Computational approaches to developmental patterning

How a reaction-diffusion signal can control spinal cord regeneration in axolotls: A modeling study

Axolotls are uniquely able to completely regenerate the spinal cord after amputation. The underlying governing mechanisms of this regenerative response have not yet been fully elucidated. We previously found that spinal cord regeneration is mainly driven by cell-cycle acceleration of ependymal cells, recruited by a hypothetical signal propagating from the injury. However, the nature of the signal and its propagation remain unknown. In this theoretical study, we investigated whether the regeneration-inducing signal can follow a reaction-diffusion process. We developed a computational model, validated it with experimental data, and showed that the signal dynamics can be understood in terms of reaction-diffusion mechanism. By developing a theory of the regenerating outgrowth in the limit of fast reaction-diffusion, we demonstrate that control of regenerative response solely relies on cell-to-signal sensitivity and the signal reaction-diffusion characteristic length. This study lays foundations for further identification of the signal controlling regeneration of the spinal cord.

A mathematical framework for understanding the spontaneous emergence of complexity applicable to growing multicellular systems

In embryonic development and organogenesis, cells sharing identical genetic codes acquire diverse gene expression states in a highly reproducible spatial distribution, crucial for multicellular formation and quantifiable through positional information. To understand the spontaneous growth of complexity, we constructed a one-dimensional division-decision model, simulating the growth of cells with identical genetic networks from a single cell. Our findings highlight the pivotal role of cell division in providing positional cues, escorting the system toward states rich in information. Moreover, we pinpointed lateral inhibition as a critical mechanism translating spatial contacts into gene expression. Our model demonstrates that the spatial arrangement resulting from cell division, combined with cell lineages, imparts positional information, specifying multiple cell states with increased complexity-illustrated through examples in C.elegans. This study constitutes a foundational step in comprehending developmental intricacies, paving the way for future quantitative formulations to construct synthetic multicellular patterns.

Mathematical modeling for developmental processes

We review several mathematical models and concepts in developmental biology that have been established over the last decade. (1) Feedback vertex set: Ascidian embryos contain cells of seven types, and cell fate is controlled by ~100 interacting genes. The "feedback vertex set" of the directed graph of the gene regulatory network consists of a small number of genes. By experimentally manipulating them, we can differentiate cells into any cell type. (2) Tissue deformation: Describing morphological changes in tissues and relating them to gene expression and other cellular processes is key in understanding morphogenesis. Expansion and anisotropy of the tissue are described by a "deformation tensor" at each location. A study on chick limb bud formation revealed that both the volume growth rate and anisotropy in deformation differed significantly between locations and stages. (3) Mechanobiology: Forces operating on each cell may alter cell shape and gene expression, which may subsequently exert forces on their surroundings. Measurements of force, tissue shape, and gene expression help us understand autonomous tissue deformation. (4) Adaptive design of development: An optimal growth schedule in fluctuating environments explains the growth response to starvation in Drosophila larvae. Adaptive placement of morphogen sources makes development robust to noises.

Regulative development as a model for origin of life and artificial life studies

Using the formal framework of the Free Energy Principle, we show how generic thermodynamic requirements on bidirectional information exchange between a system and its environment can generate complexity. This leads to the emergence of hierarchical computational architectures in systems that operate sufficiently far from thermal equilibrium. In this setting, the environment of any system increases its ability to predict system behavior by "engineering" the system towards increased morphological complexity and hence larger-scale, more macroscopic behaviors. When seen in this light, regulative development becomes an environmentally-driven process in which "parts" are assembled to produce a system with predictable behavior. We suggest on this basis that life is thermodynamically favorable and that, when designing artificial living systems, human engineers are acting like a generic "environment".

Unidirectional and phase-gated signaling synchronizes murine presomitic mesoderm cells

Oscillator systems achieve synchronization when oscillators are coupled. The presomitic mesoderm is a system of cellular oscillators, where coordinated genetic activity is necessary for proper periodic generation of somites. While Notch signaling is required for the synchronization of these cells, it is unclear what information the cells exchange and how they react to this information to align their oscillatory pace with that of their neighbors. Combining mathematical modeling and experimental data, we found that interaction between murine presomitic mesoderm cells is controlled by a phase-gated and unidirectional coupling mechanism and results in deceleration of their oscillation pace upon Notch signaling. This mechanism predicts that isolated populations of well-mixed cells synchronize, revealing a stereotypical synchronization in the mouse PSM and contradicting expectations from previously applied theoretical approaches. Collectively, our theoretical and experimental findings reveal the underlying coupling mechanisms of the presomitic mesoderm cells and provide a framework to quantitatively characterize their synchronization.

Cell state transitions: catch them if you can

The Company of Biologists' 2022 workshop on 'Cell State Transitions: Approaches, Experimental Systems and Models' brought together an international and interdisciplinary team of investigators spanning the fields of cell and developmental biology, stem cell biology, physics, mathematics and engineering to tackle the question of how cells precisely navigate between distinct identities and do so in a dynamic manner. This second edition of the workshop was organized after a successful virtual workshop on the same topic that took place in 2021.

Orchestration of tissue shape changes and gene expression patterns in development

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

Minimal Developmental Computation: A Causal Network Approach to Understand Morphogenetic Pattern Formation

What information-processing strategies and general principles are sufficient to enable self-organized morphogenesis in embryogenesis and regeneration? We designed and analyzed a minimal model of self-scaling axial patterning consisting of a cellular network that develops activity patterns within implicitly set bounds. The properties of the cells are determined by internal 'genetic' networks with an architecture shared across all cells. We used machine-learning to identify models that enable this virtual mini-embryo to pattern a typical axial gradient while simultaneously sensing the set boundaries within which to develop it from homogeneous conditions-a setting that captures the essence of early embryogenesis. Interestingly, the model revealed several features (such as planar polarity and regenerative re-scaling capacity) for which it was not directly selected, showing how these common biological design principles can emerge as a consequence of simple patterning modes. A novel "causal network" analysis of the best model furthermore revealed that the originally symmetric model dynamically integrates into intercellular causal networks characterized by broken-symmetry, long-range influence and modularity, offering an interpretable macroscale-circuit-based explanation for phenotypic patterning. This work shows how computation could occur in biological development and how machine learning approaches can generate hypotheses and deepen our understanding of how featureless tissues might develop sophisticated patterns-an essential step towards predictive control of morphogenesis in regenerative medicine or synthetic bioengineering contexts. The tools developed here also have the potential to benefit machine learning via new forms of backpropagation and by leveraging the novel distributed self-representation mechanisms to improve robustness and generalization.

Size matters: tissue size as a marker for a transition between reaction-diffusion regimes in spatio-temporal distribution of morphogens

The reaction-diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction-diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction-diffusion length, and identified two reaction-diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction-diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction-diffusion model to study tissue morphogenesis.

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Global entrainment in the brain-body-environment: retrospective and prospective views

We celebrate the 60th anniversary of Biological Cybernetics. It has also been 30 years since "Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment" was published in Biological Cybernetics (Taga et al. in Biol Cybern 65(3):147-159, 1991). I would like to look back on the creation of this paper and discuss its subsequent development and future perspectives.

Mechano-chemical feedback mediated competition for BMP signalling leads to pattern formation

Developmental patterning is thought to be regulated by conserved signalling pathways. Initial patterns are often broad before refining to only those cells that commit to a particular fate. However, the mechanisms by which pattern refinement takes place remain to be addressed. Using the posterior crossvein (PCV) of the Drosophila pupal wing as a model, into which bone morphogenetic protein (BMP) ligand is extracellularly transported to instruct vein patterning, we investigate how pattern refinement is regulated. We found that BMP signalling induces apical enrichment of Myosin II in developing crossvein cells to regulate apical constriction. Live imaging of cellular behaviour indicates that changes in cell shape are dynamic and transient, only being maintained in those cells that retain vein fate competence after refinement. Disrupting cell shape changes throughout the PCV inhibits pattern refinement. In contrast, disrupting cell shape in only a subset of vein cells can result in a loss of BMP signalling. We propose that mechano-chemical feedback leads to competition for the developmental signal which plays a critical role in pattern refinement.

A self-exciting point process to study multicellular spatial signaling patterns

Multicellular organisms rely on spatial signaling among cells to drive their organization, development, and response to stimuli. Several models have been proposed to capture the behavior of spatial signaling in multicellular systems, but existing approaches fail to capture both the autonomous behavior of single cells and the interactions of a cell with its neighbors simultaneously. We propose a spatiotemporal model of dynamic cell signaling based on Hawkes processes-self-exciting point processes-that model the signaling processes within a cell and spatial couplings between cells. With this cellular point process (CPP), we capture both the single-cell pathway activation rate and the magnitude and duration of signaling between cells relative to their spatial location. Furthermore, our model captures tissues composed of heterogeneous cell types with different bursting rates and signaling behaviors across multiple signaling proteins. We apply our model to epithelial cell systems that exhibit a range of autonomous and spatial signaling behaviors basally and under pharmacological exposure. Our model identifies known drug-induced signaling deficits, characterizes signaling changes across a wound front, and generalizes to multichannel observations.

Spatiotemporal control of cell cycle acceleration during axolotl spinal cord regeneration

Axolotls are uniquely able to resolve spinal cord injuries, but little is known about the mechanisms underlying spinal cord regeneration. We previously found that tail amputation leads to reactivation of a developmental-like program in spinal cord ependymal cells (Rodrigo Albors et al., 2015), characterized by a high-proliferation zone emerging 4 days post-amputation (Rost et al., 2016). What underlies this spatiotemporal pattern of cell proliferation, however, remained unknown. Here, we use modeling, tightly linked to experimental data, to demonstrate that this regenerative response is consistent with a signal that recruits ependymal cells during ~85 hours after amputation within ~830 μm of the injury. We adapted Fluorescent Ubiquitination-based Cell Cycle Indicator (FUCCI) technology to axolotls (AxFUCCI) to visualize cell cycles in vivo. AxFUCCI axolotls confirmed the predicted appearance time and size of the injury-induced recruitment zone and revealed cell cycle synchrony between ependymal cells. Our modeling and imaging move us closer to understanding bona fide spinal cord regeneration.

Analysis of an improved fractional-order model of boundary formation in the Drosophila large intestine dependent on Delta-Notch pathway

In this paper, an improved fractional-order model of boundary formation in the Drosophila large intestine dependent on Delta-Notch pathway is proposed for the first time. The uniqueness, nonnegativity, and boundedness of solutions are studied. In a two cells model, there are two equilibriums (no-expression of Delta and normal expression of Delta). Local asymptotic stability is proved for both cases. Stability analysis shows that the orders of the fractional-order differential equation model can significantly affect the equilibriums in the two cells model. Numerical simulations are presented to illustrate the conclusions. Next, the sensitivity of model parameters is calculated, and the calculation results show that different parameters have different sensitivities. The most and least sensitive parameters in the two cells model and the 60 cells model are verified by numerical simulations. What is more, we compare the fractional-order model with the integer-order model by simulations, and the results show that the orders can significantly affect the dynamic and the phenotypes.

A general mathematical framework for understanding the behavior of heterogeneous stem cell regeneration

Stem cell heterogeneity is essential for homeostasis in tissue development. This paper establishes a general mathematical framework to model the dynamics of stem cell regeneration with cell heterogeneity and random transitions of epigenetic states. The framework generalizes the classical G0 cell cycle model and incorporates the epigenetic states of individual cells represented by a continuous multidimensional variable. In the model, the kinetic rates of cell behaviors, including proliferation, differentiation, and apoptosis, are dependent on their epigenetic states, and the random transitions of epigenetic states between cell cycles are represented by an inheritance probability function that describes the conditional probability of cell state changes. Moreover, the model can be extended to include genotypic changes and describe the process of gene mutation-induced tumor development. The proposed mathematical framework provides a generalized formula that helps us to understand various dynamic processes of stem cell regeneration, including tissue development, degeneration, and abnormal growth.

Cell-in-the-loop pattern formation with optogenetically emulated cell-to-cell signaling

Designing and implementing synthetic biological pattern formation remains challenging due to underlying theoretical complexity as well as the difficulty of engineering multicellular networks biochemically. Here, we introduce a cell-in-the-loop approach where living cells interact through in silico signaling, establishing a new testbed to interrogate theoretical principles when internal cell dynamics are incorporated rather than modeled. We present an easy-to-use theoretical test to predict the emergence of contrasting patterns in gene expression among laterally inhibiting cells. Guided by the theory, we experimentally demonstrate spontaneous checkerboard patterning in an optogenetic setup, where cell-to-cell signaling is emulated with light inputs calculated in silico from real-time gene expression measurements. The scheme successfully produces spontaneous, persistent checkerboard patterns for systems of sixteen patches, in quantitative agreement with theoretical predictions. Our research highlights how tools from dynamical systems theory may inform our understanding of patterning, and illustrates the potential of cell-in-the-loop for engineering synthetic multicellular systems.

Stochastic pulsing of gene expression enables the generation of spatial patterns in Bacillus subtilis biofilms

Stochastic pulsing of gene expression can generate phenotypic diversity in a genetically identical population of cells, but it is unclear whether it has a role in the development of multicellular systems. Here, we show how stochastic pulsing of gene expression enables spatial patterns to form in a model multicellular system, Bacillus subtilis bacterial biofilms. We use quantitative microscopy and time-lapse imaging to observe pulses in the activity of the general stress response sigma factor σ^B in individual cells during biofilm development. Both σ^B and sporulation activity increase in a gradient, peaking at the top of the biofilm, even though σ^B represses sporulation. As predicted by a simple mathematical model, increasing σ^B expression shifts the peak of sporulation to the middle of the biofilm. Our results demonstrate how stochastic pulsing of gene expression can play a key role in pattern formation during biofilm development.

Stochastic pulsing of gene expression can generate phenotypic diversity in a genetically identical population of cells. Here, the authors show that stochastic pulsing in the expression of a sigma factor enables the formation of spatial patterns in a multicellular system, Bacillus subtilis bacterial biofilms.

Turing patterns modulation by chemical gradient in isothermal and non-isothermal conditions

Chemical gradients imposed through boundary conditions induce spatial symmetry breaking of Turing patterns in small systems.

From non-excitable single-cell to multicellular bioelectrical states supported by ion channels and gap junction proteins: Electrical potentials as distributed controllers

Endogenous bioelectric patterns within tissues are an important driver of morphogenesis and a tractable component of a number of disease states. Developing system-level understanding of the dynamics by which non-neural bioelectric circuits regulate complex downstream cascades is a key step towards both, an evolutionary understanding of ion channel genes, and novel strategies in regenerative medicine. An important capability gap is deriving rational modulation strategies targeting individual cells' bioelectric states to achieve global (tissue- or organ-level) outcomes. Here, we develop an ion channel-based model that describes multicellular states on the basis of spatio-temporal patterns of electrical potentials in aggregates of non-excitable cells. The model is of biological interest because modern techniques allow to associate bioelectrical signals with specific ion channel proteins in the cell membrane that are central to embryogenesis, regeneration, and tumorigenesis. As a complementary approach to the usual biochemical description, we have studied four biophysical questions: (i) how can single-cell bioelectrical states be established; (ii) how can a change in the cell potential caused by a transient perturbation of the cell state be maintained after the stimulus is gone (bioelectrical memory); (iii) how can a single-cell contribute to the control of multicellular ensembles based on the spatio-temporal pattern of electrical potentials; and (iv) how can oscillatory patterns arise from the single-cell bioelectrical dynamics. Experimentally, endogenous bioelectric gradients have emerged as instructive agents for morphogenetic processes. In this context, the simulations can guide new procedures that may allow a distributed control of the multicellular ensemble.

Synchronization of Bioelectric Oscillations in Networks of Nonexcitable Cells: From Single-Cell to Multicellular States

Biological networks use collective oscillations for information processing tasks. In particular, oscillatory membrane potentials have been observed in nonexcitable cells and bacterial communities where specific ion channel proteins contribute to the bioelectric coordination of large populations. We aim at describing theoretically the oscillatory spatiotemporal patterns that emerge at the multicellular level from the single-cell bioelectric dynamics. To this end, we focus on two key questions: (i) What single-cell properties are relevant to multicellular behavior? (ii) What properties defined at the multicellular level can allow an external control of the bioelectric dynamics? In particular, we explore the interplay between transcriptional and translational dynamics and membrane potential dynamics in a model multicellular ensemble, describe the spatiotemporal patterns that arise when the average electric potential allows groups of cells to act as a coordinated multicellular patch, and characterize the resulting synchronization phenomena. The simulations concern bioelectric networks and collective communication across different scales based on oscillatory and synchronization phenomena, thus shedding light on the physiological dynamics of a wide range of endogenous contexts across embryogenesis and regeneration.

From single cells to tissue self-organization

Self-organization is a process by which interacting cells organize and arrange themselves in higher order structures and patterns. To achieve this, cells must have molecular mechanisms to sense their complex local environment and interpret it to respond accordingly. A combination of cell-intrinsic and cell-extrinsic cues are decoded by the single cells dictating their behaviour, their differentiation and symmetry-breaking potential driving development, tissue remodeling and regenerative processes. A unifying property of these self-organized pattern-forming systems is the importance of fluctuations, cell-to-cell variability, or noise. Cell-to-cell variability is an inherent and emergent property of populations of cells that maximize the population performance instead of the individual cell, providing tissues the flexibility to develop and maintain homeostasis in diverse environments. In this review, we will explore the role of self-organization and cell-to-cell variability as fundamental properties of multicellularity-and the requisite of single-cell resolution for its understanding. Moreover, we will analyze how single cells generate emergent multicellular dynamics observed at the tissue level 'travelling' across different scales: spatial, temporal and functional.

A local uPAR-plasmin-TGFβ1 positive feedback loop in a qualitative computational model of angiogenic sprouting explains the in vitro effect of fibrinogen variants

In experimental assays of angiogenesis in three-dimensional fibrin matrices, a temporary scaffold formed during wound healing, the type and composition of fibrin impacts the level of sprouting. More sprouts form on high molecular weight (HMW) than on low molecular weight (LMW) fibrin. It is unclear what mechanisms regulate the number and the positions of the vascular-like structures in cell cultures. To address this question, we propose a mechanistic simulation model of endothelial cell migration and fibrin proteolysis by the plasmin system. The model is a hybrid, cell-based and continuum, computational model based on the cellular Potts model and sets of partial-differential equations. Based on the model results, we propose that a positive feedback mechanism between uPAR, plasmin and transforming growth factor β1 (TGFβ1) selects cells in the monolayer for matrix invasion. Invading cells releases TGFβ1 from the extracellular matrix through plasmin-mediated fibrin degradation. The activated TGFβ1 further stimulates fibrin degradation and keeps proteolysis active as the sprout invades the fibrin matrix. The binding capacity for TGFβ1 of LMW is reduced relative to that of HMW. This leads to reduced activation of proteolysis and, consequently, reduced cell ingrowth in LMW fibrin compared to HMW fibrin. Thus our model predicts that endothelial cells in LMW fibrin matrices compared to HMW matrices show reduced sprouting due to a lower bio-availability of TGFβ1.

Therapies for a range of medical conditions, including cancer, wound healing and diabetic retinopathy can benefit from a better control over the growth of blood vessels. The chemical properties of fibrin, the material that forms scabs in wounds and can also occur in large concentrations in tumors, can regulate the degree of blood vessel growth (angiogenesis). Angiogenesis can be mimicked in cell cultures. These allow us to modulate the chemical properties of fibrin and study the effect on angiogenesis. Fibrin occurs in high molecular weight (HMW) and in low molecular weight (LMW) forms. Interestingly, there is more ingrowth of angiogenic-like structures into HMW than in LMW fibrin, but the mechanisms are poorly understood. To get more insight into these, we constructed a computational model. Using the model, we propose and analyse a hypothetical mechanism for sprouting that could explain the differences in endothelial cell sprouting in LMW and HMW fibrin matrices. Our model suggests that cells digest fibrin, thus creating space for ingrowth. At the same time, digestion frees growth factors bound to fibrin, that activates further secretion of digestive enzymes by the cells. We propose that the resulting positive feedback loop spontaneously selects cells in the endothelial monolayer for ingrowth and helps the blood vessel sprout move deeper into the fibrin. This could be a complementary mechanism to lateral-inhibition by Delta-Notch for the selection of leader cells, also called ‘tip cells’. Our model predicts that endothelial cells in LMW fibrin compared to HMW fibrin show reduced sprouting due to a lower bio-availability of TGFβ1.

Interactions between the circadian clock and TGF-β signaling pathway in zebrafish

Background

TGF-β signaling is a cellular pathway that functions in most cells and has been shown to play a role in multiple processes, such as the immune response, cell differentiation and proliferation. Recent evidence suggests a possible interaction between TGF-β signaling and the molecular circadian oscillator. The current study aims to characterize this interaction in the zebrafish at the molecular and behavioral levels, taking advantage of the early development of a functional circadian clock and the availability of light-entrainable clock-containing cell lines.

Results

Smad3a, a TGF-β signaling-related gene, exhibited a circadian expression pattern throughout the brain of zebrafish larvae. Both pharmacological inhibition and indirect activation of TGF-β signaling in zebrafish Pac-2 cells caused a concentration dependent disruption of rhythmic promoter activity of the core clock gene Per1b. Inhibition of TGF-β signaling in intact zebrafish larvae caused a phase delay in the rhythmic expression of Per1b mRNA. TGF-β inhibition also reversibly disrupted, phase delayed and increased the period of circadian rhythms of locomotor activity in zebrafish larvae.

Conclusions

The current research provides evidence for an interaction between the TGF-β signaling pathway and the circadian clock system at the molecular and behavioral levels, and points to the importance of TGF-β signaling for normal circadian clock function. Future examination of this interaction should contribute to a better understanding of its underlying mechanisms and its influence on a variety of cellular processes including the cell cycle, with possible implications for cancer development and progression.

Modeling and analysis of the Delta-Notch dependent boundary formation in the Drosophila large intestine

BACKGROUND

The boundary formation in the Drosophila large intestine is widely studied as an important biological problem. It has been shown that the Delta-Notch signaling pathway plays an essential role in the formation of boundary cells.

RESULTS

In this paper, we propose a mathematical model for the Delta-Notch dependent boundary formation in the Drosophila large intestine in order to better interpret related experimental findings of this biological phenomenon. To achieve this, we not only perform stability analysis on the model from a theoretical point of view, but also perform numerical simulations to analyze the model with and without noises, the phenotype change with the change of Delta or Notch expression, and the perturbation influences of binding and inhibition parameters on the boundary formation.

CONCLUSIONS

By doing all these work, we can assure that our model can better interpret the biological findings related to the boundary formation in the Drosophila large intestine.

Cell Division in the Light of Modeling

Theoretical modeling is central to elucidating underlying principles of emergent properties of complex systems. In cell and developmental biology, the last 15 years have witnessed a convergence of empirical and modeling approaches for fresh perspectives. The role of cell division in coordinating size, shape, and fate in particular illustrates the ever-growing impact of modeling.

Data-driven modeling reveals cell behaviors controlling self-organization during Myxococcus xanthus development

Collective cell movement is critical to the emergent properties of many multicellular systems, including microbial self-organization in biofilms, embryogenesis, wound healing, and cancer metastasis. However, even the best-studied systems lack a complete picture of how diverse physical and chemical cues act upon individual cells to ensure coordinated multicellular behavior. Known for its social developmental cycle, the bacterium Myxococcus xanthus uses coordinated movement to generate three-dimensional aggregates called fruiting bodies. Despite extensive progress in identifying genes controlling fruiting body development, cell behaviors and cell-cell communication mechanisms that mediate aggregation are largely unknown. We developed an approach to examine emergent behaviors that couples fluorescent cell tracking with data-driven models. A unique feature of this approach is the ability to identify cell behaviors affecting the observed aggregation dynamics without full knowledge of the underlying biological mechanisms. The fluorescent cell tracking revealed large deviations in the behavior of individual cells. Our modeling method indicated that decreased cell motility inside the aggregates, a biased walk toward aggregate centroids, and alignment among neighboring cells in a radial direction to the nearest aggregate are behaviors that enhance aggregation dynamics. Our modeling method also revealed that aggregation is generally robust to perturbations in these behaviors and identified possible compensatory mechanisms. The resulting approach of directly combining behavior quantification with data-driven simulations can be applied to more complex systems of collective cell movement without prior knowledge of the cellular machinery and behavioral cues.

Shaping development by stochasticity and dynamics in gene regulation

Animal development is orchestrated by spatio-temporal gene expression programmes that drive precise lineage commitment, proliferation and migration events at the single-cell level, collectively leading to large-scale morphological change and functional specification in the whole organism. Efforts over decades have uncovered two 'seemingly contradictory' mechanisms in gene regulation governing these intricate processes: (i) stochasticity at individual gene regulatory steps in single cells and (ii) highly coordinated gene expression dynamics in the embryo. Here we discuss how these two layers of regulation arise from the molecular and the systems level, and how they might interplay to determine cell fate and to control the complex body plan. We also review recent technological advancements that enable quantitative analysis of gene regulation dynamics at single-cell, single-molecule resolution. These approaches outline next-generation experiments to decipher general principles bridging gaps between molecular dynamics in single cells and robust gene regulations in the embryo.

Applying evolutionary genetics to developmental toxicology and risk assessment

Evolutionary thinking continues to challenge our views on health and disease. Yet, there is a communication gap between evolutionary biologists and toxicologists in recognizing the connections among developmental pathways, high-throughput screening, and birth defects in humans. To increase our capability in identifying potential developmental toxicants in humans, we propose to apply evolutionary genetics to improve the experimental design and data interpretation with various in vitro and whole-organism models. We review five molecular systems of stress response and update 18 consensual cell-cell signaling pathways that are the hallmark for early development, organogenesis, and differentiation; and revisit the principles of teratology in light of recent advances in high-throughput screening, big data techniques, and systems toxicology. Multiscale systems modeling plays an integral role in the evolutionary approach to cross-species extrapolation. Phylogenetic analysis and comparative bioinformatics are both valuable tools in identifying and validating the molecular initiating events that account for adverse developmental outcomes in humans. The discordance of susceptibility between test species and humans (ontogeny) reflects their differences in evolutionary history (phylogeny). This synthesis not only can lead to novel applications in developmental toxicity and risk assessment, but also can pave the way for applying an evo-devo perspective to the study of developmental origins of health and disease.

Methods for Modeling Brassinosteroid-Mediated Signaling in Plant Development

Mathematical modeling of biological processes is a useful tool to draw conclusions that are contained in the data, but not directly reachable, as well as to make predictions and select the most efficient follow-up experiments. Here we outline a method to model systems of a few proteins that interact transcriptionally and/or posttranscriptionally, by representing the system as Ordinary Differential Equations and to study the model dynamics and stationary states. We exemplify this method by focusing on the regulation by the brassinosteroid (BR) signaling component BRASSINOSTEROID INSENSITIVE1 ETHYL METHYL SULFONATE SUPPRESSOR1 (BES1) of BRAVO, a quiescence-regulating transcription factor expressed in the quiescent cells of Arabidopsis thaliana roots. The method to extract the stationary states and the dynamics is provided as a Mathematica code and requires basic knowledge of the Mathematica software to be executed.

The interplay between genetic and bioelectrical signaling permits a spatial regionalisation of membrane potentials in model multicellular ensembles

The single cell-centred approach emphasises ion channels as specific proteins that determine individual properties, disregarding their contribution to multicellular outcomes. We simulate the interplay between genetic and bioelectrical signals in non-excitable cells from the local single-cell level to the long range multicellular ensemble. The single-cell genetic regulation is based on mean-field kinetic equations involving the mRNA and protein concentrations. The transcription rate factor is assumed to depend on the absolute value of the cell potential, which is dictated by the voltage-gated cell ion channels and the intercellular gap junctions. The interplay between genetic and electrical signals may allow translating single-cell states into multicellular states which provide spatio-temporal information. The model results have clear implications for biological processes: (i) bioelectric signals can override slightly different genetic pre-patterns; (ii) ensembles of cells initially at the same potential can undergo an electrical regionalisation because of persistent genetic differences between adjacent spatial regions; and (iii) shifts in the normal cell electrical balance could trigger significant changes in the genetic regulation.

Extracellular matrix motion and early morphogenesis

For over a century, embryologists who studied cellular motion in early amniotes generally assumed that morphogenetic movement reflected migration relative to a static extracellular matrix (ECM) scaffold. However, as we discuss in this Review, recent investigations reveal that the ECM is also moving during morphogenesis. Time-lapse studies show how convective tissue displacement patterns, as visualized by ECM markers, contribute to morphogenesis and organogenesis. Computational image analysis distinguishes between cell-autonomous (active) displacements and convection caused by large-scale (composite) tissue movements. Modern quantification of large-scale 'total' cellular motion and the accompanying ECM motion in the embryo demonstrates that a dynamic ECM is required for generation of the emergent motion patterns that drive amniote morphogenesis.

Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis

The orientation of cell division along the interphase cell long-axis, the century old Hertwig’s rule, has profound roles in tissue proliferation, morphogenesis, architecture and mechanics1,2. In epithelial tissues, the shape of the interphase cell is influenced by cell adhesion, mechanical stress, neighbour topology, and planar polarity pathways3–12. At mitosis, epithelial cells usually round up to ensure faithful chromosome segregation and to promote morphogenesis1. The mechanisms underlying interphase cell shape sensing in tissues are therefore unknown. We found that in Drosophila epithelia, tricellular junctions (TCJ) localize microtubule force generators, orienting cell division via the Dynein associated protein Mud independently of the classical Pins/Gαi pathway. Moreover, as cells round up during mitosis, TCJs serve as spatial landmarks, encoding information about interphase cell shape anisotropy to orient division in the rounded mitotic cell. Finally, experimental and simulation data show that shape and mechanical strain sensing by the TCJ emerge from a general geometric property of TCJ distributions in epithelial tissues. Thus, in addition to their function as epithelial barrier structures, TCJs serve as polarity cues promoting geometry and mechanical sensing in epithelial tissues.

An integrative approach for modeling and simulation of heterocyst pattern formation in cyanobacteria filaments

Heterocyst differentiation in cyanobacteria filaments is one of the simplest examples of cellular differentiation and pattern formation in multicellular organisms. Despite of the many experimental studies addressing the evolution and sustainment of heterocyst patterns and the knowledge of the genetic circuit underlying the behavior of single cyanobacterium under nitrogen deprivation, there is still a theoretical gap connecting these two macroscopic and microscopic processes. As an attempt to shed light on this issue, here we explore heterocyst differentiation under the paradigm of systems biology. This framework allows us to formulate the essential dynamical ingredients of the genetic circuit of a single cyanobacterium into a set of differential equations describing the time evolution of the concentrations of the relevant molecular products. As a result, we are able to study the behavior of a single cyanobacterium under different external conditions, emulating nitrogen deprivation, and simulate the dynamics of cyanobacteria filaments by coupling their respective genetic circuits via molecular diffusion. These two ingredients allow us to understand the principles by which heterocyst patterns can be generated and sustained. In particular, our results point out that, by including both diffusion and noisy external conditions in the computational model, it is possible to reproduce the main features of the formation and sustainment of heterocyst patterns in cyanobacteria filaments as observed experimentally. Finally, we discuss the validity and possible improvements of the model.

A computational model for the coordination of neural progenitor self-renewal and differentiation through Hes1 dynamics

Proper tissue development requires that stem/progenitor cells precisely coordinate cell division and differentiation in space and time. Notch-Hes1 intercellular signaling, which affects both differentiation and cell cycle progression and directs cell fate decisions at various developmental stages in many cell types, is central to this process. This study explored whether the pattern of connections among the cell cycle regulatory module, the Notch effector Hes1 and the proneural factor Ngn2 could explain salient aspects of cell fate determination in neural progenitors. A mathematical model that includes mutual interactions between Hes1, Ngn2 and G1-phase regulators was constructed and simulated at the single- and two-cell levels. By differentially regulating G1-phase progression, Hes1 and Ngn2 are shown to induce two contrasting cell cycle arrest states in early and late G1, respectively. Indeed, steady Hes1 overexpression promotes reversible quiescence by downregulating activators of G0/G1 exit and Ngn2. Ngn2 also downregulates activators of G0/G1 exit, but cooperates with Cip/Kip proteins to prevent G1/S transit, whereby it promotes G1-phase lengthening and, ultimately, contributes to reinforcing an irreversible late G1 arrest coincident with terminal differentiation. In this scheme, Hes1 oscillation in single cells is able to maintain a labile proliferation state in dynamic balance with two competing cell fate outputs associated with Hes1-mediated and Ngn2-mediated cell cycle arrest states. In Delta/Notch-connected cells, Hes1 oscillations and a lateral inhibition mechanism combine to establish heterogeneous Hes1, Ngn2 and cell cycle dynamics between proliferating neural progenitors, thereby increasing the chances of asymmetric cell fate decisions and improving the reliability of commitment to differentiation.

Every laboratory with a fluorescence microscope should consider counting molecules

Protein numbers in cells determine rates of biological processes, influence the architecture of cellular structures, reveal the stoichiometries of protein complexes, guide in vitro biochemical reconstitutions, and provide parameter values for mathematical modeling. The purpose of this essay is to increase awareness of methods for counting protein molecules using fluorescence microscopy and encourage more cell biologists to report these numbers. We address the state of the field in terms of utility and accuracy of the numbers reported and point readers to references for details of specific techniques and applications.

Do endothelial cells dream of eclectic shape?

Endothelial cells (ECs) exhibit dramatic plasticity of form at the single- and collective-cell level during new vessel growth, adult vascular homeostasis, and pathology. Understanding how, when, and why individual ECs coordinate decisions to change shape, in relation to the myriad of dynamic environmental signals, is key to understanding normal and pathological blood vessel behavior. However, this is a complex spatial and temporal problem. In this review we show that the multidisciplinary field of Adaptive Systems offers a refreshing perspective, common biological language, and straightforward toolkit that cell biologists can use to untangle the complexity of dynamic, morphogenetic systems.

Interplay between intercellular signaling and cell movement in development

Cell movement and local intercellular signaling are crucial components of morphogenesis during animal development. Intercellular signaling regulates the collective movement of a cell population via direct cell-cell contact. Cell movement, conversely, can influence local intercellular signaling by rearranging neighboring cells. Here, we first discuss theoretical models that address how intercellular signaling regulates collective cell movement during development. Examples include neural crest cell migration, convergent extension, and cell movement during vertebrate axis elongation. Second, we review theoretical studies on how cell movement may affect intercellular signaling, using the segmentation clock in zebrafish as an example. We propose that interplay between cell movement and intercellular signaling must be considered when studying morphogenesis in embryonic development.

Wnt-regulated dynamics of positional information in zebrafish somitogenesis

How signaling gradients supply positional information in a field of moving cells is an unsolved question in patterning and morphogenesis. Here, we ask how a Wnt signaling gradient regulates the dynamics of a wavefront of cellular change in a flow of cells during somitogenesis. Using time-controlled perturbations of Wnt signaling in the zebrafish embryo, we changed segment length without altering the rate of somite formation or embryonic elongation. This result implies specific Wnt regulation of the wavefront velocity. The observed Wnt signaling gradient dynamics and timing of downstream events support a model for wavefront regulation in which cell flow plays a dominant role in transporting positional information.

The role of self-organization in developmental evolution

In developmental and evolutionary biology, particular emphasis has been given to the relationship between transcription factors and the cognate cis-regulatory elements of their target genes. These constitute the gene regulatory networks that control expression and are assumed to causally determine the formation of structures and body plans. Comparative analysis has, however, established a broad sequence homology among species that nonetheless display quite different anatomies. Transgenic experiments have also confirmed that many developmentally important elements are, in fact, functionally interchangeable. Although dependent upon the appropriate degree of gene expression, the actual construction of specific structures appears not directly linked to the functions of gene products alone. Instead, the self-formation of complex patterns, due in large part to epigenetic and non-genetic determinants, remains a persisting theme in the study of ontogeny and regenerative medicine. Recent evidence indeed points to the existence of a self-organizing process, operating through a set of intrinsic rules and forces, which imposes coordination and a holistic order upon cells and tissue. This has been repeatedly demonstrated in experiments on regeneration as well as in the autonomous formation of structures in vitro. The process cannot be wholly attributed to the functional outcome of protein-protein interactions or to concentration gradients of diffusible chemicals. This phenomenon is examined here along with some of the methodological and theoretical approaches that are now used in understanding the causal basis for self-organization in development and its evolution.

Spatiotemporal control of gene expression using microfluidics

Accurate spatiotemporal regulation of genetic expression and cell microenvironment are both essential to epithelial morphogenesis during development, wound healing and cancer. In vivo, this is achieved through the interplay between intrinsic cellular properties and extrinsic signals. Amongst these, morphogen gradients induce specific concentration- and time-dependent gene expression changes that influence a target cell's fate. As systems biology attempts to understand the complex mechanisms underlying morphogenesis, the lack of experimental setup to recapitulate morphogen-induced patterning in vitro has become limiting. For this reason, we developed a versatile microfluidic-based platform to control the spatiotemporal delivery of chemical gradients to tissues grown in Petri dishes. Using this setup combined with a synthetic inducible gene expression system, we were able to restrict a target gene's expression within a confluent epithelium to bands of cells as narrow as four cell diameters with a one cell diameter accuracy. Applied to the targeted delivery of growth factor gradients to a confluent epithelium, this method further enabled the localized induction of epithelial-mesenchymal transitions and associated morphogenetic changes. Our approach paves the way for replicating in vitro the morphogen gradients observed in vivo to determine the relative contributions of known intrinsic and extrinsic factors in differential tissue patterning, during development and cancer. It could also be readily used to spatiotemporally control cell differentiation in ES/iPS cell cultures for re-engineering of complex tissues. Finally, the reversibility of the microfluidic chip assembly allows for pre- and post-treatment sample manipulations and extends the range of patternable samples to animal explants.

Testing Turing's theory of morphogenesis in chemical cells

Alan Turing, in "The Chemical Basis of Morphogenesis" [Turing AM (1952) Philos Trans R Soc Lond 237(641):37-72], described how, in circular arrays of identical biological cells, diffusion can interact with chemical reactions to generate up to six periodic spatiotemporal chemical structures. Turing proposed that one of these structures, a stationary pattern with a chemically determined wavelength, is responsible for differentiation. We quantitatively test Turing's ideas in a cellular chemical system consisting of an emulsion of aqueous droplets containing the Belousov-Zhabotinsky oscillatory chemical reactants, dispersed in oil, and demonstrate that reaction-diffusion processes lead to chemical differentiation, which drives physical morphogenesis in chemical cells. We observe five of the six structures predicted by Turing. In 2D hexagonal arrays, a seventh structure emerges, incompatible with Turing's original model, which we explain by modifying the theory to include heterogeneity.

Stochasticity in plant cellular growth and patterning

Plants, along with other multicellular organisms, have evolved specialized regulatory mechanisms to achieve proper tissue growth and morphogenesis. During development, growing tissues generate specialized cell types and complex patterns necessary for establishing the function of the organ. Tissue growth is a tightly regulated process that yields highly reproducible outcomes. Nevertheless, the underlying cellular and molecular behaviors are often stochastic. Thus, how does stochasticity, together with strict genetic regulation, give rise to reproducible tissue development? This review draws examples from plants as well as other systems to explore stochasticity in plant cell division, growth, and patterning. We conclude that stochasticity is often needed to create small differences between identical cells, which are amplified and stabilized by genetic and mechanical feedback loops to begin cell differentiation. These first few differentiating cells initiate traditional patterning mechanisms to ensure regular development.

The ecological basis of morphogenesis: branching patterns in swarming colonies of bacteria

Understanding how large-scale shapes in tissues, organs and bacterial colonies emerge from local interactions among cells and how these shapes remain stable over time are two fundamental problems in biology. Here we investigate branching morphogenesis in an experimental model system, swarming colonies of the bacterium Pseudomonas aeruginosa. We combine experiments and computer simulation to show that a simple ecological model of population dispersal can describe the emergence of branching patterns. In our system, morphogenesis depends on two counteracting processes that act on different length-scales: (1) colony expansion, which increases the likelihood of colonizing a patch at a close distance and (2) colony repulsion, which decreases the colonization likelihood over a longer distance. The two processes are included in a kernel based mathematical model using an integro-differential approach borrowed from ecological theory. Computer simulations show that the model can indeed reproduce branching, but only for a narrow range of parameter values, suggesting that P. aeruginosa has a fine-tuned physiology for branching. Simulations further show that hyperswarming, a process where highly dispersive mutants reproducibly arise within the colony and disrupt branching patterns, can be interpreted as a change in the spatial kernel.

Systems biology and the integration of mechanistic explanation and mathematical explanation

The paper discusses how systems biology is working toward complex accounts that integrate explanation in terms of mechanisms and explanation by mathematical models-which some philosophers have viewed as rival models of explanation. Systems biology is an integrative approach, and it strongly relies on mathematical modeling. Philosophical accounts of mechanisms capture integrative in the sense of multilevel and multifield explanations, yet accounts of mechanistic explanation (as the analysis of a whole in terms of its structural parts and their qualitative interactions) have failed to address how a mathematical model could contribute to such explanations. I discuss how mathematical equations can be explanatorily relevant. Several cases from systems biology are discussed to illustrate the interplay between mechanistic research and mathematical modeling, and I point to questions about qualitative phenomena (rather than the explanation of quantitative details), where quantitative models are still indispensable to the explanation. Systems biology shows that a broader philosophical conception of mechanisms is needed, which takes into account functional-dynamical aspects, interaction in complex networks with feedback loops, system-wide functional properties such as distributed functionality and robustness, and a mechanism's ability to respond to perturbations (beyond its actual operation). I offer general conclusions for philosophical accounts of explanation.