Quantitative Models of Developmental Pattern Formation

Predicting phenotypic diversity and the underlying quantitative molecular transitions

During development, signaling networks control the formation of multicellular patterns. To what extent quantitative fluctuations in these complex networks may affect multicellular phenotype remains unclear. Here, we describe a computational approach to predict and analyze the phenotypic diversity that is accessible to a developmental signaling network. Applying this framework to vulval development in C. elegans, we demonstrate that quantitative changes in the regulatory network can render ∼500 multicellular phenotypes. This phenotypic capacity is an order-of-magnitude below the theoretical upper limit for this system but yet is large enough to demonstrate that the system is not restricted to a select few outcomes. Using metrics to gauge the robustness of these phenotypes to parameter perturbations, we identify a select subset of novel phenotypes that are the most promising for experimental validation. In addition, our model calculations provide a layout of these phenotypes in network parameter space. Analyzing this landscape of multicellular phenotypes yielded two significant insights. First, we show that experimentally well-established mutant phenotypes may be rendered using non-canonical network perturbations. Second, we show that the predicted multicellular patterns include not only those observed in C. elegans, but also those occurring exclusively in other species of the Caenorhabditis genus. This result demonstrates that quantitative diversification of a common regulatory network is indeed demonstrably sufficient to generate the phenotypic differences observed across three major species within the Caenorhabditis genus. Using our computational framework, we systematically identify the quantitative changes that may have occurred in the regulatory network during the evolution of these species. Our model predictions show that significant phenotypic diversity may be sampled through quantitative variations in the regulatory network without overhauling the core network architecture. Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentally observed across multiple species, we systematically trace the quantitative regulatory changes that may have occurred during the evolution of the Caenorhabditis genus.

The diversity of metazoan life forms that we experience today arose as multicellular systems continually sampled new phenotypes that withstood ever changing selective pressures. This phenotypic diversification is driven by variations in the underlying regulatory network that instructs cells to form multicellular patterns and structures. Here, we computationally construct the phenotypic diversity that may be accessible through quantitative tuning of the regulatory network that drives multicellular patterning during C. elegans vulval development. We show that significant phenotypic diversity may be sampled through quantitative variations without overhauling the core regulatory network architecture. Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentally observed across multiple species, we systematically deduce the quantitative molecular changes that may have transpired during the evolution of the Caenorhabditis genus.

Analysis of dynamic morphogen scale invariance

During the development of some tissues, fields of multipotent cells differentiate into distinct cell types in response to the local concentration of a signalling factor called a morphogen. Typically, individual organisms within a population differ in size, but their body plans appear to be scaled versions of a common template. Similarly, closely related species may differ by three or more orders of magnitude in size, yet common structures between species scale to have similar proportions. In standard reaction-diffusion equations, the morphogen range has a length scale that depends on a balance between kinetic and transport processes and not on the length or size of the field of cells being patterned. However, as shown here for a class of morphogen-patterning systems, a number of conditions lead to scale invariance of the morphogen distribution at equilibrium and during the transient approach to equilibrium. Equilibrium scale invariance requires conservation of the total binding site number and total input flux. Dynamic scale invariance additionally requires sufficient binding to slow the diffusion of ligand. The equations derived herein can be extended to the study of other perturbations to gain further insight into the processes regulating the robustness and scaling of morphogen-mediated pattern formation.

Cell lineages and the logic of proliferative control

It is widely accepted that the growth and regeneration of tissues and organs is tightly controlled. Although experimental studies are beginning to reveal molecular mechanisms underlying such control, there is still very little known about the control strategies themselves. Here, we consider how secreted negative feedback factors ("chalones") may be used to control the output of multistage cell lineages, as exemplified by the actions of GDF11 and activin in a self-renewing neural tissue, the mammalian olfactory epithelium (OE). We begin by specifying performance objectives-what, precisely, is being controlled, and to what degree-and go on to calculate how well different types of feedback configurations, feedback sensitivities, and tissue architectures achieve control. Ultimately, we show that many features of the OE-the number of feedback loops, the cellular processes targeted by feedback, even the location of progenitor cells within the tissue-fit with expectations for the best possible control. In so doing, we also show that certain distinctions that are commonly drawn among cells and molecules-such as whether a cell is a stem cell or transit-amplifying cell, or whether a molecule is a growth inhibitor or stimulator-may be the consequences of control, and not a reflection of intrinsic differences in cellular or molecular character.

Biological systems from an engineer's point of view

The robust design of complex systems, such as cruise control, requires a careful balance of several objectives. As biological systems are no different, an engineering approach to these systems proves useful.

The Intersection of Theory and Application in Elucidating Pattern Formation in Developmental Biology

We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and vice-versa. The chosen systems - Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns - illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago'We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.'

A scenario-based approach to modeling development: a prototype model of C. elegans vulval fate specification

Studies of developmental biology are often facilitated by diagram "models" that summarize the current understanding of underlying mechanisms. The increasing complexity of our understanding of development necessitates computational models that can extend these representations to include their dynamic behavior. Here we present a prototype model of Caenorhabditis elegans vulval precursor cell fate specification that represents many processes crucial for this developmental event but that are hard to integrate using other modeling methodologies. We demonstrate the integrative capabilities of our methodology by comprehensively incorporating the contents of three seminal papers, showing that this methodology can lead to comprehensive models of developmental biology. The prototype computational model was built and is run using a language (Live Sequence Charts) and tool (the Play-Engine) that facilitate the same conceptual processes biologists use to construct and probe diagram-type models. We demonstrate that this modeling approach permits rigorous tests of mutual consistency between experimental data and mechanistic hypotheses and can identify specific conflicting results, providing a useful approach to probe developmental systems.

Compensated optimal grids for elliptic boundary-value problems

A method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary. The method is an extension of the optimal grid approach for elliptic problems, based on optimal rational approximation of the associated Neumann-to-Dirichlet map in Fourier space. It is shown that, using certain types of boundary discretization, one can go from second-order accurate schemes to essentially spectrally accurate schemes in two-dimensional problems, and to fourth-order accurate schemes in three-dimensional problems without any increase in the computational complexity. The main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite-difference stencil discretizing the differential operator on the boundary. We illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication.

Decoding the quantitative nature of TGF-beta/Smad signaling

How transforming growth factor-beta (TGF-beta) signaling elicits diverse cell responses remains elusive, despite the major molecular components of the pathway being known. We contend that understanding TGF-beta biology requires mathematical models to decipher the quantitative nature of TGF-beta/Smad signaling and to account for its complexity. Here, we review mathematical models of TGF-beta superfamily signaling that predict how robustness is achieved in bone-morphogenetic-protein signaling in the Drosophila embryo, how changes in receptor-trafficking dynamics can be exploited by cancer cells and how the basic mechanisms of TGF-beta/Smad signaling conspire to promote Smad accumulation in the nucleus. These studies demonstrate the power of mathematical modeling for understanding TGF-beta biology.

Dynamics of maternal morphogen gradients in Drosophila

The first direct studies of morphogen gradients were done in the end of 1980s, in the early Drosophila embryo, which is patterned under the action of four maternally determined morphogens. Since the early studies of maternal morphogens were done with fixed embryos, they were viewed as relatively static signals. Several recent studies analyze dynamics of the anterior, dorsoventral, and terminal patterning signals. The results of these quantitative studies provide critical tests of classical models and reveal new modes of morphogen regulation and readout in one of the most extensively studied patterning systems.

Mathematical modeling identifies Smad nucleocytoplasmic shuttling as a dynamic signal-interpreting system

TGF-beta-induced Smad signal transduction from the membrane into the nucleus is not linear and unidirectional, but rather a dynamic network that couples Smad phosphorylation and dephosphorylation through continuous nucleocytoplasmic shuttling of Smads. To understand the quantitative behavior of this network, we have developed a tightly constrained computational model, exploiting the interplay between mathematical modeling and experimental strategies. The model simultaneously reproduces four distinct datasets with excellent accuracy and provides mechanistic insights into how the network operates. We use the model to make predictions about the outcome of fluorescence recovery after photobleaching experiments and the behavior of a functionally impaired Smad2 mutant, which we then verify experimentally. Successful model performance strongly supports the hypothesis of a dynamic maintenance of Smad nuclear accumulation during active signaling. The presented work establishes Smad nucleocytoplasmic shuttling as a dynamic network that flexibly transmits quantitative features of the extracellular TGF-beta signal, such as its duration and intensity, into the nucleus.

Cell population-based model of dermal wound invasion with heterogeneous intracellular signaling properties

A deterministic model of dermal wound invasion, which accounts for the platelet-derived growth factor (PDGF) gradient sensing mechanism in fibroblasts mediated by cell surface receptors and the phosphoinositide 3-kinase (PI3K) signal transduction pathway, was previously described (Biophys J 2006; 90:2297-308). Here, we extend that work and implement a hybrid modeling strategy that treats fibroblasts as discrete entities endowed with heterogeneous properties, namely receptor, PI3K and 3' phosphoinositide phosphatase expression levels. Analysis of the model suggests that the wound environment fosters the advancement of cells within the population that are better fit to migrate and/or proliferate in response to PDGF stimulation. Thus, cell-to-cell variability results in a significantly higher rate of wound invasion as compared with the deterministic model, in a manner that depends on the way in which individual cell properties are sampled or inherited upon cell division.

Modeling the bicoid gradient: diffusion and reversible nuclear trapping of a stable protein

The Bicoid gradient in the Drosophila embryo provided the first example of a morphogen gradient studied at the molecular level. The exponential shape of the Bicoid gradient had always been interpreted within the framework of the localized production, diffusion, and degradation model. We propose an alternative mechanism, which assumes no Bicoid degradation. The medium where the Bicoid gradient is formed and interpreted is very dynamic. Most notably, the number of nuclei changes over three orders of magnitude from fertilization, when Bicoid synthesis is initiated, to nuclear cycle 14 when most of the measurements were taken. We demonstrate that a model based on Bicoid diffusion and nucleocytoplasmic shuttling in the presence of the growing number of nuclei can account for most of the properties of the Bicoid concentration profile. Consistent with experimental observations, the Bicoid gradient in our model is established before nuclei migrate to the periphery of the embryo and remains stable during subsequent nuclear divisions.

TGFbeta-SMAD signal transduction: molecular specificity and functional flexibility

Ligands of the transforming growth factor-beta (TGFbeta) superfamily of growth factors initiate signal transduction through a bewildering complexity of ligand-receptor interactions. Signalling then converges to nuclear accumulation of transcriptionally active SMAD complexes and gives rise to a plethora of specific functional responses in both embryos and adult organisms. Current research is focused on the mechanisms that regulate SMAD activity to evoke cell-type-specific and context-dependent transcriptional programmes. An equally important challenge is understanding the functional role of signal strength and duration. How are these quantitative aspects of the extracellular signal regulated? How are they then sensed and interpreted, and how do they affect responses?

Modeling homeoprotein intercellular transfer unveils a parsimonious mechanism for gradient and boundary formation in early brain development

Morphogens are molecules inducing morphogenetic responses from cells and cell ensembles. The concept of morphogen is related to that of positional value, as the generation of morphological and physiological characteristics is function of position. Based on the observation that homeoproteins, a category of transcription factors with morphogenetic functions, traffic between abutting cells and, very often, regulate their own expression, we develop here a biophysical model of homeoprotein propagation and study the associated mathematical equations. This mode of cell signaling can generate domains of homeoprotein expression. We study both the transient and steady-state regimes and, in this latter regime, we obtain various morphogenetic gradients, depending on the value of some parameters, such as morphogen synthesis, degradation rates and efficiency of intercellular passage. The same equations, applied to pairs of homeoproteins with auto-activation and reciprocal inhibition properties, account for border formation. They also allow us to compute how specific perturbations can either be buffered or lead to modifications in the position of borders between adjacent areas. The model developed here, based on experimental data, and avoids theoretical obstacles associated with pluricellularity. It extends the idea that Bicoid homeoprotein is a morphogen in the fly embryo syncitium to most homeoproteins and to pluricellular systems. Because the position of borders between brain areas is of primary physiological importance, our model might lead to original views regarding epigenetic inter-individual variations and the origin of neurological and psychiatric diseases. In addition, it provides new hypotheses regarding the molecular basis of brain evolution.

Modeling tissue morphogenesis and cancer in 3D

Three-dimensional (3D) in vitro models span the gap between two-dimensional cell cultures and whole-animal systems. By mimicking features of the in vivo environment and taking advantage of the same tools used to study cells in traditional cell culture, 3D models provide unique perspectives on the behavior of stem cells, developing tissues and organs, and tumors. These models may help to accelerate translational research in cancer biology and tissue engineering.

Efficient parameter estimation for spatio-temporal models of pattern formation: case study of Drosophila melanogaster

MOTIVATION

Diffusable and non-diffusable gene products play a major role in body plan formation. A quantitative understanding of the spatio-temporal patterns formed in body plan formation, by using simulation models is an important addition to experimental observation. The inverse modelling approach consists of describing the body plan formation by a rule-based model, and fitting the model parameters to real observed data. In body plan formation, the data are usually obtained from fluorescent immunohistochemistry or in situ hybridizations. Inferring model parameters by comparing such data to those from simulation is a major computational bottleneck. An important aspect in this process is the choice of method used for parameter estimation. When no information on parameters is available, parameter estimation is mostly done by means of heuristic algorithms.

RESULTS

We show that parameter estimation for pattern formation models can be efficiently performed using an evolution strategy (ES). As a case study we use a quantitative spatio-temporal model of the regulatory network for early development in Drosophila melanogaster. In order to estimate the parameters, the simulated results are compared to a time series of gene products involved in the network obtained with immunohistochemistry. We demonstrate that a (mu,lambda)-ES can be used to find good quality solutions in the parameter estimation. We also show that an ES with multiple populations is 5-140 times as fast as parallel simulated annealing for this case study, and that combining ES with a local search results in an efficient parameter estimation method.

A propagating ERKII switch forms zones of elevated dendritic activation correlated with plasticity

Strong inputs to neurons trigger complex biochemical events leading to synaptic plasticity. These biochemical events occur at many spatial scales, ranging from submicron dendritic spines to signals that propagate hundreds of microns from dendrites to the nucleus. ERKII is an important signaling molecule that is involved in many aspects of plasticity, including local excitability, communication with the nucleus, and control of local protein synthesis. We observed that ERKII activation spreads long distances in apical dendrites of stimulated hippocampal CA1 pyramidal neurons. We combined experiments and models to show that this >100 mum spread was too large to be explained by biochemical reaction-diffusion effects. We show that two modes of calcium entry along the dendrite contribute to the extensive activation of ERKII. We predict the occurrence of feedback between biophysical events resulting in calcium entry, and biochemical events resulting in ERKII activation. This feedback causes a switch-like propagation of ERKII activation, coupled with enhanced electrical excitability, along the apical dendrite. We propose that this propagating switch forms zones on dendrites in which plasticity is facilitated.

A propagating ERKII switch forms zones of elevated dendritic activation correlated with plasticity

Strong inputs to neurons trigger complex biochemical events leading to synaptic plasticity. These biochemical events occur at many spatial scales, ranging from submicron dendritic spines to signals that propagate hundreds of microns from dendrites to the nucleus. ERKII is an important signaling molecule that is involved in many aspects of plasticity, including local excitability, communication with the nucleus, and control of local protein synthesis. We observed that ERKII activation spreads long distances in apical dendrites of stimulated hippocampal CA1 pyramidal neurons. We combined experiments and models to show that this >100 mum spread was too large to be explained by biochemical reaction-diffusion effects. We show that two modes of calcium entry along the dendrite contribute to the extensive activation of ERKII. We predict the occurrence of feedback between biophysical events resulting in calcium entry, and biochemical events resulting in ERKII activation. This feedback causes a switch-like propagation of ERKII activation, coupled with enhanced electrical excitability, along the apical dendrite. We propose that this propagating switch forms zones on dendrites in which plasticity is facilitated.

Enhancer responses to similarly distributed antagonistic gradients in development

Formation of spatial gene expression patterns in development depends on transcriptional responses mediated by gene control regions, enhancers. Here, we explore possible responses of enhancers to overlapping gradients of antagonistic transcriptional regulators in the Drosophila embryo. Using quantitative models based on enhancer structure, we demonstrate how a pair of antagonistic transcription factor gradients with similar or even identical spatial distributions can lead to the formation of distinct gene expression domains along the embryo axes. The described mechanisms are sufficient to explain the formation of the anterior and the posterior knirps expression, the posterior hunchback expression domain, and the lateral stripes of rhomboid expression and of other ventral neurogenic ectodermal genes. The considered principles of interaction between antagonistic gradients at the enhancer level can also be applied to diverse developmental processes, such as domain specification in imaginal discs, or even eyespot pattern formation in the butterfly wing.

The early development of the fruit fly embryo depends on an intricate but well-studied gene regulatory network. In fly eggs, maternally deposited gene products—morphogenes—form spatial concentration gradients. The graded distribution of the maternal morphogenes initiates a cascade of gene interactions leading to embryo development. Gradients of activators and repressors regulating common target genes may produce different outcomes depending on molecular mechanisms, mediating their function. Here, we describe quantitative mathematical models for the interplay between gradients of positive and negative transcriptional regulators—proteins, activating or repressing their target genes through binding the gene's regulatory DNA sequences. We predict possible spatial outcomes of the transcriptional antagonistic interactions in fly development and consider examples where the predicted cases may take place.

A synthetic biology challenge: making cells compute

Advances in biology and engineering have enabled the reprogramming of cells with well-defined functions, leading to the emergence of synthetic biology. Early successes in this nascent field suggest its potential to impact diverse areas. Here, we examine the feasibility of engineering circuits for cell-based computation. We illustrate the basic concepts by describing the mapping of several computational problems to engineered gene circuits. Revolving around these examples and past studies, we discuss technologies and computational methods available to design, test, and optimize gene circuits. We conclude with discussion of challenges involved in a typical design cycle, as well as those specific to cellular computation.

Morpheus Unbound: Reimagining the Morphogen Gradient

The theory that the spatial organization of cell fate is orchestrated by gradients of diffusing molecules was a major contribution to 20th century developmental biology. Although the existence of morphogens is no longer in doubt, studies on the formation and function of their gradients have yielded far more puzzles than answers. On close inspection, every morphogen gradient seems to use a rich array of regulatory mechanisms, suggesting that the tasks carried out by such systems are far more extensive than previously thought.