The many bits of positional information

Information Processing in Biochemical Networks

Living systems are characterized by controlled flows of matter, energy, and information. While the biophysics community has productively engaged with the first two, addressing information flows has been more challenging, with some scattered success in evolutionary theory and a more coherent track record in neuroscience. Nevertheless, interdisciplinary work of the past two decades at the interface of biophysics, quantitative biology, and engineering has led to an emerging mathematical language for describing information flows at the molecular scale. This is where the central processes of life unfold: from detection and transduction of environmental signals to the readout or copying of genetic information and the triggering of adaptive cellular responses. Such processes are coordinated by complex biochemical reaction networks that operate at room temperature, are out of equilibrium, and use low copy numbers of diverse molecular species with limited interaction specificity. Here we review how flows of information through biochemical networks can be formalized using information-theoretic quantities, quantified from data, and computed within various modeling frameworks. Optimization of information flows is presented as a candidate design principle that navigates the relevant time, energy, crosstalk, and metabolic constraints to predict reliable cellular signaling and gene regulation architectures built of individually noisy components.

Limits to positional information in boundary-driven systems

Chemical gradients can be used by a particle to determine its position. This positional information is of crucial importance, for example, in developmental biology in the formation of patterns in an embryo. The central goal of this paper is to study the fundamental physical limits on how much positional information can be stored inside a system. To achieve this, we study positional information for boundary-driven systems, and derive, in the near-equilibrium regime, a universal expression involving only the chemical potential and density gradients of the system. We also conjecture that this expression serves as an upper bound on the positional information of boundary-driven systems beyond linear response. To support this claim, we test it on a broad range of solvable boundary-driven systems.

Conservation of symmetry breaking at the level of chromatin accessibility between fly species with unrelated anterior determinants

Establishing the anterior-posterior body axis is a fundamental process during embryogenesis, and the fruit fly, Drosophila melanogaster, provides one of the best-known case studies of this process. In Drosophila, localized mRNA of bicoid serves as anterior determinant (AD). Bicoid engages in a concentration-dependent competition with nucleosomes and initiates symmetry-breaking along the AP axis by promoting chromatin accessibility at the loci of transcription factor (TF) genes that are expressed in the anterior of the embryo. However, ADs of other fly species are unrelated and structurally distinct, and little is known about how they function. We addressed this question in the moth fly, Clogmia albipunctata, in which a maternally expressed transcript isoform of the pair-rule segmentation gene odd-paired is localized in the anterior egg and has been co-opted as AD. We provide a de novo assembly and annotation of the Clogmia genome and describe how knockdown of zelda and maternal odd-paired transcript affect chromatin accessibility and expression of TF-encoding loci. The results of these experiments suggest direct roles of Cal-Zld in opening and closing chromatin during nuclear cleavage cycles and show that Clogmia's maternal odd-paired activity promotes chromatin accessibility and anterior expression during the early phase of zygotic genome activation at Clogmia's homeobrain and sloppy-paired loci. We conclude that unrelated dipteran ADs initiate anterior-posterior axis-specification at the level of enhancer accessibility and that homeobrain and sloppy-paired homologs may serve a more widely conserved role in the initiation of anterior pattern formation given their early anterior expression and function in head development in other insects.

Deriving a genetic regulatory network from an optimization principle

Many biological systems operate near the physical limits to their performance, suggesting that aspects of their behavior and underlying mechanisms could be derived from optimization principles. However, such principles have often been applied only in simplified models. Here, we explore a detailed mechanistic model of the gap gene network in the Drosophila embryo, optimizing its 50+ parameters to maximize the information that gene expression levels provide about nuclear positions. This optimization is conducted under realistic constraints, such as limits on the number of available molecules. Remarkably, the optimal networks we derive closely match the architecture and spatial gene expression profiles observed in the real organism. Our framework quantifies the tradeoffs involved in maximizing functional performance and allows for the exploration of alternative network configurations, addressing the question of which features are necessary and which are contingent. Our results suggest that multiple solutions to the optimization problem might exist across closely related organisms, offering insights into the evolution of gene regulatory networks.

Dynamics of positional information in the vertebrate neural tube

In developing embryos, cells acquire distinct identities depending on their position in a tissue. Secreted signalling molecules, known as morphogens, act as long-range cues to provide the spatial information that controls these cell fate decisions. In several tissues, both the level and the duration of morphogen signalling appear to be important for determining cell fates. This is the case in the forming vertebrate nervous system where antiparallel morphogen gradients pattern the dorsal-ventral axis by partitioning the tissue into sharply delineated domains of molecularly distinct neural progenitors. How information in the gradients is decoded to generate precisely positioned boundaries of gene expression remains an open question. Here, we adopt tools from information theory to quantify the positional information in the neural tube and investigate how temporal changes in signalling could influence positional precision. The results reveal that the use of signalling dynamics, as well as the signalling level, substantially increases the precision possible for the estimation of position from morphogen gradients. This analysis links the dynamics of opposing morphogen gradients with precise pattern formation and provides an explanation for why time is used to impart positional information.

Stable developmental patterns of gene expression without morphogen gradients

Gene expression patterns in developing organisms are established by groups of cross-regulating target genes that are driven by morphogen gradients. As development progresses, morphogen activity is reduced, leaving the emergent pattern without stabilizing positional cues and at risk of rapid deterioration due to the inherently noisy biochemical processes at the cellular level. But remarkably, gene expression patterns remain spatially stable and reproducible over long developmental time spans in many biological systems. Here we combine spatial-stochastic simulations with an enhanced sampling method (Non-Stationary Forward Flux Sampling) and a recently developed stability theory to address how spatiotemporal integrity of a gene expression pattern is maintained in developing tissue lacking morphogen gradients. Using a minimal embryo model consisting of spatially coupled biochemical reactor volumes, we study a prototypical stripe pattern in which weak cross-repression between nearest neighbor expression domains alternates with strong repression between next-nearest neighbor domains, inspired by the gap gene system in the Drosophila embryo. We find that tuning of the weak repressive interactions to an optimal level can prolong stability of the expression patterns by orders of magnitude, enabling stable patterns over developmentally relevant times in the absence of morphogen gradients. The optimal parameter regime found in simulations of the embryo model closely agrees with the predictions of our coarse-grained stability theory. To elucidate the origin of stability, we analyze a reduced phase space defined by two measures of pattern asymmetry. We find that in the optimal regime, intact patterns are protected via restoring forces that counteract random perturbations and give rise to a metastable basin. Together, our results demonstrate that metastable attractors can emerge as a property of stochastic gene expression patterns even without system-wide positional cues, provided that the gene regulatory interactions shaping the pattern are optimally tuned.

Scale invariance in early embryonic development

The expression of a few key genes determines the body plan of the fruit fly. We show that the spatial expression patterns for several of these genes scale precisely with embryo size. Discrete positional markers such as the peaks in striped patterns or the boundaries of expression domains have positions along the embryo's major axis proportional to embryo length, accurate to within 1%. Further, the information (in bits) that graded patterns of expression provide about a cell's position can be decomposed into information about fractional or scaled position and information about absolute position or embryo length; all information available is about scaled position, with [Formula: see text]2% error. These findings imply that the underlying genetic network's behavior exhibits scale invariance in a more precise mathematical sense. We argue that models that can explain this scale invariance also have a "zero mode" in the dynamics of gene expression, and this connects to observations on the spatial correlation of fluctuations in expression levels.

AI-powered simulation-based inference of a genuinely spatial-stochastic gene regulation model of early mouse embryogenesis

Understanding how multicellular organisms reliably orchestrate cell-fate decisions is a central challenge in developmental biology, particularly in early mammalian development, where tissue-level differentiation arises from seemingly cell-autonomous mechanisms. In this study, we present a multi-scale, spatial-stochastic simulation framework for mouse embryogenesis, focusing on inner cell mass (ICM) differentiation into epiblast (EPI) and primitive endoderm (PRE) at the blastocyst stage. Our framework models key regulatory and tissue-scale interactions in a biophysically realistic fashion, capturing the inherent stochasticity of intracellular gene expression and intercellular signaling, while efficiently simulating these processes by advancing event-driven simulation techniques. Leveraging the power of Simulation-Based Inference (SBI) through the AI-driven Sequential Neural Posterior Estimation (SNPE) algorithm, we conduct a large-scale Bayesian inferential analysis to identify parameter sets that faithfully reproduce experimentally observed features of ICM specification. Our results reveal mechanistic insights into how the combined action of autocrine and paracrine FGF4 signaling coordinates stochastic gene expression at the cellular scale to achieve robust and reproducible ICM patterning at the tissue scale. We further demonstrate that the ICM exhibits a specific time window of sensitivity to exogenous FGF4, enabling lineage proportions to be adjusted based on timing and dosage, thereby extending current experimental findings and providing quantitative predictions for both mutant and wild-type ICM systems. Notably, FGF4 signaling not only ensures correct EPI-PRE lineage proportions but also enhances ICM resilience to perturbations, reducing fate-proportioning errors by 10-20% compared to a purely cell-autonomous system. Additionally, we uncover a surprising role for variability in intracellular initial conditions, showing that high gene-expression heterogeneity can improve both the accuracy and precision of cell-fate proportioning, which remains robust when fewer than 25% of the ICM population experiences perturbed initial conditions. Our work offers a comprehensive, spatial-stochastic description of the biochemical processes driving ICM differentiation and identifies the necessary conditions for its robust unfolding. It also provides a framework for future exploration of similar spatial-stochastic systems in developmental biology.

The evolution of developmental biology through conceptual and technological revolutions

Developmental biology-the study of the processes by which cells, tissues, and organisms develop and change over time-has entered a new golden age. After the molecular genetics revolution in the 80s and 90s and the diversification of the field in the early 21st century, we have entered a phase when powerful technologies provide new approaches and open unexplored avenues. Progress in the field has been accelerated by advances in genomics, imaging, engineering, and computational biology and by emerging model systems ranging from tardigrades to organoids. We summarize how revolutionary technologies have led to remarkable progress in understanding animal development. We describe how classic questions in gene regulation, pattern formation, morphogenesis, organogenesis, and stem cell biology are being revisited. We discuss the connections of development with evolution, self-organization, metabolism, time, and ecology. We speculate how developmental biology might evolve in an era of synthetic biology, artificial intelligence, and human engineering.

Information content and optimization of self-organized developmental systems

A key feature of many developmental systems is their ability to self-organize spatial patterns of functionally distinct cell fates. To ensure proper biological function, such patterns must be established reproducibly, by controlling and even harnessing intrinsic and extrinsic fluctuations. While the relevant molecular processes are increasingly well understood, we lack a principled framework to quantify the performance of such stochastic self-organizing systems. To that end, we introduce an information-theoretic measure for self-organized fate specification during embryonic development. We show that the proposed measure assesses the total information content of fate patterns and decomposes it into interpretable contributions corresponding to the positional and correlational information. By optimizing the proposed measure, our framework provides a normative theory for developmental circuits, which we demonstrate on lateral inhibition, cell type proportioning, and reaction-diffusion models of self-organization. This paves a way toward a classification of developmental systems based on a common information-theoretic language, thereby organizing the zoo of implicated chemical and mechanical signaling processes.

Responses of organ precursors to correct and incorrect inductive signals

During embryonic development, the inductive molecules produced by local origins normally arrive at their target tissues in a nondirectional, diffusion manner. The target organ precursor cells must correctly interpret these inductive signals to ensure proper specification/differentiation, which is dependent on two prerequisites: (i) obtaining cell-intrinsic competence; and (ii) receiving correct inductive signals while resisting incorrect ones. Gain of intrinsic competence could avoid a large number of misinductions because the incompetent cells are nonresponsive to inductive signals. However, in cases of different precursor cells with similar competence and located in close proximity, resistance to incorrect inductive signals is essential for accurate determination of cell fate. Here we outline the mechanisms of how organ precursors respond to correct and incorrect inductive signals.

Cellular remodeling and JAK inhibition promote zygotic gene expression in the Ciona germline

Transcription control is a major determinant of cell fate decisions in somatic tissues. By contrast, early germline fate specification in numerous vertebrate and invertebrate species relies extensively on RNA-level regulation, exerted on asymmetrically inherited maternal supplies, with little-to-no zygotic transcription. However delayed, a maternal-to-zygotic transition is nevertheless poised to complete the deployment of pre-gametic programs in the germline. Here, we focus on early germline specification in the tunicate Ciona to study zygotic genome activation. We first demonstrate that a peculiar cellular remodeling event excludes localized postplasmic Pem-1 mRNA, which encodes the general inhibitor of transcription. Subsequently, zygotic transcription begins in Pem-1-negative primordial germ cells (PGCs), as revealed by histochemical detection of elongating RNA Polymerase II, and nascent Mef2 transcripts. In addition, we uncover a provisional antagonism between JAK and MEK/BMPRI/GSK3 signaling, which controls the onset of zygotic gene expression, following cellular remodeling of PGCs. We propose a 2-step model for the onset of zygotic transcription in the Ciona germline and discuss the significance of germ plasm dislocation and remodeling in the context of developmental fate specification.

Automatic design of gene regulatory mechanisms for spatial pattern formation

Gene regulatory mechanisms (GRMs) control the formation of spatial and temporal expression patterns that can serve as regulatory signals for the development of complex shapes. Synthetic developmental biology aims to engineer such genetic circuits for understanding and producing desired multicellular spatial patterns. However, designing synthetic GRMs for complex, multi-dimensional spatial patterns is a current challenge due to the nonlinear interactions and feedback loops in genetic circuits. Here we present a methodology to automatically design GRMs that can produce any given two-dimensional spatial pattern. The proposed approach uses two orthogonal morphogen gradients acting as positional information signals in a multicellular tissue area or culture, which constitutes a continuous field of engineered cells implementing the same designed GRM. To efficiently design both the circuit network and the interaction mechanisms-including the number of genes necessary for the formation of the target spatial pattern-we developed an automated algorithm based on high-performance evolutionary computation. The tolerance of the algorithm can be configured to design GRMs that are either simple to produce approximate patterns or complex to produce precise patterns. We demonstrate the approach by automatically designing GRMs that can produce a diverse set of synthetic spatial expression patterns by interpreting just two orthogonal morphogen gradients. The proposed framework offers a versatile approach to systematically design and discover complex genetic circuits producing spatial patterns.

Finding the last bits of positional information

In a developing embryo, information about the position of cells is encoded in the concentrations of morphogen molecules. In the fruit fly, the local concentrations of just a handful of proteins encoded by the gap genes are sufficient to specify position with a precision comparable to the spacing between cells along the anterior-posterior axis. This matches the precision of downstream events such as the striped patterns of expression in the pair-rule genes, but is not quite sufficient to define unique identities for individual cells. We demonstrate theoretically that this information gap can be bridged if positional errors are spatially correlated, with correlation lengths ~ 20% of the embryo length. We then show experimentally that these correlations are present, with the required strength, in the fluctuating positions of the pair-rule stripes, and this can be traced back to the gap genes. Taking account of these correlations, the available information matches the information needed for unique cellular specification, within error bars of ~ 2%. These observation support a precisionist view of information flow through the underlying genetic networks, in which accurate signals are available from the start and preserved as they are transformed into the final spatial patterns.

Scale invariance in early embryonic development

The body plan of the fruit fly is determined by the expression of just a handful of genes. We show that the spatial patterns of expression for several of these genes scale precisely with the size of the embryo. Concretely, discrete positional markers such as the peaks in striped patterns have absolute positions along the anterior-posterior axis that are proportional to embryo length, with better than 1% accuracy. Further, the information (in bits) that graded patterns of expression provide about position can be decomposed into information about fractional or scaled position and information about absolute position or embryo length; all of the available information is about scaled position, again with ~ 1% accuracy. These observations suggest that the underlying genetic network exhibits scale invariance in a deeper mathematical sense. Taking this mathematical statement seriously requires that the network dynamics have a zero mode, which connects to many other observations on this system.

Automatic design of gene regulatory mechanisms for spatial pattern formation

Synthetic developmental biology aims to engineer gene regulatory mechanisms (GRMs) for understanding and producing desired multicellular patterns and shapes. However, designing GRMs for spatial patterns is a current challenge due to the nonlinear interactions and feedback loops in genetic circuits. Here we present a methodology to automatically design GRMs that can produce any given spatial pattern. The proposed approach uses two orthogonal morphogen gradients acting as positional information signals in a multicellular tissue area or culture, which constitutes a continuous field of engineered cells implementing the same designed GRM. To efficiently design both the circuit network and the interaction mechanisms-including the number of genes necessary for the formation of the target pattern-we developed an automated algorithm based on high-performance evolutionary computation. The tolerance of the algorithm can be configured to design GRMs that are either simple to produce approximate patterns or complex to produce precise patterns. We demonstrate the approach by automatically designing GRMs that can produce a diverse set of synthetic spatial expression patterns by interpreting just two orthogonal morphogen gradients. The proposed framework offers a versatile approach to systematically design and discover pattern-producing genetic circuits.

A bistable autoregulatory module in the developing embryo commits cells to binary expression fates

Bistable autoactivation has been proposed as a mechanism for cells to adopt binary fates during embryonic development. However, it is unclear whether the autoactivating modules found within developmental gene regulatory networks are bistable, unless their parameters are quantitatively determined. Here, we combine in vivo live imaging with mathematical modeling to dissect the binary cell fate dynamics of the fruit fly pair-rule gene fushi tarazu (ftz), which is regulated by two known enhancers: the early (non-autoregulating) element and the autoregulatory element. Live imaging of transcription and protein concentration in the blastoderm revealed that binary Ftz fates are achieved as Ftz expression rapidly transitions from being dictated by the early element to the autoregulatory element. Moreover, we discovered that Ftz concentration alone is insufficient to activate the autoregulatory element, and that this element only becomes responsive to Ftz at a prescribed developmental time. Based on these observations, we developed a dynamical systems model and quantitated its kinetic parameters directly from experimental measurements. Our model demonstrated that the ftz autoregulatory module is indeed bistable and that the early element transiently establishes the content of the binary cell fate decision to which the autoregulatory module then commits. Further in silico analysis revealed that the autoregulatory element locks the Ftz fate quickly, within 35 min of exposure to the transient signal of the early element. Overall, our work confirms the widely held hypothesis that autoregulation can establish developmental fates through bistability and, most importantly, provides a framework for the quantitative dissection of cellular decision-making.

Integrative approaches to study enhancer-promoter communication

The spatiotemporal control of gene expression in complex multicellular organisms relies on noncoding regulatory sequences such as enhancers, which activate transcription of target genes often over large genomic distances. Despite the advances in the identification and characterization of enhancers, the principles and mechanisms by which enhancers select and control their target genes remain largely unknown. Here, we review recent interdisciplinary and quantitative approaches based on emerging techniques that aim to address open questions in the field, notably how regulatory information is encoded in the DNA sequence, how this information is transferred from enhancers to promoters, and how these processes are regulated in time.

Self-organized collective cell behaviors as design principles for synthetic developmental biology

Over the past two decades, molecular cell biology has graduated from a mostly analytic science to one with substantial synthetic capability. This success is built on a deep understanding of the structure and function of biomolecules and molecular mechanisms. For synthetic biology to achieve similar success at the scale of tissues and organs, an equally deep understanding of the principles of development is required. Here, we review some of the central concepts and recent progress in tissue patterning, morphogenesis and collective cell migration and discuss their value for synthetic developmental biology, emphasizing in particular the power of (guided) self-organization and the role of theoretical advances in making developmental insights applicable in synthesis.

Early radial positional information in the cochlea is optimized by a precise linear BMP gradient and enhanced by SOX2

Positional information encoded in signaling molecules is essential for early patterning in the prosensory domain of the developing cochlea. The sensory epithelium, the organ of Corti, contains an exquisite repeating pattern of hair cells and supporting cells. This requires precision in the morphogen signals that set the initial radial compartment boundaries, but this has not been investigated. To measure gradient formation and morphogenetic precision in developing cochlea, we developed a quantitative image analysis procedure measuring SOX2 and pSMAD1/5/9 profiles in mouse embryos at embryonic day (E)12.5, E13.5, and E14.5. Intriguingly, we found that the pSMAD1/5/9 profile forms a linear gradient up to the medial ~ 75% of the PSD from the pSMAD1/5/9 peak in the lateral edge during E12.5 and E13.5. This is a surprising activity readout for a diffusive BMP4 ligand secreted from a tightly constrained lateral region since morphogens typically form exponential or power-law gradient shapes. This is meaningful for gradient interpretation because while linear profiles offer the theoretically highest information content and distributed precision for patterning, a linear morphogen gradient has not yet been observed. Furthermore, this is unique to the cochlear epithelium as the pSMAD1/5/9 gradient is exponential in the surrounding mesenchyme. In addition to the information-optimized linear profile, we found that while pSMAD1/5/9 is stable during this timeframe, an accompanying gradient of SOX2 shifts dynamically. Last, through joint decoding maps of pSMAD1/5/9 and SOX2, we see that there is a high-fidelity mapping between signaling activity and position in the regions that will become Kölliker's organ and the organ of Corti. Mapping is ambiguous in the prosensory domain precursory to the outer sulcus. Altogether, this research provides new insights into the precision of early morphogenetic patterning cues in the radial cochlea prosensory domain.

Cellular compartmentalisation and receptor promiscuity as a strategy for accurate and robust inference of position during morphogenesis

Precise spatial patterning of cell fate during morphogenesis requires accurate inference of cellular position. In making such inferences from morphogen profiles, cells must contend with inherent stochasticity in morphogen production, transport, sensing and signalling. Motivated by the multitude of signalling mechanisms in various developmental contexts, we show how cells may utilise multiple tiers of processing (compartmentalisation) and parallel branches (multiple receptor types), together with feedback control, to bring about fidelity in morphogenetic decoding of their positions within a developing tissue. By simultaneously deploying specific and nonspecific receptors, cells achieve a more accurate and robust inference. We explore these ideas in the patterning of Drosophila melanogaster wing imaginal disc by Wingless morphogen signalling, where multiple endocytic pathways participate in decoding the morphogen gradient. The geometry of the inference landscape in the high dimensional space of parameters provides a measure for robustness and delineates stiff and sloppy directions. This distributed information processing at the scale of the cell highlights how local cell autonomous control facilitates global tissue scale design.

Stability of Pattern Formation in Systems with Dynamic Source Regions

We explain the principles of gene expression pattern stabilization in systems of interacting, diffusible morphogens, with dynamically established source regions. Using a reaction-diffusion model with a step-function production term, we identify the phase transition between low-precision indeterminate patterning and the phase in which a traveling, well-defined contact zone between two domains is formed. Our model analytically explains single- and two-gene domain dynamics and provides pattern stability conditions for all possible two-gene regulatory network motifs.

Orchestration of tissue shape changes and gene expression patterns in development

In development, tissue shape changes and gene expression patterns give rise to morphogenesis. Understanding tissue shape changes requires the analysis of mechanical properties of the tissue such as tissue rigidity, cell influx from neighboring tissues, cell shape changes and cell proliferation. Local and global gene expression patterns can be influenced by neighbor exchange and tissue shape changes. Here we review recent studies on the mechanisms for tissue elongation and its influences on dynamic gene expression patterns by focusing on vertebrate somitogenesis. We first introduce mechanical and biochemical properties of the segmenting tissue that drive tissue elongation. Then, we discuss patterning in the presence of cell mixing, scaling of signaling gradients, and dynamic phase waves of rhythmic gene expression under tissue shape changes. We also highlight the importance of theoretical approaches to address the relation between tissue shape changes and patterning.

Scaling dictates the decoder structure

Despite fluctuations in embryo size within a species, the spatial gene expression pattern and hence the embryonic structure can nonetheless maintain the correct proportion to the embryo size. This is known as the scaling phenomenon. For morphogen-induced patterning of gene expression, the positional information encoded in the local morphogen concentrations is decoded by the downstream genetic network (the decoder). In this paper, we show that the requirement of scaling sets severe constraints on the geometric structure of such a local decoder, which in turn enables deduction of mutants' behavior and extraction of regulation information without going into any molecular details. We demonstrate that the Drosophila gap gene system achieves scaling in the way consistent with our theory-the decoder geometry required by scaling correctly accounts for the observed gap gene expression pattern in nearly all maternal morphogen mutants. Furthermore, the regulation logic and the coding/decoding strategy of the gap gene system can also be revealed from the decoder geometry. Our work provides a general theoretical framework for a large class of problems where scaling output is achieved by non-scaling inputs and a local decoder, as well as a unified understanding of scaling, mutants' behavior, and gene regulation for the Drosophila gap gene system.

Localization of signaling receptors maximizes cellular information acquisition in spatially structured natural environments

Cells in natural environments, such as tissue or soil, sense and respond to extracellular ligands with intricately structured and non-monotonic spatial distributions, sculpted by processes such as fluid flow and substrate adhesion. In this work, we show that spatial sensing and navigation can be optimized by adapting the spatial organization of signaling pathways to the spatial structure of the environment. We develop an information-theoretic framework for computing the optimal spatial organization of a sensing system for a given signaling environment. We find that receptor localization previously observed in cells maximizes information acquisition in simulated natural contexts, including tissue and soil. Specifically, information acquisition is maximized when receptors form localized patches at regions of maximal ligand concentration. Receptor localization extends naturally to produce a dynamic protocol for continuously redistributing signaling receptors, which when implemented using simple feedback, boosts cell navigation efficiency by 30-fold.

Latent space of a small genetic network: Geometry of dynamics and information

The high-dimensional character of most biological systems presents genuine challenges for modeling and prediction. Here we propose a neural network-based approach for dimensionality reduction and analysis of biological gene expression data, using, as a case study, a well-known genetic network in the early Drosophila embryo, the gap gene patterning system. We build an autoencoder compressing the dynamics of spatial gap gene expression into a two-dimensional (2D) latent map. The resulting 2D dynamics suggests an almost linear model, with a small bare set of essential interactions. Maternally defined spatial modes control gap genes positioning, without the classically assumed intricate set of repressive gap gene interactions. This, surprisingly, predicts minimal changes of neighboring gap domains when knocking out gap genes, consistent with previous observations. Latent space geometries in maternal mutants are also consistent with the existence of such spatial modes. Finally, we show how positional information is well defined and interpretable as a polar angle in latent space. Our work illustrates how optimization of small neural networks on medium-sized biological datasets is sufficiently informative to capture essential underlying mechanisms of network function.

Relationship between epithelial organization and morphogen interpretation

Despite molecular noise and genetic differences between individuals, developmental outcomes are remarkably constant. Decades of research has focused on the underlying mechanisms that ensure this precision and robustness. Recent quantifications of chemical gradients and epithelial cell shapes provide novel insights into the basis of precise development. In this review, we argue that these two aspects may be linked in epithelial morphogenesis.

Patterning, From Conifers to Consciousness: Turing's Theory and Order From Fluctuations

This is a brief account of Turing's ideas on biological pattern and the events that led to their wider acceptance by biologists as a valid way to investigate developmental pattern, and of the value of theory more generally in biology. Periodic patterns have played a key role in this process, especially 2D arrays of oriented stripes, which proved a disappointment in theoretical terms in the case of Drosophila segmentation, but a boost to theory as applied to skin patterns in fish and model chemical reactions. The concept of "order from fluctuations" is a key component of Turing's theory, wherein pattern arises by selective amplification of spatial components concealed in the random disorder of molecular and/or cellular processes. For biological examples, a crucial point from an analytical standpoint is knowing the nature of the fluctuations, where the amplifier resides, and the timescale over which selective amplification occurs. The answer clarifies the difference between "inelegant" examples such as Drosophila segmentation, which is perhaps better understood as a programmatic assembly process, and "elegant" ones expressible in equations like Turing's: that the fluctuations and selection process occur predominantly in evolutionary time for the former, but in real time for the latter, and likewise for error suppression, which for Drosophila is historical, in being lodged firmly in past evolutionary events. The prospects for a further extension of Turing's ideas to the complexities of brain development and consciousness is discussed, where a case can be made that it could well be in neuroscience that his ideas find their most important application.

Collective gradient sensing with limited positional information

Eukaryotic cells sense chemical gradients to decide where and when to move. Clusters of cells can sense gradients more accurately than individual cells by integrating measurements of the concentration made across the cluster. Is this gradient-sensing accuracy impeded when cells have limited knowledge of their position within the cluster, i.e., limited positional information? We apply maximum likelihood estimation to study gradient-sensing accuracy of a cluster of cells with finite positional information. If cells must estimate their location within the cluster, this lowers the accuracy of collective gradient sensing. We compare our results with a tug-of-war model where cells respond to the gradient by polarizing away from their neighbors without relying on their positional information. As the cell positional uncertainty increases, there is a trade-off where the tug-of-war model responds more accurately to the chemical gradient. However, for sufficiently large cell clusters or sufficiently shallow chemical gradients, the tug-of-war model will always be suboptimal to one that integrates information from all cells, even if positional uncertainty is high.

Precision of morphogen gradients in neural tube development

Morphogen gradients encode positional information during development. How high patterning precision is achieved despite natural variation in both the morphogen gradients and in the readout process, is still largely elusive. Here, we show that the positional error of gradients in the mouse neural tube has previously been overestimated, and that the reported accuracy of the central progenitor domain boundaries in the mouse neural tube can be achieved with a single gradient, rather than requiring the simultaneous readout of opposing gradients. Consistently and independently, numerical simulations based on measured molecular noise levels likewise result in lower gradient variabilities than reported. Finally, we show that the patterning mechanism yields progenitor cell numbers with even greater precision than boundary positions, as gradient amplitude changes do not affect interior progenitor domain sizes. We conclude that single gradients can yield the observed developmental precision, which provides prospects for tissue engineering.

Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries formed during animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. The accuracy of boundary location is limited by the fluctuations of morphogen concentration that thresholds the expression level of target gene. Producing more morphogen molecules, which gives rise to smaller relative fluctuations, would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-depletion model of morphogen profile formation, the morphogen molecules synthesized from a local source display an exponentially decaying concentration profile with a characteristic length λ. Our theory suggests that in order to attain a precise profile with the minimal cost, λ should be roughly half the distance to the target boundary position from the source. Remarkably, we find that the profiles of morphogens that pattern the Drosophila embryo and wing imaginal disk are formed with nearly optimal λ. Our finding underscores the cost-effectiveness of precise morphogen profile formation in Drosophila development.

The early Drosophila embryo as a model system for quantitative biology

With the rise of new tools, from controlled genetic manipulations and optogenetics to improved microscopy, it is now possible to make clear, quantitative and reproducible measurements of biological processes. The humble fruit fly Drosophila melanogaster, with its ease of genetic manipulation combined with excellent imaging accessibility, has become a major model system for performing quantitative in vivo measurements. Such measurements are driving a new wave of interest from physicists and engineers, who are developing a range of testable dynamic models of active systems to understand fundamental biological processes. The reproducibility of the early Drosophila embryo has been crucial for understanding how biological systems are robust to unavoidable noise during development. Insights from quantitative in vivo experiments in the Drosophila embryo are having an impact on our understanding of critical biological processes, such as how cells make decisions and how complex tissue shape emerges. Here, to highlight the power of using Drosophila embryogenesis for quantitative biology, I focus on three main areas: (1) formation and robustness of morphogen gradients; (2) how gene regulatory networks ensure precise boundary formation; and (3) how mechanical interactions drive packing and tissue folding. I further discuss how such data has driven advances in modelling.