When it pays to rush: interpreting morphogen gradients prior to steady-state

Multi-Grid Reaction-Diffusion Master Equation: Applications to Morphogen Gradient Modelling

The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem.

Single-cell directional sensing from just a few receptor binding events

Identifying the directionality of signaling sources from noisy input to membrane receptors is an essential task performed by many cell types. A variety of models have been proposed to explain directional sensing in cells. However, many of these require significant computational and memory capacities for the cell. We propose and analyze a simple mechanism in which a cell adopts the direction associated with the first few membrane binding events. This model yields an accurate angular estimate to the source long before steady state is reached in biologically relevant scenarios. Our proposed mechanism allows for reliable estimates of the directionality of external signals using temporal information and assumes minimal computational capacities of the cell.

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.

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.

French flag gradients and Turing reaction-diffusion versus differentiation waves as models of morphogenesis

The Turing reaction-diffusion model and the French Flag Model are widely accepted in the field of development as the best models for explaining embryogenesis. Virtually all current attempts to understand cell differentiation in embryos begin and end with the assumption that some combination of these two models works. The result may become a bias in embryogenesis in assuming the problem has been solved by these two-chemical substance-based models. Neither model is applied consistently. We review the differences between the French Flag, Turing reaction-diffusion model, and a mechanochemical model called the differentiation wave/cell state splitter model. The cytoskeletal cell state splitter and the embryonic differentiation waves was first proposed in 1987 as a combined physics and chemistry model for cell differentiation in embryos, based on empirical observations on urodele amphibian embryos. We hope that the development of theory can be advanced and observations relevant to distinguishing the embryonic differentiation wave model from the French Flag model and reaction-diffusion equations will be taken up by experimentalists. Experimentalists rely on mathematical biologists for theory, and therefore depend on them for what parameters they choose to measure and ignore. Therefore, mathematical biologists need to fully understand the distinctions between these three models.

Bicoid gradient formation mechanism and dynamics revealed by protein lifetime analysis

Embryogenesis relies on instructions provided by spatially organized signaling molecules known as morphogens. Understanding the principles behind morphogen distribution and how cells interpret locally this information remains a major challenge in developmental biology. Here, we introduce morphogen-age measurements as a novel approach to test models of morphogen gradient formation. Using a tandem fluorescent timer as a protein age sensor, we find a gradient of increasing age of Bicoid along the anterior-posterior axis in the early Drosophila embryo. Quantitative analysis of the protein age distribution across the embryo reveals that the synthesis-diffusion-degradation model is the most likely model underlying Bicoid gradient formation, and rules out other hypotheses for gradient formation. Moreover, we show that the timer can detect transitions in the dynamics associated with syncytial cellularization. Our results provide new insight into Bicoid gradient formation and demonstrate how morphogen-age information can complement knowledge about movement, abundance, and distribution, which should be widely applicable to other systems.

The organelle of differentiation in embryos: the cell state splitter

The cell state splitter is a membraneless organelle at the apical end of each epithelial cell in a developing embryo. It consists of a microfilament ring and an intermediate filament ring subtending a microtubule mat. The microtubules and microfilament ring are in mechanical opposition as in a tensegrity structure. The cell state splitter is bistable, perturbations causing it to contract or expand radially. The intermediate filament ring provides metastability against small perturbations. Once this snap-through organelle is triggered, it initiates signal transduction to the nucleus, which changes gene expression in one of two readied manners, causing its cell to undergo a step of determination and subsequent differentiation. The cell state splitter also triggers the cell state splitters of adjacent cells to respond, resulting in a differentiation wave. Embryogenesis may be represented then as a bifurcating differentiation tree, each edge representing one cell type. In combination with the differentiation waves they propagate, cell state splitters explain the spatiotemporal course of differentiation in the developing embryo. This review is excerpted from and elaborates on "Embryogenesis Explained" (World Scientific Publishing, Singapore, 2016).

Spatiotemporal analysis of different mechanisms for interpreting morphogen gradients

During development, multicellular organisms must accurately control both temporal and spatial aspects of tissue patterning. This is often achieved using morphogens, signaling molecules that form spatially varying concentrations and so encode positional information. Typical analysis of morphogens assumes that spatial information is decoded in steady state by measuring the value of the morphogen concentration. However, recent experimental work suggests that both pre-steady-state readout and measurement of spatial and temporal derivatives of the morphogen concentration can play important roles in defining boundaries. Here, we undertake a detailed theoretical and numerical study of the accuracy of patterning-both in space and time-in models where readout is provided not by the morphogen concentration but by its spatial and temporal derivatives. In both cases we find that accurate patterning can be achieved, with sometimes even smaller errors than directly reading the morphogen concentration. We further demonstrate that such models provide other potential benefits to the system, such as the ability to switch on and off gene response with a high degree of spatiotemporal accuracy. Finally, we discuss how such derivatives might be calculated biologically and examine these models in relation to Sonic Hedgehog signaling in the vertebrate central nervous system. We show that, when coupled to a downstream transcriptional network, pre-steady-state measurement of the temporal change in the Shh morphogen is a plausible mechanism for determining precise gene boundaries in both space and time.

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.

A large number of receptors may reduce cellular response time variation

Cells often have tens of thousands of receptors, even though only a few activated receptors can trigger full cellular responses. Reasons for the overabundance of receptors remain unclear. We suggest that, under certain conditions, the large number of receptors can result in a competition among receptors to be the first to activate the cell. The competition decreases the variability of the time to cellular activation, and hence results in a more synchronous activation of cells. We argue that, in simple models, this variability reduction does not necessarily interfere with the receptor specificity to ligands achieved by the kinetic proofreading mechanism. Thus cells can be activated accurately in time and specifically to certain signals. We predict the minimum number of receptors needed to reduce the coefficient of variation for the time to activation following binding of a specific ligand. Furthermore, we predict the maximum number of receptors so that the kinetic proofreading mechanism still can improve the specificity of the activation. These predictions fall in line with experimentally reported receptor numbers for multiple systems.

Physical interpretation of mean local accumulation time of morphogen gradient formation

The paper deals with a reaction-diffusion problem that arises in developmental biology when describing the formation of the concentration profiles of signaling molecules, called morphogens, which control gene expression and, hence, cell differentiation. The mean local accumulation time, which is the mean time required to reach the steady state at a fixed point of a patterned tissue, is an important characteristic of the formation process. We show that this time is a sum of two times, the conditional mean first-passage time from the source to the observation point and the mean local accumulation time in the situation when the source is localized at the observation point.

Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain

Morphogens provide positional information for spatial patterns of gene expression during development. However, stochastic effects such as local fluctuations in morphogen concentration and noise in signal transduction make it difficult for cells to respond to their positions accurately enough to generate sharp boundaries between gene expression domains. During development of rhombomeres in the zebrafish hindbrain, the morphogen retinoic acid (RA) induces expression of hoxb1a in rhombomere 4 (r4) and krox20 in r3 and r5. Fluorescent in situ hybridization reveals rough edges around these gene expression domains, in which cells co-express hoxb1a and krox20 on either side of the boundary, and these sharpen within a few hours. Computational analysis of spatial stochastic models shows, surprisingly, that noise in hoxb1a/krox20 expression actually promotes sharpening of boundaries between adjacent segments. In particular, fluctuations in RA initially induce a rough boundary that requires noise in hoxb1a/krox20 expression to sharpen. This finding suggests a novel noise attenuation mechanism that relies on intracellular noise to induce switching and coordinate cellular decisions during developmental patterning.

Ordinary differential equation for local accumulation time

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation.

Improved readout precision of the Bicoid morphogen gradient by early decoding

Transcription factors (TFs) bind to specific DNA sequences to induce or repress gene expression. Expression levels can be tuned by changing TF concentrations, but the precision of such tuning is limited, since the fraction of time a TF occupies its binding site is subject to stochastic fluctuations. Bicoid (Bcd) is a TF that patterns the early Drosophila embryo by establishing an anterior-to-posterior concentration gradient and activating specific gene targets ("gap genes") in a concentration-dependent manner. Recently, the Bcd gradient and its in-vivo diffusion were quantified in live embryos, raising a quandary: the precision by which the Bcd target genes are defined (one-cell resolution) appeared to exceed the physical limits set by the stochastic binding of Bcd to DNA. We hypothesize that early readout of Bcd could account for the observed precision. Specifically, we consider the possibility that gap genes begin to be expressed earlier than typically measured experimentally, at a time when the distance between the nuclei is large. At this time, the difference in Bcd concentration between adjacent nuclei is large, enabling better tolerance for measurement imprecision. We show that such early decoding can indeed increase the accuracy of gap-gene expression, and that the initial pattern can be stabilized during subsequent divisions.

Morphogen gradients: from generation to interpretation

Morphogens are long-range signaling molecules that pattern developing tissues in a concentration-dependent manner. The graded activity of morphogens within tissues exposes cells to different signal levels and leads to region-specific transcriptional responses and cell fates. In its simplest incarnation, a morphogen signal forms a gradient by diffusion from a local source and clearance in surrounding tissues. Responding cells often transduce morphogen levels in a linear fashion, which results in the graded activation of transcriptional effectors. The concentration-dependent expression of morphogen target genes is achieved by their different binding affinities for transcriptional effectors as well as inputs from other transcriptional regulators. Morphogen distribution and interpretation are the result of complex interactions between the morphogen and responding tissues. The response to a morphogen is dependent not simply on morphogen concentration but also on the duration of morphogen exposure and the state of the target cells. In this review, we describe the morphogen concept and discuss the mechanisms that underlie the generation, modulation, and interpretation of morphogen gradients.

Analysis of mutants with altered shh activity and posterior digit loss supports a biphasic model for shh function as a morphogen and mitogen

Sonic hedgehog (Shh) controls the number and type of digits formed. Using a conditional genetic approach for timed removal of Shh, we previously proposed a biphasic model of Shh function: a transient patterning phase, during which digit progenitors are specified, and an extended proliferative phase, during which expansion of progenitor pools enables digit formation. Other models favor a close integration of digit patterning and expansion, with sequential promotion to more posterior identity over time, apparently supported by some mutants with selective posterior digit loss. To further test these models, we analyzed the dynamics of Shh activity in several oligodactylous mutants with different types of digit loss. The profile of Shh activity and phenotypic outcome in these mutants supports a biphasic over an integrated temporal model. Eomesodermin expression, as an independent marker of posterior digit identity, confirmed that proper digit 4 specification requires only the transient phase of Shh activity.

Local kinetics of morphogen gradients

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production. We present analytical results that characterize the dynamics of this process and are in quantitative agreement with numerical solutions of the underlying nonlinear equations. The derived results provide an explicit connection between the parameters of the problem and the time needed to reach a steady state value at a given position. Our approach can be used for the quantitative analysis of tissue patterning by morphogen gradients, a subject of active research in biophysics and developmental biology.

How long does it take to establish a morphogen gradient?

A morphogen gradient is defined as a concentration field of a molecule that acts as a dose-dependent regulator of cell differentiation. One of the key questions in studies of morphogen gradients is whether they reach steady states on timescales relevant for developmental patterning. We propose a systematic approach for addressing this question and illustrate it by analyzing several models that account for diffusion and degradation of locally produced chemical signals.

Shaping a morphogen gradient for positional precision

Morphogen gradients, which provide positional information to cells in a developing tissue, could in principle adopt any nonuniform profile. To our knowledge, how the profile of a morphogen gradient affects positional precision has not been well studied experimentally. Here, we compare the positional precision provided by the Drosophila morphogenetic protein Bicoid (Bcd) in wild-type (wt) embryos with embryos lacking an interacting cofactor. The Bcd gradient in the latter case exhibits decreased positional precision around mid-embryo compared with its wt counterpart. The domain boundary of Hunchback (Hb), a target activated by Bcd, becomes more variable in mutant embryos. By considering embryo-to-embryo, internal, and measurement fluctuations, we dissect mathematically the relevant sources of fluctuations that contribute to the error in positional information. Using this approach, we show that the defect in Hb boundary positioning in mutant embryos is directly reflective of an altered Bcd gradient profile with increasing flatness toward mid-embryo. Furthermore, we find that noise in the Bcd input signal is dominated by internal fluctuations but, due to time and spatial averaging, the spatial precision of the Hb boundary is primarily affected by embryo-to-embryo variations. Our results demonstrate that the positional information provided by the wt Bcd gradient profile is highly precise and necessary for patterning precision.

Positional information and patterning revisited

The concept of positional information proposes that cells acquire positional values as in a coordinate system, which they interpret by developing in particular ways to give rise to spatial patterns. Some of the best evidence for positional information comes from regeneration experiments, and the patterning of the leg and antenna in Drosophila and the vertebrate limb. Central problems are how positional information is set up, how it is recorded, and then how it is interpreted by the cells. A number of models have been proposed for the setting up of positional gradients, and most are based on diffusion of a morphogen and its interactions with extracellular molecules. It is argued that diffusion may not be reliable mechanism. There are also mechanisms based on timing. There is no good evidence for the quantitative aspects of any of the gradients and details how they are set up. The way in which a signalling gradient regulates differential gene expression in a concentration-dependent manner also raises several mechanistic issues.