Positional information, positional error, and readout precision in morphogenesis: a mathematical framework

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.

Precise and scalable self-organization in mammalian pseudo-embryos

Gene expression is inherently noisy, posing a challenge to understanding how precise and reproducible patterns of gene expression emerge in mammals. Here we investigate this phenomenon using gastruloids, a three-dimensional in vitro model for early mammalian development. Our study reveals intrinsic reproducibility in the self-organization of gastruloids, encompassing growth dynamics and gene expression patterns. We observe a remarkable degree of control over gene expression along the main body axis, with pattern boundaries positioned with single-cell precision. Furthermore, as gastruloids grow, both their physical proportions and gene expression patterns scale proportionally with system size. Notably, these properties emerge spontaneously in self-organizing cell aggregates, distinct from many in vivo systems constrained by fixed boundary conditions. Our findings shed light on the intricacies of developmental precision, reproducibility and size scaling within a mammalian system, suggesting that these phenomena might constitute fundamental features of multicellularity.

Transcription factor clusters as information transfer agents

Deciphering how genes interpret information from the concentration of transcription factors (TFs) within the cell nucleus remains a fundamental question in gene regulation. Recent advancements have unveiled the heterogeneous distribution of TF molecules in the nucleus, posing challenges to the precise decoding of concentration signals. To explore this phenomenon, we employ high-resolution single-cell imaging of a fluorescently tagged TF protein, Bicoid, in living fly embryos. We show that accumulation of Bicoid in submicron clusters preserves the spatial information of the maternal Bicoid gradient, and that cluster intensity, size, and frequency offer remarkably precise spatial cues. We further discover that various known gene targets of Bicoid activation colocalize with clusters and that for the target gene Hunchback, this colocalization is dependent on its enhancer binding affinity. Modeling information transfer through these clusters suggests that clustering offers a more rapid sensing mechanism for global nuclear concentrations than freely diffusing TF molecules detected by simple enhancers.

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.

An archetype and scaling of developmental tissue dynamics across species

Morphometric studies have revealed the existence of simple geometric relationships among various animal shapes. However, we have little knowledge of the mathematical principles behind the morphogenetic dynamics that form the organ/body shapes of different species. Here, we address this issue by focusing on limb morphogenesis in Gallus gallus domesticus (chicken) and Xenopus laevis (African clawed frog). To compare the deformation dynamics between tissues with different sizes/shapes as well as their developmental rates, we introduce a species-specific rescaled spatial coordinate and a common clock necessary for cross-species synchronization of developmental times. We find that tissue dynamics are well conserved across species under this spacetime coordinate system, at least from the early stages of development through the phase when basic digit patterning is established. For this developmental period, we also reveal that the tissue dynamics of both species are mapped with each other through a time-variant linear transformation in real physical space, from which hypotheses on a species-independent archetype of tissue dynamics and morphogenetic scaling are proposed.

Functional analysis of the Drosophila eve locus in response to non-canonical combinations of gap gene expression levels

Transcription factor combinations play a key role in shaping cellular identity. However, the precise relationship between specific combinations and downstream effects remains elusive. Here, we investigate this relationship within the context of the Drosophila eve locus, which is controlled by gap genes. We measure spatiotemporal levels of four gap genes in heterozygous and homozygous gap mutant embryos and correlate them with the striped eve activity pattern. Although changes in gap gene expression extend beyond the manipulated gene, the spatial patterns of Eve expression closely mirror canonical activation levels in wild type. Interestingly, some combinations deviate from the wild-type repertoire but still drive eve activation. Although in homozygous mutants some Eve stripes exhibit partial penetrance, stripes consistently emerge at reproducible positions, even with varying gap gene levels. Our findings suggest a robust molecular canalization of cell fates in gap mutants and provide insights into the regulatory constraints governing multi-enhancer gene loci.

Optimal control of gene regulatory networks for morphogen-driven tissue patterning

The generation of distinct cell types in developing tissues depends on establishing spatial patterns of gene expression. Often, this is directed by spatially graded chemical signals-known as morphogens. In the "French Flag model," morphogen concentration instructs cells to acquire specific fates. How this mechanism produces timely and organized cell-fate decisions, despite the presence of changing morphogen levels, molecular noise, and individual variability, is unclear. Moreover, feedback is present at various levels in developing tissues, breaking the link between morphogen concentration, signaling activity, and position. Here, we develop an alternative framework using optimal control theory to tackle the problem of morphogen-driven patterning: intracellular signaling is derived as the control strategy that guides cells to the correct fate while minimizing a combination of signaling levels and time. This approach recovers experimentally observed properties of patterning strategies and offers insight into design principles that produce timely, precise, and reproducible morphogen patterning.

2D effects enhance precision of gradient-based tissue patterning

Robust embryonic development requires pattern formation with high spatial accuracy. In epithelial tissues that are patterned by morphogen gradients, the emerging patterns achieve levels of precision that have recently been explained by a simple one-dimensional reaction-diffusion model with kinetic noise. Here, we show that patterning precision is even greater if transverse diffusion effects are at play in such tissues. The positional error, a measure for spatial patterning accuracy, decreases in wider tissues but then saturates beyond a width of about ten cells. This demonstrates that the precision of gradient-based patterning in two- or higher-dimensional systems can be even greater than predicted by 1D models, and further attests to the potential of noisy morphogen gradients for high-precision tissue patterning.

Channel capacity of the ribosome

Translation is one of the most fundamental processes in the biological cell. Because of the central role that translation plays across all domains of life, the enzyme that carries out this process, the ribosome, is required to process information with high accuracy. This accuracy often approaches values near unity experimentally. In this paper, we model the ribosome as an information channel and demonstrate mathematically that this biological machine has information-processing capabilities that have not been recognized previously. In particular, we calculate bounds on the ribosome's theoretical Shannon capacity and numerically approximate this capacity. Finally, by incorporating estimates on the ribosome's operation time, we show that the ribosome operates at speeds safely below its capacity, allowing the ribosome to process information with an arbitrary degree of error. Our results show that the ribosome achieves a high accuracy in line with purely information-theoretic means.

Laying out the mammalian body plan to a T

In this issue of Developmental Cell, Schülle et al. address the mechanism that lays down the mammalian body plan and show that the first step in this process is associated with the activity of two T-box transcription factors whose interactions outline territories that will be further elaborated during gastrulation.

Embodied cognitive morphogenesis as a route to intelligent systems

The embryological view of development is that coordinated gene expression, cellular physics and migration provides the basis for phenotypic complexity. This stands in contrast with the prevailing view of embodied cognition, which claims that informational feedback between organisms and their environment is key to the emergence of intelligent behaviours. We aim to unite these two perspectives as embodied cognitive morphogenesis, in which morphogenetic symmetry breaking produces specialized organismal subsystems which serve as a substrate for the emergence of autonomous behaviours. As embodied cognitive morphogenesis produces fluctuating phenotypic asymmetry and the emergence of information processing subsystems, we observe three distinct properties: acquisition, generativity and transformation. Using a generic organismal agent, such properties are captured through models such as tensegrity networks, differentiation trees and embodied hypernetworks, providing a means to identify the context of various symmetry-breaking events in developmental time. Related concepts that help us define this phenotype further include concepts such as modularity, homeostasis and 4E (embodied, enactive, embedded and extended) cognition. We conclude by considering these autonomous developmental systems as a process called connectogenesis, connecting various parts of the emerged phenotype into an approach useful for the analysis of organisms and the design of bioinspired computational agents.

Microbial models of development: Inspiration for engineering self-assembled synthetic multicellularity

While the field of synthetic developmental biology has traditionally focused on the study of the rich developmental processes seen in metazoan systems, an attractive alternate source of inspiration comes from microbial developmental models. Microbes face unique lifestyle challenges when forming emergent multicellular collectives. As a result, the solutions they employ can inspire the design of novel multicellular systems. In this review, we dissect the strategies employed in multicellular development by two model microbial systems: the cellular slime mold Dictyostelium discoideum and the biofilm-forming bacterium Bacillus subtilis. Both microbes face similar challenges but often have different solutions, both from metazoan systems and from each other, to create emergent multicellularity. These challenges include assembling and sustaining a critical mass of participating individuals to support development, regulating entry into development, and assigning cell fates. The mechanisms these microbial systems exploit to robustly coordinate development under a wide range of conditions offer inspiration for a new toolbox of solutions to the synthetic development community. Additionally, recreating these phenomena synthetically offers a pathway to understanding the key principles underlying how these behaviors are coordinated naturally.

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.

The rate of thermodynamic cost against adiabatic and nonadiabatic fluctuations of a single gene circuit in Drosophila embryos

We study the stochastic dynamics of the externally regulating　gene circuit as an example of eve-skipped gene stripe in the development of Drosophila. Three gene regulation regimes are　considered: adiabatic phase when the switching rate of the gene from the OFF to the ON state is faster than the rate of mRNA degradation; nonadiabatic phase when the switching rate from the OFF to the ON state is slower than that of the mRNA degradation; the bursting phase when the gene switching is fast and transcription is very fast, while the ON state probability is very low. We found that the rate of　thermodynamic cost quantified by the entropy production rate can suppress the fluctuations of the gene circuit. Higher(lower) rate of thermodynamic cost leads to reduced (increased) fluctuations on the number of gene products in the adiabatic (nonadiabatic) regime. We also found that higher thermodynamic cost is often required to sustain the emergence of more gene states and therefore more heterogeneity coming from genetic mutations or epigenetics. We also study the stability of the gene state using the mean first passage time from one state to another. We found the monotonic decrease in time, i.e. on the stability of the state, in the transition from the nonadiabatic to the adiabatic regimes. Therefore, as the higher rate of thermodynamic cost suppresses the fluctuations, higher stability requires higher thermodynamics cost to maintain.

Synthetic cell-based materials extract positional information from morphogen gradients

Biomaterials composed of synthetic cells have the potential to adapt and differentiate guided by physicochemical environmental cues. Inspired by biological systems in development, which extract positional information (PI) from morphogen gradients in the presence of uncertainties, we here investigate how well synthetic cells can determine their position within a multicellular structure. To calculate PI, we created and analyzed a large number of synthetic cellular assemblies composed of emulsion droplets connected via lipid bilayer membranes. These droplets contained cell-free feedback gene circuits that responded to gradients of a genetic inducer acting as a morphogen. PI is found to be limited by gene expression noise and affected by the temporal evolution of the morphogen gradient and the cell-free expression system itself. The generation of PI can be rationalized by computational modeling of the system. We scale our approach using three-dimensional printing and demonstrate morphogen-based differentiation in larger tissue-like assemblies.

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.

Time and space in segmentation

Arthropod segmentation and vertebrate somitogenesis are leading fields in the experimental and theoretical interrogation of developmental patterning. However, despite the sophistication of current research, basic conceptual issues remain unresolved. These include: (i) the mechanistic origins of spatial organization within the segment addition zone (SAZ); (ii) the mechanistic origins of segment polarization; (iii) the mechanistic origins of axial variation; and (iv) the evolutionary origins of simultaneous patterning. Here, I explore these problems using coarse-grained models of cross-regulating dynamical processes. In the morphogenetic framework of a row of cells undergoing axial elongation, I simulate interactions between an 'oscillator', a 'switch' and up to three 'timers', successfully reproducing essential patterning behaviours of segmenting systems. By comparing the output of these largely cell-autonomous models to variants that incorporate positional information, I find that scaling relationships, wave patterns and patterning dynamics all depend on whether the SAZ is regulated by temporal or spatial information. I also identify three mechanisms for polarizing oscillator output, all of which functionally implicate the oscillator frequency profile. Finally, I demonstrate significant dynamical and regulatory continuity between sequential and simultaneous modes of segmentation. I discuss these results in the context of the experimental literature.

Precision of tissue patterning is controlled by dynamical properties of gene regulatory networks

During development, gene regulatory networks allocate cell fates by partitioning tissues into spatially organised domains of gene expression. How the sharp boundaries that delineate these gene expression patterns arise, despite the stochasticity associated with gene regulation, is poorly understood. We show, in the vertebrate neural tube, using perturbations of coding and regulatory regions, that the structure of the regulatory network contributes to boundary precision. This is achieved, not by reducing noise in individual genes, but by the configuration of the network modulating the ability of stochastic fluctuations to initiate gene expression changes. We use a computational screen to identify network properties that influence boundary precision, revealing two dynamical mechanisms by which small gene circuits attenuate the effect of noise in order to increase patterning precision. These results highlight design principles of gene regulatory networks that produce precise patterns of gene expression.

The many bits of positional information

Half a century after Lewis Wolpert's seminal conceptual advance on how cellular fates distribute in space, we provide a brief historical perspective on how the concept of positional information emerged and influenced the field of developmental biology and beyond. We focus on a modern interpretation of this concept in terms of information theory, largely centered on its application to cell specification in the early Drosophila embryo. We argue that a true physical variable (position) is encoded in local concentrations of patterning molecules, that this mapping is stochastic, and that the processes by which positions and corresponding cell fates are determined based on these concentrations need to take such stochasticity into account. With this approach, we shift the focus from biological mechanisms, molecules, genes and pathways to quantitative systems-level questions: where does positional information reside, how it is transformed and accessed during development, and what fundamental limits it is subject to?

Single Cell Characterization of a Synthetic Bacterial Clock with a Hybrid Feedback Loop Containing dCas9-sgRNA

Genetic networks that generate oscillations in gene expression activity are found in a wide range of organisms throughout all kingdoms of life. Oscillatory dynamics facilitates the temporal orchestration of metabolic and growth processes inside cells and organisms, as well as the synchronization of such processes with periodically occurring changes in the environment. Synthetic oscillator gene circuits such as the "repressilator" can perform similar functions in bacteria. Until recently, such circuits were mainly based on a relatively small set of well-characterized transcriptional repressors and activators. A promising, sequence-programmable alternative for gene regulation is given by CRISPR interference (CRISPRi), which enables transcriptional repression of nearly arbitrary gene targets directed by short guide RNA molecules. In order to demonstrate the use of CRISPRi in the context of dynamic gene circuits, we here replaced one of the nodes of a repressilator circuit by the RNA-guided dCas9 protein. Using single cell experiments in microfluidic reactors we show that this system displays robust relaxation oscillations over multiple periods and over several days. With a period of ≈14 bacterial generations, our oscillator is similar in speed as previously reported oscillators. Using an information-theoretic approach for the analysis of the single cell data, the potential of the circuit to act as a synthetic pacemaker for cellular processes is evaluated. We also observe that the oscillator appears to affect cellular growth, leading to variations in growth rate with the oscillator's frequency.

Mesoderm patterning by a dynamic gradient of retinoic acid signalling

Retinoic acid (RA), derived from vitamin A, is a major teratogen, clinically recognized in 1983. Identification of its natural presence in the embryo and dissection of its molecular mechanism of action became possible in the animal model with the advent of molecular biology, starting with the cloning of its nuclear receptor. In normal development, the dose of RA is tightly controlled to regulate organ formation. Its production depends on enzymes, which have a dynamic expression profile during embryonic development. As a small molecule, it diffuses rapidly and acts as a morphogen. Here, we review advances in deciphering how endogenously produced RA provides positional information to cells. We compare three mesodermal tissues, the limb, the somites and the heart, and discuss how RA signalling regulates antero-posterior and left-right patterning. A common principle is the establishment of its spatio-temporal dynamics by positive and negative feedback mechanisms and by antagonistic signalling by FGF. However, the response is cell-specific, pointing to the existence of cofactors and effectors, which are as yet incompletely characterized. This article is part of a discussion meeting issue 'Contemporary morphogenesis'.

Stochasticity and determinism in cell fate decisions

During development, cells need to make decisions about their fate in order to ensure that the correct numbers and types of cells are established at the correct time and place in the embryo. Such cell fate decisions are often classified as deterministic or stochastic. However, although these terms are clearly defined in a mathematical sense, they are sometimes used ambiguously in biological contexts. Here, we provide some suggestions on how to clarify the definitions and usage of the terms stochastic and deterministic in biological experiments. We discuss the frameworks within which such clear definitions make sense and highlight when certain ambiguity prevails. As an example, we examine how these terms are used in studies of neuronal cell fate decisions and point out areas in which definitions and interpretations have changed and matured over time. We hope that this Review will provide some clarification and inspire discussion on the use of terminology in relation to fate decisions.

Dynamic positional information: Patterning mechanism versus precision in gradient-driven systems

There is much talk about information in biology. In developmental biology, this takes the form of "positional information," especially in the context of morphogen-based pattern formation. Unfortunately, the concept of "information" is rarely defined in any precise manner. Here, we provide two alternative interpretations of "positional information," and examine the complementary meanings and uses of each concept. Positional information defined as Shannon information helps us understand decoding and error propagation in patterning systems. General relativistic positional information, in contrast, provides a metric to assess the output of pattern-forming mechanisms. Both interpretations provide powerful conceptual tools that do not compete, but are best used in combination to gain a proper mechanistic understanding of robust patterning.

Estimating information in time-varying signals

Across diverse biological systems-ranging from neural networks to intracellular signaling and genetic regulatory networks-the information about changes in the environment is frequently encoded in the full temporal dynamics of the network nodes. A pressing data-analysis challenge has thus been to efficiently estimate the amount of information that these dynamics convey from experimental data. Here we develop and evaluate decoding-based estimation methods to lower bound the mutual information about a finite set of inputs, encoded in single-cell high-dimensional time series data. For biological reaction networks governed by the chemical Master equation, we derive model-based information approximations and analytical upper bounds, against which we benchmark our proposed model-free decoding estimators. In contrast to the frequently-used k-nearest-neighbor estimator, decoding-based estimators robustly extract a large fraction of the available information from high-dimensional trajectories with a realistic number of data samples. We apply these estimators to previously published data on Erk and Ca2+ signaling in mammalian cells and to yeast stress-response, and find that substantial amount of information about environmental state can be encoded by non-trivial response statistics even in stationary signals. We argue that these single-cell, decoding-based information estimates, rather than the commonly-used tests for significant differences between selected population response statistics, provide a proper and unbiased measure for the performance of biological signaling networks.

Information-theoretic analysis of multivariate single-cell signaling responses

Mathematical methods of information theory appear to provide a useful language to describe how stimuli are encoded in activities of signaling effectors. Exploring the information-theoretic perspective, however, remains conceptually, experimentally and computationally challenging. Specifically, existing computational tools enable efficient analysis of relatively simple systems, usually with one input and output only. Moreover, their robust and readily applicable implementations are missing. Here, we propose a novel algorithm, SLEMI—statistical learning based estimation of mutual information, to analyze signaling systems with high-dimensional outputs and a large number of input values. Our approach is efficient in terms of computational time as well as sample size needed for accurate estimation. Analysis of the NF-κB single—cell signaling responses to TNF-α reveals that NF-κB signaling dynamics improves discrimination of high concentrations of TNF-α with a relatively modest impact on discrimination of low concentrations. Provided R-package allows the approach to be used by computational biologists with only elementary knowledge of information theory.

In light of single-cell, live-imaging experiments understanding of how cells transmit information about identity and quantity of stimuli is incomplete. When exposed to the same stimulus individual cells exhibit substantial cell-to-cell heterogeneity. Besides, stimuli have been shown to regulate temporal profiles of signaling effectors. Therefore, it is, for instance, not entirely clear whether single-cell responses are binary or contain more information about the quantity of stimuli. The above questions resulted in a considerable interest to study cellular signaling within the framework of information theory. Unfortunately, the utilization of the information-theoretic perspective is handicapped in part by the lack of suitable methods that account for multivariate signaling data. Here, we propose a novel algorithm that breaks a considerable computational barrier by allowing the effective information-theoretic analysis of highly-dimensional single-cell measurements. Our approach is computationally efficient, robust and straightforward to use. Moreover, we provide a simple R-package implementation.

Measuring Dorsoventral Pattern and Morphogen Signaling Profiles in the Growing Neural Tube

Developmental processes are inherently dynamic and understanding them requires quantitative measurements of gene and protein expression levels in space and time. While live imaging is a powerful approach for obtaining such data, it is still a challenge to apply it over long periods of time to large tissues, such as the embryonic spinal cord in mouse and chick. Nevertheless, dynamics of gene expression and signaling activity patterns in this organ can be studied by collecting tissue sections at different developmental stages. In combination with immunohistochemistry, this allows for measuring the levels of multiple developmental regulators in a quantitative manner with high spatiotemporal resolution. The mean protein expression levels over time, as well as embryo-to-embryo variability can be analyzed. A key aspect of the approach is the ability to compare protein levels across different samples. This requires a number of considerations in sample preparation, imaging and data analysis. Here we present a protocol for obtaining time course data of dorsoventral expression patterns from mouse and chick neural tube in the first 3 days of neural tube development. The described workflow starts from embryo dissection and ends with a processed dataset. Software scripts for data analysis are included. The protocol is adaptable and instructions that allow the user to modify different steps are provided. Thus, the procedure can be altered for analysis of time-lapse images and applied to systems other than the neural tube.

Agent-based modeling of morphogenetic systems: Advantages and challenges

The complexity of morphogenesis poses a fundamental challenge to understanding the mechanisms governing the formation of biological patterns and structures. Over the past century, numerous processes have been identified as critically contributing to morphogenetic events, but the interplay between the various components and aspects of pattern formation have been much harder to grasp. The combination of traditional biology with mathematical and computational methods has had a profound effect on our current understanding of morphogenesis and led to significant insights and advancements in the field. In particular, the theoretical concepts of reaction–diffusion systems and positional information, proposed by Alan Turing and Lewis Wolpert, respectively, dramatically influenced our general view of morphogenesis, although typically in isolation from one another. In recent years, agent-based modeling has been emerging as a consolidation and implementation of the two theories within a single framework. Agent-based models (ABMs) are unique in their ability to integrate combinations of heterogeneous processes and investigate their respective dynamics, especially in the context of spatial phenomena. In this review, we highlight the benefits and technical challenges associated with ABMs as tools for examining morphogenetic events. These models display unparalleled flexibility for studying various morphogenetic phenomena at multiple levels and have the important advantage of informing future experimental work, including the targeted engineering of tissues and organs.

Optimal Decoding of Cellular Identities in a Genetic Network

In developing organisms, spatially prescribed cell identities are thought to be determined by the expression levels of multiple genes. Quantitative tests of this idea, however, require a theoretical framework capable of exposing the rules and precision of cell specification over developmental time. We use the gap gene network in the early fly embryo as an example to show how expression levels of the four gap genes can be jointly decoded into an optimal specification of position with 1% accuracy. The decoder correctly predicts, with no free parameters, the dynamics of pair-rule expression patterns at different developmental time points and in various mutant backgrounds. Precise cellular identities are thus available at the earliest stages of development, contrasting the prevailing view of positional information being slowly refined across successive layers of the patterning network. Our results suggest that developmental enhancers closely approximate a mathematically optimal decoding strategy.

Precision in a rush: Trade-offs between reproducibility and steepness of the hunchback expression pattern

Fly development amazes us by the precision and reproducibility of gene expression, especially since the initial expression patterns are established during very short nuclear cycles. Recent live imaging of hunchback promoter dynamics shows a stable steep binary expression pattern established within the three minute interphase of nuclear cycle 11. Considering expression models of different complexity, we explore the trade-off between the ability of a regulatory system to produce a steep boundary and minimize expression variability between different nuclei. We show how a limited readout time imposed by short developmental cycles affects the gene’s ability to read positional information along the embryo’s anterior posterior axis and express reliably. Comparing our theoretical results to real-time monitoring of the hunchback transcription dynamics in live flies, we discuss possible regulatory strategies, suggesting an important role for additional binding sites, gradients or non-equilibrium binding and modified transcription factor search strategies.

Despite very limited time, organisms develop in reproducible ways. In the early stages of fly development the information about maternal signals is read out in a few minutes to produce steep and precise gene expression patterns. Motivated by recent live imaging experiments in fly embryos, we explore the consequences of the trade-off between a rushed but reproducible readout and a steep expression pattern on the regulatory modules of gene expression. We show that the current view of one anterior gradient morphogen binding to six binding sites is quantitatively inconsistent with the experimental data given the short readout time, suggesting other regulatory features.

Single transcript imaging to assay gene expression in wholemount Drosophila melanogaster tissues

Single molecule Fluorescence in situ Hybridization (smFISH) for mRNA provides a powerful quantitative handle on expression from endogenous gene loci. While the method has been widely applied in cells in culture, applications to primary tissue samples remain fewer, and often use involved cryosectioning. Even apart from quantitative access to absolute transcript counts in specific tissue volumes, many other advantages of smFISH can be envisaged in tissue samples. Primary among these are the ability to report on subtle differences in expression among different cell types within a tissue, and the ability to correlate the expression from different target genes. Here, we present a modified method of smFISH applicable on various primary wholemount tissues from the fruit fly Drosophila melanogaster, and show the efficacy of the method in a variety of larval and adult tissue, and embryos. We also combine smFISH in tissue with immunofluorescence to demonstrate the possibility of capturing transcriptional and translational aspects of gene expression in the same tissue. Given the widespread use of Drosophila melanogaster as a model system in Developmental Biology and Genetics, such methods are likely to be of wide interest and could yield rich information about gene expression in tissues from this organism.

Morphogen gradient reconstitution reveals Hedgehog pathway design principles

In developing tissues, cells estimate their spatial position by sensing graded concentrations of diffusible signaling proteins called morphogens. Morphogen-sensing pathways exhibit diverse molecular architectures, whose roles in controlling patterning dynamics and precision have been unclear. In this work, combining cell-based in vitro gradient reconstitution, genetic rewiring, and mathematical modeling, we systematically analyzed the distinctive architectural features of the Sonic Hedgehog pathway. We found that the combination of double-negative regulatory logic and negative feedback through the PTCH receptor accelerates gradient formation and improves robustness to variation in the morphogen production rate compared with alternative designs. The ability to isolate morphogen patterning from concurrent developmental processes and to compare the patterning behaviors of alternative, rewired pathway architectures offers a powerful way to understand and engineer multicellular patterning.

Physical Limits on the Precision of Mitotic Spindle Positioning by Microtubule Pushing forces: Mechanics of mitotic spindle positioning

Tissues are shaped and patterned by mechanical and chemical processes. A key mechanical process is the positioning of the mitotic spindle, which determines the size and location of the daughter cells within the tissue. Recent force and position-fluctuation measurements indicate that pushing forces, mediated by the polymerization of astral microtubules against- the cell cortex, maintain the mitotic spindle at the cell center in Caenorhabditis elegans embryos. The magnitude of the centering forces suggests that the physical limit on the accuracy and precision of this centering mechanism is determined by the number of pushing microtubules rather than by thermally driven fluctuations. In cells that divide asymmetrically, anti-centering, pulling forces generated by cortically located dyneins, in conjunction with microtubule depolymerization, oppose the pushing forces to drive spindle displacements away from the center. Thus, a balance of centering pushing forces and anti-centering pulling forces localize the mitotic spindles within dividing C. elegans cells.

Dynamic patterning by the Drosophila pair-rule network reconciles long-germ and short-germ segmentation

Drosophila segmentation is a well-established paradigm for developmental pattern formation. However, the later stages of segment patterning, regulated by the "pair-rule" genes, are still not well understood at the system level. Building on established genetic interactions, I construct a logical model of the Drosophila pair-rule system that takes into account the demonstrated stage-specific architecture of the pair-rule gene network. Simulation of this model can accurately recapitulate the observed spatiotemporal expression of the pair-rule genes, but only when the system is provided with dynamic "gap" inputs. This result suggests that dynamic shifts of pair-rule stripes are essential for segment patterning in the trunk and provides a functional role for observed posterior-to-anterior gap domain shifts that occur during cellularisation. The model also suggests revised patterning mechanisms for the parasegment boundaries and explains the aetiology of the even-skipped null mutant phenotype. Strikingly, a slightly modified version of the model is able to pattern segments in either simultaneous or sequential modes, depending only on initial conditions. This suggests that fundamentally similar mechanisms may underlie segmentation in short-germ and long-germ arthropods.

Decoding of position in the developing neural tube from antiparallel morphogen gradients

Like many developing tissues, the vertebrate neural tube is patterned by antiparallel morphogen gradients. To understand how these inputs are interpreted, we measured morphogen signaling and target gene expression in mouse embryos and chick ex vivo assays. From these data, we derived and validated a characteristic decoding map that relates morphogen input to the positional identity of neural progenitors. Analysis of the observed responses indicates that the underlying interpretation strategy minimizes patterning errors in response to the joint input of noisy opposing gradients. We reverse-engineered a transcriptional network that provides a mechanistic basis for the observed cell fate decisions and accounts for the precision and dynamics of pattern formation. Together, our data link opposing gradient dynamics in a growing tissue to precise pattern formation.

Genetic control of encoding strategy in a food-sensing neural circuit

Neuroendocrine circuits encode environmental information via changes in gene expression and other biochemical activities to regulate physiological responses. Previously, we showed that daf-7 TGFβ and tph-1 tryptophan hydroxylase expression in specific neurons encode food abundance to modulate lifespan in Caenorhabditis elegans, and uncovered cross- and self-regulation among these genes (Entchev et al., 2015). Here, we now extend these findings by showing that these interactions between daf-7 and tph-1 regulate redundancy and synergy among neurons in food encoding through coordinated control of circuit-level signal and noise properties. Our analysis further shows that daf-7 and tph-1 contribute to most of the food-responsiveness in the modulation of lifespan. We applied a computational model to capture the general coding features of this system. This model agrees with our previous genetic analysis and highlights the consequences of redundancy and synergy during information transmission, suggesting a rationale for the regulation of these information processing features.

Beyond the French Flag Model: Exploiting Spatial and Gene Regulatory Interactions for Positional Information

A crucial step in the early development of multicellular organisms involves the establishment of spatial patterns of gene expression which later direct proliferating cells to take on different cell fates. These patterns enable the cells to infer their global position within a tissue or an organism by reading out local gene expression levels. The patterning system is thus said to encode positional information, a concept that was formalized recently in the framework of information theory. Here we introduce a toy model of patterning in one spatial dimension, which can be seen as an extension of Wolpert’s paradigmatic “French Flag” model, to patterning by several interacting, spatially coupled genes subject to intrinsic and extrinsic noise. Our model, a variant of an Ising spin system, allows us to systematically explore expression patterns that optimally encode positional information. We find that optimal patterning systems use positional cues, as in the French Flag model, together with gene-gene interactions to generate combinatorial codes for position which we call “Counter” patterns. Counter patterns can also be stabilized against noise and variations in system size or morphogen dosage by longer-range spatial interactions of the type invoked in the Turing model. The simple setup proposed here qualitatively captures many of the experimentally observed properties of biological patterning systems and allows them to be studied in a single, theoretically consistent framework.

Unipolar distributions of junctional Myosin II identify cell stripe boundaries that drive cell intercalation throughout Drosophila axis extension

Convergence and extension movements elongate tissues during development. Drosophila germ-band extension (GBE) is one example, which requires active cell rearrangements driven by Myosin II planar polarisation. Here, we develop novel computational methods to analyse the spatiotemporal dynamics of Myosin II during GBE, at the scale of the tissue. We show that initial Myosin II bipolar cell polarization gives way to unipolar enrichment at parasegmental boundaries and two further boundaries within each parasegment, concomitant with a doubling of cell number as the tissue elongates. These boundaries are the primary sites of cell intercalation, behaving as mechanical barriers and providing a mechanism for how cells remain ordered during GBE. Enrichment at parasegment boundaries during GBE is independent of Wingless signaling, suggesting pair-rule gene control. Our results are consistent with recent work showing that a combinatorial code of Toll-like receptors downstream of pair-rule genes contributes to Myosin II polarization via local cell-cell interactions. We propose an updated cell-cell interaction model for Myosin II polarization that we tested in a vertex-based simulation.

Extending the dynamic range of transcription factor action by translational regulation

A crucial step in the regulation of gene expression is binding of transcription factor (TF) proteins to regulatory sites along the DNA. But transcription factors act at nanomolar concentrations, and noise due to random arrival of these molecules at their binding sites can severely limit the precision of regulation. Recent work on the optimization of information flow through regulatory networks indicates that the lower end of the dynamic range of concentrations is simply inaccessible, overwhelmed by the impact of this noise. Motivated by the behavior of homeodomain proteins, such as the maternal morphogen Bicoid in the fruit fly embryo, we suggest a scheme in which transcription factors also act as indirect translational regulators, binding to the mRNA of other regulatory proteins. Intuitively, each mRNA molecule acts as an independent sensor of the input concentration, and averaging over these multiple sensors reduces the noise. We analyze information flow through this scheme and identify conditions under which it outperforms direct transcriptional regulation. Our results suggest that the dual role of homeodomain proteins is not just a historical accident, but a solution to a crucial physics problem in the regulation of gene expression.

Morphogen rules: design principles of gradient-mediated embryo patterning

The Drosophila blastoderm and the vertebrate neural tube are archetypal examples of morphogen-patterned tissues that create precise spatial patterns of different cell types. In both tissues, pattern formation is dependent on molecular gradients that emanate from opposite poles. Despite distinct evolutionary origins and differences in time scales, cell biology and molecular players, both tissues exhibit striking similarities in the regulatory systems that establish gene expression patterns that foreshadow the arrangement of cell types. First, signaling gradients establish initial conditions that polarize the tissue, but there is no strict correspondence between specific morphogen thresholds and boundary positions. Second, gradients initiate transcriptional networks that integrate broadly distributed activators and localized repressors to generate patterns of gene expression. Third, the correct positioning of boundaries depends on the temporal and spatial dynamics of the transcriptional networks. These similarities reveal design principles that are likely to be broadly applicable to morphogen-patterned tissues.

Only accessible information is useful: insights from gradient-mediated patterning

Information theory is gaining popularity as a tool to characterize performance of biological systems. However, information is commonly quantified without reference to whether or how a system could extract and use it; as a result, information-theoretic quantities are easily misinterpreted. Here, we take the example of pattern-forming developmental systems which are commonly structured as cascades of sequential gene expression steps. Such a multi-tiered structure appears to constitute sub-optimal use of the positional information provided by the input morphogen because noise is added at each tier. However, one must distinguish between the total information in a morphogen and information that can be usefully extracted and interpreted by downstream elements. We demonstrate that quantifying the information that is accessible to the system naturally explains the prevalence of multi-tiered network architectures as a consequence of the noise inherent to the control of gene expression. We support our argument with empirical observations from patterning along the major body axis of the fruit fly embryo. We use this example to highlight the limitations of the standard information-theoretic characterization of biological signalling, which are frequently de-emphasized, and illustrate how they can be resolved.

Optimizing information flow in small genetic networks. IV. Spatial coupling

We typically think of cells as responding to external signals independently by regulating their gene expression levels, yet they often locally exchange information and coordinate. Can such spatial coupling be of benefit for conveying signals subject to gene regulatory noise? Here we extend our information-theoretic framework for gene regulation to spatially extended systems. As an example, we consider a lattice of nuclei responding to a concentration field of a transcriptional regulator (the input) by expressing a single diffusible target gene. When input concentrations are low, diffusive coupling markedly improves information transmission; optimal gene activation functions also systematically change. A qualitatively different regulatory strategy emerges where individual cells respond to the input in a nearly steplike fashion that is subsequently averaged out by strong diffusion. While motivated by early patterning events in the Drosophila embryo, our framework is generically applicable to spatially coupled stochastic gene expression models.