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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Collective and single cell behavior in epithelial contact inhibition.

Control of cell proliferation is a fundamental aspect of tissue physiology central to morphogenesis, wound healing, and cancer. Although many of the molecular genetic factors are now known, the system level regulation of growth is still poorly understood. A simple form of inhibition of cell proliferation is encountered in vitro in normally differentiating epithelial cell cultures and is known as "contact inhibition." The study presented here provides a quantitative characterization of contact inhibition dynamics on tissue-wide and single cell levels. Using long-term tracking of cultured Madin-Darby canine kidney cells we demonstrate that inhibition of cell division in a confluent monolayer follows inhibition of cell motility and sets in when mechanical constraint on local expansion causes divisions to reduce cell area. We quantify cell motility and cell cycle statistics in the low density confluent regime and their change across the transition to epithelial morphology which occurs with increasing cell density. We then study the dynamics of cell area distribution arising through reductive division, determine the average mitotic rate as a function of cell size, and demonstrate that complete arrest of mitosis occurs when cell area falls below a critical value. We also present a simple computational model of growth mechanics which captures all aspects of the observed behavior. Our measurements and analysis show that contact inhibition is a consequence of mechanical interaction and constraint rather than interfacial contact alone, and define quantitative phenotypes that can guide future studies of molecular mechanisms underlying contact inhibition.

Correlated evolution of nearby residues in Drosophilid proteins.

Here we investigate the correlations between coding sequence substitutions as a function of their separation along the protein sequence. We consider both substitutions between the reference genomes of several Drosophilids as well as polymorphisms in a population sample of Zimbabwean Drosophila melanogaster. We find that amino acid substitutions are "clustered" along the protein sequence, that is, the frequency of additional substitutions is strongly enhanced within ≈10 residues of a first such substitution. No such clustering is observed for synonymous substitutions, supporting a "correlation length" associated with selection on proteins as the causative mechanism. Clustering is stronger between substitutions that arose in the same lineage than it is between substitutions that arose in different lineages. We consider several possible origins of clustering, concluding that epistasis (interactions between amino acids within a protein that affect function) and positional heterogeneity in the strength of purifying selection are primarily responsible. The role of epistasis is directly supported by the tendency of nearby substitutions that arose on the same lineage to preserve the total charge of the residues within the correlation length and by the preferential cosegregation of neighboring derived alleles in our population sample. We interpret the observed length scale of clustering as a statistical reflection of the functional locality (or modularity) of proteins: amino acids that are near each other on the protein backbone are more likely to contribute to, and collaborate toward, a common subfunction.

A dynamical model of ommatidial crystal formation.

The crystalline photoreceptor lattice in the Drosophila eye is a paradigm for pattern formation during development. During eye development, activation of proneural genes at a moving front adds new columns to a regular lattice of R8 photoreceptors. We present a mathematical model of the governing activator-inhibitor system, which indicates that the dynamics of positive induction play a central role in the selection of certain cells as R8s. The "switch and template" patterning mechanism we observe is mathematically very different from the well-known Turing instability. Unlike a standard lateral inhibition model, our picture implies that R8s are defined before the appearance of the complete group of proneural cells. The model reproduces the full time course of proneural gene expression and accounts for specific features of the refinement of proneural groups that had resisted explanation. It moreover predicts that perturbing the normal template can lead to eyes containing stripes of R8 cells. We observed these stripes experimentally after manipulation of the Notch and scabrous genes. Our results suggest an alternative to the generally assumed mode of operation for lateral inhibition during development; more generally, they hint at a broader role for bistable switches in the initial establishment of patterns as well as in their maintenance.

Mechanical stress inference for two dimensional cell arrays.

Many morphogenetic processes involve mechanical rearrangements of epithelial tissues that are driven by precisely regulated cytoskeletal forces and cell adhesion. The mechanical state of the cell and intercellular adhesion are not only the targets of regulation, but are themselves the likely signals that coordinate developmental process. Yet, because it is difficult to directly measure mechanical stress in vivo on sub-cellular scale, little is understood about the role of mechanics in development. Here we present an alternative approach which takes advantage of the recent progress in live imaging of morphogenetic processes and uses computational analysis of high resolution images of epithelial tissues to infer relative magnitude of forces acting within and between cells. We model intracellular stress in terms of bulk pressure and interfacial tension, allowing these parameters to vary from cell to cell and from interface to interface. Assuming that epithelial cell layers are close to mechanical equilibrium, we use the observed geometry of the two dimensional cell array to infer interfacial tensions and intracellular pressures. Here we present the mathematical formulation of the proposed Mechanical Inverse method and apply it to the analysis of epithelial cell layers observed at the onset of ventral furrow formation in the Drosophila embryo and in the process of hair-cell determination in the avian cochlea. The analysis reveals mechanical anisotropy in the former process and mechanical heterogeneity, correlated with cell differentiation, in the latter process. The proposed method opens a way for quantitative and detailed experimental tests of models of cell and tissue mechanics.

'Infotaxis' as a strategy for searching without gradients.

Chemotactic bacteria rely on local concentration gradients to guide them towards the source of a nutrient. Such local cues pointing towards the location of the source are not always available at macroscopic scales because mixing in a flowing medium breaks up regions of high concentration into random and disconnected patches. Thus, animals sensing odours in air or water detect them only intermittently as patches sweep by on the wind or currents. A macroscopic searcher must devise a strategy of movement based on sporadic cues and partial information. Here we propose a search algorithm, which we call 'infotaxis', designed to work under such conditions. Any search process can be thought of as acquisition of information on source location; for infotaxis, information plays a role similar to concentration in chemotaxis. The infotaxis strategy locally maximizes the expected rate of information gain. We demonstrate its efficiency using a computational model of odour plume propagation and experimental data on mixing flows. Infotactic trajectories feature 'zigzagging' and 'casting' paths similar to those observed in the flight of moths. The proposed search algorithm is relevant to the design of olfactory robots, but the general idea of infotaxis can be applied more broadly in the context of searching with sparse information.

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.

Competition between recombination and epistasis can cause a transition from allele to genotype selection.

Biochemical and regulatory interactions central to biological networks are expected to cause extensive genetic interactions or epistasis affecting the heritability of complex traits and the distribution of genotypes in populations. However, the inference of epistasis from the observed phenotype-genotype correlation is impeded by statistical difficulties, while the theoretical understanding of the effects of epistasis remains limited, in turn limiting our ability to interpret data. Of particular interest is the biologically relevant situation of numerous interacting genetic loci with small individual contributions to fitness. Here, we present a computational model of selection dynamics involving many epistatic loci in a recombining population. We demonstrate that a large number of polymorphic interacting loci can, despite frequent recombination, exhibit cooperative behavior that locks alleles into favorable genotypes leading to a population consisting of a set of competing clones. When the recombination rate exceeds a certain critical value that depends on the strength of epistasis, this "genotype selection" regime disappears in an abrupt transition, giving way to "allele selection"--the regime where different loci are only weakly correlated as expected in sexually reproducing populations. We show that large populations attain highest fitness at a recombination rate just below critical. Clustering of interacting sets of genes on a chromosome leads to the emergence of an intermediate regime, where blocks of cooperating alleles lock into genetic modules. These haplotype blocks disappear in a second transition to pure allele selection. Our results demonstrate that the collective effect of many weak epistatic interactions can have dramatic effects on the population structure.

Emergent gene order in a model of modular polyketide synthases.

Polyketides are a class of biologically active heteropolymers produced by assembly line-like multiprotein complexes of modular polyketide synthases (PKS). The polyketide product is encoded in the order of the PKS proteins in the assembly line, suggesting that polyketide diversity derives from combinatorial rearrangement of these PKS complexes. Remarkably, the order of PKS genes on the chromosome follows the order of PKS proteins in the assembly line: This fact is commonly referred to as "collinearity". Here we propose an evolutionary origin for collinearity and demonstrate the mechanism by using a computational model of PKS evolution in a population. Assuming continuous evolutionary pressure for novel polyketides, and that new polyketide pathways are formed by horizontal transfer/recombination of PKS-encoding DNA, we demonstrate the existence of a broad range of parameters for which collinearity emerges spontaneously. Collinearity confers no fitness advantage in our model; it is established and maintained through a "secondary selection" mechanism, as a trait which increases the probability of forming long, novel PKS complexes through recombination. Consequently, collinearity hitchhikes on the successful genotypes which periodically sweep through the evolving population. In addition to computer simulation of a simplified model of PKS evolution, we provide a mathematical framework describing the secondary selection mechanism, which generalizes beyond the context of the present model.

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.