives epithelial morphogenesis, the process wherein cells collectively rearrange to produce tissue-scale deformations that determine organismal shape. However, quantitative understanding of tissue mechanics is impaired by the difficulty of direct measurement of stress in-vivo. This difficulty has spurred the development of image-based inference algorithms that estimate stress from snapshots of epithelial geometry. Such methods are challenged by sensitivity to measurement error and thus require accurate geometric segmentation for practical use. We overcome this difficulty by introducing a novel approach - the Variational Method of Stress Inference (VMSI) - which exploits the fundamental duality between stress and geometry at equilibrium of discrete mechanical networks that model confluent cellular layers. We approximate the apical geometry of an epithelial tissue by a 2D tiling with Circular Arc Polygons (CAP) in which arcs represent intercellular interfaces defined by the balance of local line tension and pressure differentials between adjacent cells. The mechanical equilibrium of such networks imposes extensive local constraints on CAP geometry. These constraints provide the foundation of VMSI which, starting with images of epithelial monolayers, simultaneously approximates both tissue geometry and internal forces, subject to the constraint of equilibrium. We find VMSI to be more robust than previous methods. Specifically, the VMSI performance is validated by the comparison of the predicted cellular and mesoscopic scale stress with the measured myosin II patterns during early Drosophila embryogenesis. VMSI prediction of mesoscopic stress tensor correlates at the 80% level with the measured myosin distribution and reveals that most of the myosin activity in that case is involved in a static internal force balance within the epithelial layer. In addition to insight into cell mechanics, this study provides a practical method for non-destructive estimation of stress in live epithelial tissue.

Phylodynamic theory of persistence, extinction and speciation of rapidly adapting pathogens.

Rapidly evolving pathogens like influenza viruses can persist by changing their antigenic properties fast enough to evade the adaptive immunity, yet they rarely split into diverging lineages. By mapping the multi-strain Susceptible-Infected-Recovered model onto the traveling wave model of adapting populations, we demonstrate that persistence of a rapidly evolving, Red-Queen-like state of the pathogen population requires long-ranged cross-immunity and sufficiently large population sizes. This state is unstable and the population goes extinct or 'speciates' into two pathogen strains with antigenic divergence beyond the range of cross-inhibition. However, in a certain range of evolutionary parameters, a single cross-inhibiting population can exist for times long compared to the time to the most recent common ancestor ([Formula: see text]) and gives rise to phylogenetic patterns typical of influenza virus. We demonstrate that the rate of speciation is related to fluctuations of [Formula: see text] and construct a 'phase diagram' identifying different phylodynamic regimes as a function of evolutionary parameters.

Global morphogenetic flow is accurately predicted by the spatial distribution of myosin motors.

During embryogenesis tissue layers undergo morphogenetic flow rearranging and folding into specific shapes. While developmental biology has identified key genes and local cellular processes, global coordination of tissue remodeling at the organ scale remains unclear. Here, we combine in toto light-sheet microscopy of the Drosophila embryo with quantitative analysis and physical modeling to relate cellular flow with the patterns of force generation during the gastrulation process. We find that the complex spatio-temporal flow pattern can be predicted from the measured meso-scale myosin density and anisotropy using a simple, effective viscous model of the tissue, achieving close to 90% accuracy with one time dependent and two constant parameters. Our analysis uncovers the importance of a) spatial modulation of myosin distribution on the scale of the embryo and b) the non-locality of its effect due to mechanical interaction of cells, demonstrating the need for the global perspective in the study of morphogenetic flow.

Mechanical control of growth: ideas, facts and challenges.

In his classic book On Growth and Form, D'Arcy Thompson discussed the necessity of a physical and mathematical approach to understanding the relationship between growth and form. The past century has seen extraordinary advances in our understanding of biological components and processes contributing to organismal morphogenesis, but the mathematical and physical principles involved have not received comparable attention. The most obvious entry of physics into morphogenesis is via tissue mechanics. In this Review, we discuss the fundamental role of mechanical interactions between cells induced by growth in shaping a tissue. Non-uniform growth can lead to accumulation of mechanical stress, which in the context of two-dimensional sheets of tissue can specify the shape it assumes in three dimensions. A special class of growth patterns - conformal growth - does not lead to the accumulation of stress and can generate a rich variety of planar tissue shapes. Conversely, mechanical stress can provide a regulatory feedback signal into the growth control circuit. Both theory and experiment support a key role for mechanical interactions in shaping tissues and, via mechanical feedback, controlling epithelial growth.

Active Tension Network model suggests an exotic mechanical state realized in epithelial tissues.

Mechanical interactions play a crucial role in epithelial morphogenesis, yet understanding the complex mechanisms through which stress and deformation affect cell behavior remains an open problem. Here we formulate and analyze the Active Tension Network (ATN) model, which assumes that the mechanical balance of cells within a tissue is dominated by cortical tension and introduces tension-dependent active remodeling of the cortex. We find that ATNs exhibit unusual mechanical properties. Specifically, an ATN behaves as a fluid at short times, but at long times supports external tension like a solid. Furthermore, an ATN has an extensively degenerate equilibrium mechanical state associated with a discrete conformal - "isogonal" - deformation of cells. The ATN model predicts a constraint on equilibrium cell geometries, which we demonstrate to approximately hold in certain epithelial tissues. We further show that isogonal modes are observed in the fruit y embryo, accounting for the striking variability of apical areas of ventral cells and helping understand the early phase of gastrulation. Living matter realizes new and exotic mechanical states, the study of which helps to understand biological phenomena.

Inferring Cell-State Transition Dynamics from Lineage Trees and Endpoint Single-Cell Measurements.

As they proliferate, living cells undergo transitions between specific molecularly and developmentally distinct states. Despite the functional centrality of these transitions in multicellular organisms, it has remained challenging to determine which transitions occur and at what rates without perturbations and cell engineering. Here, we introduce kin correlation analysis (KCA) and show that quantitative cell-state transition dynamics can be inferred, without direct observation, from the clustering of cell states on pedigrees (lineage trees). Combining KCA with pedigrees obtained from time-lapse imaging and endpoint single-molecule RNA-fluorescence in situ hybridization (RNA-FISH) measurements of gene expression, we determined the cell-state transition network of mouse embryonic stem (ES) cells. This analysis revealed that mouse ES cells exhibit stochastic and reversible transitions along a linear chain of states ranging from 2C-like to epiblast-like. Our approach is broadly applicable and may be applied to systems with irreversible transitions and non-stationary dynamics, such as in cancer and development.

Differential growth triggers mechanical feedback that elevates Hippo signaling.

Mechanical stress can influence cell proliferation in vitro, but whether it makes a significant contribution to growth control in vivo, and how it is modulated and experienced by cells within developing tissues, has remained unclear. Here we report that differential growth reduces cytoskeletal tension along cell junctions within faster-growing cells. We propose a theoretical model to explain the observed reduction of tension within faster-growing clones, supporting it by computer simulations based on a generalized vertex model. This reduced tension modulates a biomechanical Hippo pathway, decreasing recruitment of Ajuba LIM protein and the Hippo pathway kinase Warts, and decreasing the activity of the growth-promoting transcription factor Yorkie. These observations provide a specific mechanism for a mechanical feedback that contributes to evenly distributed growth, and we show that genetically suppressing mechanical feedback alters patterns of cell proliferation in the developing Drosophila wing. By providing experimental support for the induction of mechanical stress by differential growth, and a molecular mechanism linking this stress to the regulation of growth in developing organs, our results confirm and extend the mechanical feedback hypothesis.

Leaf growth is conformal.

Growth pattern dynamics lie at the heart of morphogenesis. Here, we investigate the growth of plant leaves. We compute the conformal transformation that maps the contour of a leaf at a given stage onto the contour of the same leaf at a later stage. Based on the mapping we predict the local displacement field in the leaf blade and find it to agree with the experimentally measured displacement field to 92%. This approach is applicable to any two-dimensional system with locally isotropic growth, enabling the deduction of the whole growth field just from observation of the tissue contour.

Prediction, dynamics, and visualization of antigenic phenotypes of seasonal influenza viruses.

Human seasonal influenza viruses evolve rapidly, enabling the virus population to evade immunity and reinfect previously infected individuals. Antigenic properties are largely determined by the surface glycoprotein hemagglutinin (HA), and amino acid substitutions at exposed epitope sites in HA mediate loss of recognition by antibodies. Here, we show that antigenic differences measured through serological assay data are well described by a sum of antigenic changes along the path connecting viruses in a phylogenetic tree. This mapping onto the tree allows prediction of antigenicity from HA sequence data alone. The mapping can further be used to make predictions about the makeup of the future A(H3N2) seasonal influenza virus population, and we compare predictions between models with serological and sequence data. To make timely model output readily available, we developed a web browser-based application that visualizes antigenic data on a continuously updated phylogeny.

Inferring epigenetic dynamics from kin correlations.

Populations of isogenic embryonic stem cells or clonal bacteria often exhibit extensive phenotypic heterogeneity that arises from intrinsic stochastic dynamics of cells. The phenotypic state of a cell can be transmitted epigenetically in cell division, leading to correlations in the states of cells related by descent. The extent of these correlations is determined by the rates of transitions between the phenotypic states. Therefore, a snapshot of the phenotypes of a collection of cells with known genealogical structure contains information on phenotypic dynamics. Here, we use a model of phenotypic dynamics on a genealogical tree to define an inference method that allows extraction of an approximate probabilistic description of the dynamics from observed phenotype correlations as a function of the degree of kinship. The approach is tested and validated on the example of Pyoverdine dynamics in Pseudomonas aeruginosa colonies. Interestingly, we find that correlations among pairs and triples of distant relatives have a simple but nontrivial structure indicating that observed phenotypic dynamics on the genealogical tree is approximately conformal--a symmetry characteristic of critical behavior in physical systems. The proposed inference method is sufficiently general to be applied in any system where lineage information is available.

Predicting evolution from the shape of genealogical trees.

Given a sample of genome sequences from an asexual population, can one predict its evolutionary future? Here we demonstrate that the branching patterns of reconstructed genealogical trees contains information about the relative fitness of the sampled sequences and that this information can be used to predict successful strains. Our approach is based on the assumption that evolution proceeds by accumulation of small effect mutations, does not require species specific input and can be applied to any asexual population under persistent selection pressure. We demonstrate its performance using historical data on seasonal influenza A/H3N2 virus. We predict the progenitor lineage of the upcoming influenza season with near optimal performance in 30% of cases and make informative predictions in 16 out of 19 years. Beyond providing a tool for prediction, our ability to make informative predictions implies persistent fitness variation among circulating influenza A/H3N2 viruses.

How to infer relative fitness from a sample of genomic sequences.

Mounting evidence suggests that natural populations can harbor extensive fitness diversity with numerous genomic loci under selection. It is also known that genealogical trees for populations under selection are quantifiably different from those expected under neutral evolution and described statistically by Kingman's coalescent. While differences in the statistical structure of genealogies have long been used as a test for the presence of selection, the full extent of the information that they contain has not been exploited. Here we demonstrate that the shape of the reconstructed genealogical tree for a moderately large number of random genomic samples taken from a fitness diverse, but otherwise unstructured, asexual population can be used to predict the relative fitness of individuals within the sample. To achieve this we define a heuristic algorithm, which we test in silico, using simulations of a Wright-Fisher model for a realistic range of mutation rates and selection strength. Our inferred fitness ranking is based on a linear discriminator that identifies rapidly coalescing lineages in the reconstructed tree. Inferred fitness ranking correlates strongly with actual fitness, with a genome in the top 10% ranked being in the top 20% fittest with false discovery rate of 0.1-0.3, depending on the mutation/selection parameters. The ranking also enables us to predict the genotypes that future populations inherit from the present one. While the inference accuracy increases monotonically with sample size, samples of 200 nearly saturate the performance. We propose that our approach can be used for inferring relative fitness of genomes obtained in single-cell sequencing of tumors and in monitoring viral outbreaks.

Collective polarization model for gradient sensing via Dachsous-Fat intercellular signaling.

Dachsous-Fat signaling via the Hippo pathway influences proliferation during Drosophila development, and some of its mammalian homologs are tumor suppressors, highlighting its role as a universal growth regulator. The Fat/Hippo pathway responds to morphogen gradients and influences the in-plane polarization of cells and orientation of divisions, linking growth with tissue patterning. Remarkably, the Fat pathway transduces a growth signal through the polarization of transmembrane complexes that responds to both morphogen level and gradient. Dissection of these complex phenotypes requires a quantitative model that provides a systematic characterization of the pathway. In the absence of detailed knowledge of molecular interactions, we take a phenomenological approach that considers a broad class of simple models, which are sufficiently constrained by observations to enable insight into possible mechanisms. We predict two modes of local/cooperative interactions among Fat-Dachsous complexes, which are necessary for the collective polarization of tissues and enhanced sensitivity to weak gradients. Collective polarization convolves level and gradient of input signals, reproducing known phenotypes while generating falsifiable predictions. Our construction of a simplified signal transduction map allows a generalization of the positional value model and emphasizes the important role intercellular interactions play in growth and patterning of tissues.

Coalescence and genetic diversity in sexual populations under selection.

In sexual populations, selection operates neither on the whole genome, which is repeatedly taken apart and reassembled by recombination, nor on individual alleles that are tightly linked to the chromosomal neighborhood. The resulting interference between linked alleles reduces the efficiency of selection and distorts patterns of genetic diversity. Inference of evolutionary history from diversity shaped by linked selection requires an understanding of these patterns. Here, we present a simple but powerful scaling analysis identifying the unit of selection as the genomic "linkage block" with a characteristic length, , determined in a self-consistent manner by the condition that the rate of recombination within the block is comparable to the fitness differences between different alleles of the block. We find that an asexual model with the strength of selection tuned to that of the linkage block provides an excellent description of genetic diversity and the site frequency spectra compared with computer simulations. This linkage block approximation is accurate for the entire spectrum of strength of selection and is particularly powerful in scenarios with many weakly selected loci. The latter limit allows us to characterize coalescence, genetic diversity, and the speed of adaptation in the infinitesimal model of quantitative genetics.