Regulation of neuronal differentiation at the neurogenic wavefront

Modelling Population-Level Hes1 Dynamics: Insights from a Multi-framework Approach

Mathematical models of living cells have been successively refined with advancements in experimental techniques. A main concern is striking a balance between modelling power and the tractability of the associated mathematical analysis. In this work we model the dynamics for the transcription factor Hairy and enhancer of split-1 (Hes1), whose expression oscillates during neural development, and which critically enables stable fate decision in the embryonic brain. We design, parametrise, and analyse a detailed spatial model using ordinary differential equations (ODEs) over a grid capturing both transient oscillatory behaviour and fate decision on a population-level. We also investigate the relationship between this ODE model and a more realistic grid-based model involving intrinsic noise using mostly directly biologically motivated parameters. While we focus specifically on Hes1 in neural development, the approach of linking deterministic and stochastic grid-based models shows promise in modelling various biological processes taking place in a cell population. In this context, our work stresses the importance of the interpretability of complex computational models into a framework which is amenable to mathematical analysis.

Control of spatio-temporal patterning via cell growth in a multicellular synthetic gene circuit

A major goal in synthetic development is to build gene regulatory circuits that control patterning. In natural development, an interplay between mechanical and chemical communication shapes the dynamics of multicellular gene regulatory circuits. For synthetic circuits, how non-genetic properties of the growth environment impact circuit behavior remains poorly explored. Here, we first describe an occurrence of mechano-chemical coupling in synthetic Notch (synNotch) patterning circuits: high cell density decreases synNotch-gated gene expression in different cellular systems in vitro. We then construct, both in vitro and in silico, a synNotch-based signal propagation circuit whose outcome can be regulated by cell density. Spatial and temporal patterning outcomes of this circuit can be predicted and controlled via modulation of cell proliferation, initial cell density, and/or spatial distribution of cell density. Our work demonstrates that synthetic patterning circuit outcome can be controlled via cellular growth, providing a means for programming multicellular circuit patterning outcomes.

Contact-mediated signaling enables disorder-driven transitions in cellular assemblies

We show that, when cells communicate by contact-mediated interactions, heterogeneity in cell shapes and sizes leads to qualitatively distinct collective behavior in the tissue. For intercellular coupling that implements lateral inhibition, such disorder-driven transitions can substantially alter the asymptotic pattern of differentiated cells by modulating their fate choice through changes in the neighborhood geometry. In addition, when contact-induced signals influence inherent cellular oscillations, disorder leads to the emergence of functionally relevant partially-ordered dynamical states.

Ordered hexagonal patterns via notch-delta signaling

Many developmental processes in biology utilize notch-delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, exploiting this 'juxtacrine' property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high delta at their center, as opposed to those with central high notch levels (referred to as anti-hexagons). We show that outside this range of parameters, in particular for lowcis-coupling, a novel kind of pattern is produced, where high delta cells have high notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.

Defect patterns on the curved surface of fish retinae suggest a mechanism of cone mosaic formation

The outer epithelial layer of zebrafish retinae contains a crystalline array of cone photoreceptors, called the cone mosaic. As this mosaic grows by mitotic addition of new photoreceptors at the rim of the hemispheric retina, topological defects, called "Y-Junctions", form to maintain approximately constant cell spacing. The generation of topological defects due to growth on a curved surface is a distinct feature of the cone mosaic not seen in other well-studied biological patterns like the R8 photoreceptor array in the Drosophila compound eye. Since defects can provide insight into cell-cell interactions responsible for pattern formation, here we characterize the arrangement of cones in individual Y-Junction cores as well as the spatial distribution of Y-junctions across entire retinae. We find that for individual Y-junctions, the distribution of cones near the core corresponds closely to structures observed in physical crystals. In addition, Y-Junctions are organized into lines, called grain boundaries, from the retinal center to the periphery. In physical crystals, regardless of the initial distribution of defects, defects can coalesce into grain boundaries via the mobility of individual particles. By imaging in live fish, we demonstrate that grain boundaries in the cone mosaic instead appear during initial mosaic formation, without requiring defect motion. Motivated by this observation, we show that a computational model of repulsive cell-cell interactions generates a mosaic with grain boundaries. In contrast to paradigmatic models of fate specification in mostly motionless cell packings, this finding emphasizes the role of cell motion, guided by cell-cell interactions during differentiation, in forming biological crystals. Such a route to the formation of regular patterns may be especially valuable in situations, like growth on a curved surface, where the resulting long-ranged, elastic, effective interactions between defects can help to group them into grain boundaries.

Landau theory for cellular patterns driven by lateral inhibition interaction

The phenomenology of Landau theory with spatial coupling through diffusion has been widely used in the study of phase transitions and patterning. Here we follow this theory and apply it to study theoretically and numerically continuous and discontinuous transitions to periodic spatial cellular patterns driven by lateral inhibition coupling. As opposed to diffusion, lateral inhibition coupling drives differences between adjacent cells. We analyze the appearance of errors in these patterns (disordered metastable states) and propose mechanisms to prevent them. These mechanisms are based on a temporal-dependent lateral inhibition coupling strength, which can be mediated, among others, by gradients of diffusing molecules. The simplicity and generality of the framework used herein is expected to facilitate future analyses of additional phenomena taking place through lateral inhibition interactions in more complex scenarios.

Regulation of Proneural Wave Propagation Through a Combination of Notch-Mediated Lateral Inhibition and EGF-Mediated Reaction Diffusion

Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt-and-pepper pattern formation during various biological processes. In many cases, Notch signaling acts together with other signaling systems. However, it is not clear what happens when Notch signaling is combined with other signaling systems. Mathematical modeling and the use of a simple biological model system will be essential to address this uncertainty. A wave of differentiation in the Drosophila visual center, the "proneural wave," accompanies the activity of the Notch and EGF signaling pathways. Although all of the Notch signaling components required for lateral inhibition are involved in the proneural wave, no salt-and-pepper pattern is found during the progression of the proneural wave. Instead, Notch is activated along the wave front and regulates proneural wave progression. How does Notch signaling control wave propagation without forming a salt-and-pepper pattern? A mathematical model of the proneural wave, based on biological evidence, has demonstrated that Notch-mediated lateral inhibition is implemented within the proneural wave and that the diffusible action of EGF cancels salt-and-pepper pattern formation. The results from numerical simulation have been confirmed by genetic experiments in vivo and suggest that the combination of Notch-mediated lateral inhibition and EGF-mediated reaction diffusion enables a novel function of Notch signaling that regulates propagation of the proneural wave. Similar mechanisms may play important roles in diverse biological processes found in animal development and cancer pathogenesis.

The proneural wave in the Drosophila optic lobe is driven by an excitable reaction-diffusion mechanism

In living organisms, self-organised waves of signalling activity propagate spatiotemporal information within tissues. During the development of the largest component of the visual processing centre of the Drosophila brain, a travelling wave of proneural gene expression initiates neurogenesis in the larval optic lobe primordium and drives the sequential transition of neuroepithelial cells into neuroblasts. Here, we propose that this 'proneural wave' is driven by an excitable reaction-diffusion system involving epidermal growth factor receptor (EGFR) signalling interacting with the proneural gene l'sc. Within this framework, a propagating transition zone emerges from molecular feedback and diffusion. Ectopic activation of EGFR signalling in clones within the neuroepithelium demonstrates that a transition wave can be excited anywhere in the tissue by inducing signalling activity, consistent with a key prediction of the model. Our model illuminates the physical and molecular underpinnings of proneural wave progression and suggests a generic mechanism for regulating the sequential differentiation of tissues.

High-throughput identification of factors promoting neuronal differentiation of human neural progenitor cells in microscale 3D cell culture

Identification of conditions for guided and specific differentiation of human stem cell and progenitor cells is important for continued development and engineering of in vitro cell culture systems for use in regenerative medicine, drug discovery, and human toxicology. Three-dimensional (3D) and organotypic cell culture models have been used increasingly for in vitro cell culture because they may better model endogenous tissue environments. However, detailed studies of stem cell differentiation within 3D cultures remain limited, particularly with respect to high-throughput screening. Herein, we demonstrate the use of a microarray chip-based platform to screen, in high-throughput, individual and paired effects of 12 soluble factors on the neuronal differentiation of a human neural progenitor cell line (ReNcell VM) encapsulated in microscale 3D Matrigel cultures. Dose-response analysis of selected combinations from the initial combinatorial screen revealed that the combined treatment of all-trans retinoic acid (RA) with the glycogen synthase kinase 3 inhibitor CHIR-99021 (CHIR) enhances neurogenesis while simultaneously decreases astrocyte differentiation, whereas the combined treatment of brain-derived neurotrophic factor and the small azide neuropathiazol enhances the differentiation into neurons and astrocytes. Subtype specification analysis of RA- and CHIR-differentiated cultures revealed that enhanced neurogenesis was not biased toward a specific neuronal subtype. Together, these results demonstrate a high-throughput screening platform for rapid evaluation of differentiation conditions in a 3D environment, which will aid the development and application of 3D stem cell culture models.

Combining a Toggle Switch and a Repressilator within the AC-DC Circuit Generates Distinct Dynamical Behaviors

Although the structure of a genetically encoded regulatory circuit is an important determinant of its function, the relationship between circuit topology and the dynamical behaviors it can exhibit is not well understood. Here, we explore the range of behaviors available to the AC-DC circuit. This circuit consists of three genes connected as a combination of a toggle switch and a repressilator. Using dynamical systems theory, we show that the AC-DC circuit exhibits both oscillations and bistability within the same region of parameter space; this generates emergent behaviors not available to either the toggle switch or the repressilator alone. The AC-DC circuit can switch on oscillations via two distinct mechanisms, one of which induces coherence into ensembles of oscillators. In addition, we show that in the presence of noise, the AC-DC circuit can behave as an excitable system capable of spatial signal propagation or coherence resonance. Together, these results demonstrate how combinations of simple motifs can exhibit multiple complex behaviors.

Modeling the Notch Response

NOTCH signaling regulates developmental processes in all tissues and all organisms across the animal kingdom. It is often involved in coordinating the differentiation of neighboring cells into different cell types. As our knowledge on the structural, molecular and cellular properties of the NOTCH pathway expands, there is a greater need for quantitative methodologies to get a better understanding of the processes controlled by NOTCH signaling. In recent years, theoretical and computational approaches to NOTCH signaling and NOTCH mediated patterning are gaining popularity. Mathematical models of NOTCH mediated patterning provide insight into complex and counterintuitive behaviors and can help generate predictions that can guide experiments. In this chapter, we review the recent advances in modeling NOTCH mediated patterning processes. We discuss new modeling approaches to lateral inhibition patterning that take into account cis-interactions between NOTCH receptors and ligands, signaling through long cellular protrusions, cell division processes, and coupling to external signals. We also describe models of somitogenesis, where NOTCH signaling is used for synchronizing cellular oscillations. We then discuss modeling approaches that consider the effect of cell morphology on NOTCH signaling and NOTCH mediated patterning. Finally, we consider models of boundary formation and how they are influenced by the combinatorial action of multiple ligands. Together, these topics cover the main advances in the field of modeling the NOTCH response.

DISC1 Regulates the Proliferation and Migration of Mouse Neural Stem/Progenitor Cells through Pax5, Sox2, Dll1 and Neurog2

Background: Disrupted-in-schizophrenia 1 (DISC1) regulates neurogenesis and is a genetic risk factor for major psychiatric disorders. However, how DISC1 dysfunction affects neurogenesis and cell cycle progression at the molecular level is still unknown. Here, we investigated the role of DISC1 in regulating proliferation, migration, cell cycle progression and apoptosis in mouse neural stem/progenitor cells (MNSPCs) in vitro. Methods: MNSPCs were isolated and cultured from mouse fetal hippocampi. Retroviral vectors or siRNAs were used to manipulate DISC1 expression in MNSPCs. Proliferation, migration, cell cycle progression and apoptosis of altered MNSPCs were analyzed in cell proliferation assays (MTS), transwell system and flow cytometry. A neurogenesis specific polymerase chain reaction (PCR) array was used to identify genes downstream of DISC1, and functional analysis was performed through transfection of expression plasmids and siRNAs. Results: Loss of DISC1 reduced proliferation and migration of MNSPCs, while an increase in DISC1 led to increased proliferation and migration. Meanwhile, an increase in the proportion of cells in G0/G1 phase was concomitant with reduced levels of DISC1, but significant changes were not observed in the number MNSPCs undergoing apoptosis. Paired box gene 5 (Pax5), sex determining region Y-box 2 (Sox2), delta-like1 (Dll1) and Neurogenin2 (Neurog2) emerged as candidate molecules downstream of DISC1, and rescue experiments demonstrated that increased or decreased expression of either molecule regulated proliferation and migration in DISC1-altered MNSPCs. Conclusion: These results suggest that Pax5, Sox2, Dll1 and Neurog2 mediate DISC1 activity in MNSPC proliferation and migration.

Diversity of fate outcomes in cell pairs under lateral inhibition

Cell fate determination by lateral inhibition via Notch/Delta signalling has been extensively studied. Most formalised models consider Notch/Delta interactions in fields of cells, with parameters that typically lead to symmetry breaking of signalling states between neighbouring cells, commonly resulting in salt-and-pepper fate patterns. Here we consider the case of signalling between isolated cell pairs, and find that the bifurcation properties of a standard mathematical model of lateral inhibition can lead to stable symmetric signalling states. We apply this model to the adult intestinal stem cell (ISC) of Drosophila, whose fate is stochastic but dependent on the Notch/Delta pathway. We observe a correlation between signalling state in cell pairs and their contact area. We interpret this behaviour in terms of the properties of our model in the presence of population variability in contact areas, which affects the effective signalling threshold of individual cells. Our results suggest that the dynamics of Notch/Delta signalling can contribute to explain stochasticity in stem cell fate decisions, and that the standard model for lateral inhibition can account for a wider range of developmental outcomes than previously considered.

Notch-mediated lateral inhibition regulates proneural wave propagation when combined with EGF-mediated reaction diffusion

Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt and pepper patterns during various developmental processes. However, how Notch signaling behaves in combination with other signaling systems remains elusive. The wave of differentiation in the Drosophila visual center or "proneural wave" accompanies Notch activity that is propagated without the formation of a salt and pepper pattern, implying that Notch does not form a feedback loop of lateral inhibition during this process. However, mathematical modeling and genetic analysis clearly showed that Notch-mediated lateral inhibition is implemented within the proneural wave. Because partial reduction in EGF signaling causes the formation of the salt and pepper pattern, it is most likely that EGF diffusion cancels salt and pepper pattern formation in silico and in vivo. Moreover, the combination of Notch-mediated lateral inhibition and EGF-mediated reaction diffusion enables a function of Notch signaling that regulates propagation of the wave of differentiation.

Pattern selection by dynamical biochemical signals

The development of multicellular organisms involves cells to decide their fate upon the action of biochemical signals. This decision is often spatiotemporally coordinated such that a spatial pattern arises. The dynamics that drive pattern formation usually involve genetic nonlinear interactions and positive feedback loops. These complex dynamics may enable multiple stable patterns for the same conditions. Under these circumstances, pattern formation in a developing tissue involves a selection process: why is a certain pattern formed and not another stable one? Herein we computationally address this issue in the context of the Notch signaling pathway. We characterize a dynamical mechanism for developmental selection of a specific pattern through spatiotemporal changes of the control parameters of the dynamics, in contrast to commonly studied situations in which initial conditions and noise determine which pattern is selected among multiple stable ones. This mechanism can be understood as a path along the parameter space driven by a sequence of biochemical signals. We characterize the selection process for three different scenarios of this dynamical mechanism that can take place during development: the signal either 1) acts in all the cells at the same time, 2) acts only within a cluster of cells, or 3) propagates along the tissue. We found that key elements for pattern selection are the destabilization of the initial pattern, the subsequent exploration of other patterns determined by the spatiotemporal symmetry of the parameter changes, and the speeds of the path compared to the timescales of the pattern formation process itself. Each scenario enables the selection of different types of patterns and creates these elements in distinct ways, resulting in different features. Our approach extends the concept of selection involved in cellular decision-making, usually applied to cell-autonomous decisions, to systems that collectively make decisions through cell-to-cell interactions.

Signaling Delays Preclude Defects in Lateral Inhibition Patterning

Lateral inhibition represents a well-studied example of biology's ability to self-organize multicellular spatial patterns with single-cell precision. Despite established biochemical mechanisms for lateral inhibition (e.g., Delta-Notch), it remains unclear how cell-cell signaling delays inherent to these mechanisms affect patterning outcomes. We investigate a compact model of lateral inhibition highlighting these delays and find, remarkably, that long delays can ensure defect-free patterning. This effect is underscored by an interplay with synchronous oscillations, cis interactions, and signaling strength. Our results suggest that signaling delays, though previously posited as a source of developmental defects, may in fact be a general regulatory knob for tuning developmental robustness.

Genetic oscillators in development

In development, morphogenetic processes are strictly coordinated in time. Cells in a developing tissue would need mechanisms for time-keeping. One such time-keeping mechanism is to use oscillations of gene expression. Oscillatory gene expression can be generated by transcriptional/translational feedback loops, usually referred to as a genetic oscillator. In this review article, we discuss genetic oscillators in the presence of developmental processes such as cell division, cell movement and cell differentiation. We first introduce the gene regulatory network for generating a rhythm of gene expression. We then discuss how developmental processes influence genetic oscillators. Examples include vertebrate somitogenesis and neural progenitor cell differentiation, as well as the circadian clock for comparison. To understand the behaviors of genetic oscillators in development, it is necessary to consider both gene expression dynamics and cellular behaviors simultaneously. Theoretical modeling combined with live imaging at single-cell resolution will be a powerful tool to analyze genetic oscillators in development.

Ligand-dependent Notch signaling strength orchestrates lateral induction and lateral inhibition in the developing inner ear

During inner ear development, Notch exhibits two modes of operation: lateral induction, which is associated with prosensory specification, and lateral inhibition, which is involved in hair cell determination. These mechanisms depend respectively on two different ligands, jagged 1 (Jag1) and delta 1 (Dl1), that rely on a common signaling cascade initiated after Notch activation. In the chicken otocyst, expression of Jag1 and the Notch target Hey1 correlates well with lateral induction, whereas both Jag1 and Dl1 are expressed during lateral inhibition, as are Notch targets Hey1 and Hes5. Here, we show that Jag1 drives lower levels of Notch activity than Dl1, which results in the differential expression of Hey1 and Hes5. In addition, Jag1 interferes with the ability of Dl1 to elicit high levels of Notch activity. Modeling the sensory epithelium when the two ligands are expressed together shows that ligand regulation, differential signaling strength and ligand competition are crucial to allow the two modes of operation and for establishing the alternate pattern of hair cells and supporting cells. Jag1, while driving lateral induction on its own, facilitates patterning by lateral inhibition in the presence of Dl1. This novel behavior emerges from Jag1 acting as a competitive inhibitor of Dl1 for Notch signaling. Both modeling and experiments show that hair cell patterning is very robust. The model suggests that autoactivation of proneural factor Atoh1, upstream of Dl1, is a fundamental component for robustness. The results stress the importance of the levels of Notch signaling and ligand competition for Notch function.

Competition in notch signaling with cis enriches cell fate decisions

Notch signaling is involved in cell fate choices during the embryonic development of Metazoa. Commonly, Notch signaling arises from the binding of the Notch receptor to its ligands in adjacent cells driving cell-to-cell communication. Yet, cell-autonomous control of Notch signaling through both ligand-dependent and ligand-independent mechanisms is known to occur as well. Examples include Notch signaling arising in the absence of ligand binding, and cis-inhibition of Notch signaling by titration of the Notch receptor upon binding to its ligands within a single cell. Increasing experimental evidences support that the binding of the Notch receptor with its ligands within a cell (cis-interactions) can also trigger a cell-autonomous Notch signal (cis-signaling), whose potential effects on cell fate decisions and patterning remain poorly understood. To address this question, herein we mathematically and computationally investigate the cell states arising from the combination of cis-signaling with additional Notch signaling sources, which are either cell-autonomous or involve cell-to-cell communication. Our study shows that cis-signaling can switch from driving cis-activation to effectively perform cis-inhibition and identifies under which conditions this switch occurs. This switch relies on the competition between Notch signaling sources, which share the same receptor but differ in their signaling efficiency. We propose that the role of cis-interactions and their signaling on fine-grained patterning and cell fate decisions is dependent on whether they drive cis-inhibition or cis-activation, which could be controlled during development. Specifically, cis-inhibition and not cis-activation facilitates patterning and enriches it by modulating the ratio of cells in the high-ligand expression state, by enabling additional periodic patterns like stripes and by allowing localized patterning highly sensitive to the precursor state and cell-autonomous bistability. Our study exemplifies the complexity of regulations when multiple signaling sources share the same receptor and provides the tools for their characterization.

Modeling Notch signaling: a practical tutorial

Theoretical and computational approaches for understanding different aspects of Notch signaling and Notch dependent patterning are gaining popularity in recent years. These in silico methodologies can provide dynamic insights that are often not intuitive and may help guide experiments aimed at elucidating these processes. This chapter is an introductory tutorial intended to allow someone with basic mathematical and computational knowledge to explore new mathematical models of Notch-mediated processes and perform numerical simulations of these models. In particular, we explain how to define and simulate models of lateral inhibition patterning processes. We provide a Matlab code for simulating various lateral inhibition models in a simple and intuitive manner, and show how to present the results from the computational models. This code can be used as a starting point for exploring more specific models that include additional aspects of the Notch pathway and its regulation.

Modeling gene expression in time and space

Cell populations rarely exhibit gene-expression profiles that are homogeneous in time and space. In the temporal domain, dynamical behaviors such as oscillations and pulses of protein production pervade cell biology, underlying phenomena as diverse as circadian rhythmicity, cell cycle control, stress and damage responses, and stem-cell pluripotency. In multicellular populations, spatial heterogeneities are crucial for decision making and development, among many other functions. Cells need to exquisitely coordinate this temporal and spatial variation to survive. Although the spatiotemporal character of gene expression is challenging to quantify experimentally at the level of individual cells, it is beneficial from the modeling viewpoint, because it provides strong constraints that can be probed by theoretically analyzing mathematical models of candidate gene and protein circuits. Here, we review recent examples of temporal dynamics and spatial patterning in gene expression to show how modeling such phenomenology can help us unravel the molecular mechanisms of cellular function.