Signaling Delays Preclude Defects in Lateral Inhibition Patterning

Dynamical Modeling and Qualitative Analysis of a Delayed Model for CD8 T Cells in Response to Viral Antigens

Although the immune effector CD8 T cells play a crucial role in clearance of viruses, the mechanisms underlying the dynamics of how CD8 T cells respond to viral infection remain largely unexplored. Here, we develop a delayed model that incorporates CD8 T cells and infected cells to investigate the functional role of CD8 T cells in persistent virus infection. Bifurcation analysis reveals that the model has four steady states that can finely divide the progressions of viral infection into four states, and endows the model with bistability that has ability to achieve the switch from one state to another. Furthermore, analytical and numerical methods find that the time delay resulting from incubation period of virus can induce a stable low-infection steady state to be oscillatory, coexisting with a stable high-infection steady state in phase space. In particular, a novel mechanism to achieve the switch between two stable steady states, time-delay-based switch, is proposed, where the initial conditions and other parameters of the model remain unchanged. Moreover, our model predicts that, for a certain range of initial antigen load: 1) under a longer incubation period, the lower the initial antigen load, the easier the virus infection will evolve into severe state; while the higher the initial antigen load, the easier it is for the virus infection to be effectively controlled and 2) only when the incubation period is small, the lower the initial antigen load, the easier it is to effectively control the infection progression. Our results are consistent with multiple experimental observations, which may facilitate the understanding of the dynamical and physiological mechanisms of CD8 T cells in response to viral infections.

The alternate ligand Jagged enhances the robustness of Notch signaling patterns

The Notch pathway, an example of juxtacrine signaling, is an evolutionary conserved cell-cell communication mechanism. It governs emergent spatiotemporal patterning in tissues during development, wound healing and tumorigenesis. Communication occurs when Notch receptors of one cell bind to either of its ligands, Delta/Jagged of the neighboring cell. In general, Delta-mediated signaling drives neighboring cells to have an opposite fate (lateral inhibition) whereas Jagged-mediated signaling drives cells to maintain similar fates (lateral induction). Here, by deriving and solving a reduced set of 12 coupled ordinary differential equations for the Notch-Delta-Jagged system on a hexagonal grid of cells, we determine the allowed states across different parameter sets. We also show that Jagged (at low dose) acts synergistically with Delta to enable more robust pattern formation by making the neighboring cell states more distinct from each other, despite its lateral induction property. Our findings extend our understanding of the possible synergistic role of Jagged with Delta which had been previously proposed through experiments and models in the context of chick inner ear development. Finally, we show how Jagged can help to expand the bistable (both uniform and hexagon phases are stable) region, where a local perturbation can spread over time in an ordered manner to create a biologically relevant, perfectly ordered lateral inhibition pattern.

Stochastic fluctuations promote ordered pattern formation of cells in the Notch-Delta signaling pathway

The Notch-Delta signaling pathway mediates cell differentiation implicated in many regulatory processes including spatiotemporal patterning in tissues by promoting alternate cell fates between neighboring cells. At the multicellular level, this "lateral inhibition” principle leads to checkerboard patterns with alternation of Sender and Receiver cells. While it is well known that stochasticity modulates cell fate specification, little is known about how stochastic fluctuations at the cellular level propagate during multicell pattern formation. Here, we model stochastic fluctuations in the Notch-Delta pathway in the presence of two different noise types–shot and white–for a multicell system. Our results show that intermediate fluctuations reduce disorder and guide the multicell lattice toward checkerboard-like patterns. By further analyzing cell fate transition events, we demonstrate that intermediate noise amplitudes provide enough perturbation to facilitate “proofreading” of disordered patterns and cause cells to switch to the correct ordered state (Sender surrounded by Receivers, and vice versa). Conversely, high noise can override environmental signals coming from neighboring cells and lead to switching between ordered and disordered patterns. Therefore, in analogy with spin glass systems, intermediate noise levels allow the multicell Notch system to escape frustrated patterns and relax towards the lower energy checkerboard pattern while at large noise levels the system is unable to find this ordered base of attraction.

The Notch pathway is involved in many biological processes and is known to form precise spatial patterns alternating Sender and Receiver cell states. Quantifying the implications of stochastic fluctuations provided insight that patterns formed in Notch-mediated pathways must follow a predetermined path towards checkerboard or exist in a noisy environment which promotes order through error correction.

We model Notch pattern formation stochastically and analyze the spatiotemporal dynamics. Our results show multicellular systems equilibrate towards ordered systems, but mistakes in the initial lattice propagate causing the systems to relax into frustrated systems. Only through existing in a noisy environment are the systems able to relax into the checkerboard pattern. Analyzing the temporal dynamics confirms, in environments with intermediate noise, the “incorrect” cells (Sender in a Sender environment, and vice versa) can be flipped to the correct state (Sender in a Receiver environment, and vice versa). Comparing with the spin glass energy landscape, we suggest the multicellular model follows a rugged landscape to form patterns with stochastic fluctuations required to enforce order throughout the system.

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.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

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.

Engineering and modeling of multicellular morphologies and patterns

Synthetic multicellular (MC) systems have the capacity to increase our understanding of biofilms and higher organisms, and to serve as engineering platforms for developing complex products in the areas of medicine, biosynthesis and smart materials. Here we provide an interdisciplinary perspective and review on emerging approaches to engineer and model MC systems. We lay out definitions for key terms in the field and identify toolboxes of standardized parts which can be combined into various MC algorithms to achieve specific outcomes. Many essential parts and algorithms have been demonstrated in some form. As key next milestones for the field, we foresee the improvement of these parts and their adaptation to more biological systems, the demonstration of more complex algorithms, the advancement of quantitative modeling approaches and compilers to support rational MC engineering, and implementation of MC engineering for practical applications.

Constraints on somite formation in developing embryos

Segment formation in vertebrate embryos is a stunning example of biological self-organization. Here, we present an idealized framework, in which we treat the presomitic mesoderm (PSM) as a one-dimensional line of oscillators. We use the framework to derive constraints that connect the size of somites, and the timing of their formation, to the growth of the PSM and the gradient of the somitogenesis clock period across the PSM. Our analysis recapitulates the observations made recently in ex vivo cultures of mouse PSM cells, and makes predictions for how perturbations, such as increased Wnt levels, would alter somite widths. Finally, our analysis makes testable predictions for the shape of the phase profile and somite widths at different stages of PSM growth. In particular, we show that the phase profile is robustly concave when the PSM length is steady and slightly convex in an important special case when it is decreasing exponentially. In both cases, the phase profile scales with the PSM length; in the latter case, it scales dynamically. This has important consequences for the velocity of the waves that traverse the PSM and trigger somite formation, as well as the effect of errors in phase measurement on somite widths.

Embryonic lateral inhibition as optical modes: An analytical framework for mesoscopic pattern formation

Cellular checkerboard patterns are observed at many stages of embryonic development. We study an analytically tractable model for lateral inhibition and show that the steady states are analogous to optical phonons at the Γ point, which have the wave number k=0. We study the cases of cells arranged in linear and hexagonal lattices. To determine how the final pattern is selected it is necessary to take into account the granularity of the pattern and, analogously to solid-state physics, to redefine the basis and lattice sites in terms of a periodic crystal. The sites and basis are determined by looking at the symmetries of inhibitory interactions between cells. The redefined basis for cells placed in a linear lattice is composed by two cells which are embedded in another linear lattice, while for cells placed in a hexagonal lattice the redefined basis consists of three cells embedded in another hexagonal lattice. The pattern in hexagonal lattices can be driven into three different states: two of those states are periodic checkerboards and a third in which both periodic states coexist. These observations provides new predictions for experiments.

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.

A Network Model to Explore the Effect of the Micro-environment on Endothelial Cell Behavior during Angiogenesis

Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-function mutations. These properties make our model a suitable platform for the understanding of the molecular mechanisms underlying some pathologies. For example, it is possible to follow the changes in the activation patterns caused by mutations that promote Tip EC behavior and inhibit Phalanx EC behavior, that lead to the conditions associated with retinal vascular disorders and tumor vascularization. Moreover, the model describes how mutations that promote Phalanx EC behavior are associated with the development of arteriovenous and venous malformations. These results suggest that the network model that we propose has the potential to be used in the study of how the modulation of the EC extracellular micro-environment may improve the outcome of vascular disease treatments.

A Model for Adult Organ Resizing Demonstrates Stem Cell Scaling through a Tunable Commitment Rate

Many adult organs grow or shrink to accommodate different physiological demands. Often, as total cell number changes, stem cell number changes proportionally in a phenomenon called "stem cell scaling". The cellular behaviors that give rise to scaling are unknown. Here we study two complementary theoretical models of the adult Drosophila midgut, a stem cell-based organ with known resizing dynamics. First, we derive a differential equations model of midgut resizing and show that the in vivo kinetics of growth can be recapitulated if the rate of fate commitment depends on the tissue's stem cell proportion. Second, we develop a 2D simulation of the midgut and find that proportion-dependent commitment rate and stem cell scaling can arise phenomenologically from the stem cells' exploration of physical tissue space during its lifetime. Together, these models provide a biophysical understanding of how stem cell scaling is maintained during organ growth and shrinkage.

Delta-Notch signalling in segmentation

Modular body organization is found widely across multicellular organisms, and some of them form repetitive modular structures via the process of segmentation. It's vastly interesting to understand how these regularly repeated structures are robustly generated from the underlying noise in biomolecular interactions. Recent studies from arthropods reveal similarities in segmentation mechanisms with vertebrates, and raise the possibility that the three phylogenetic clades, annelids, arthropods and chordates, might share homology in this process from a bilaterian ancestor. Here, we discuss vertebrate segmentation with particular emphasis on the role of the Notch intercellular signalling pathway. We introduce vertebrate segmentation and Notch signalling, pointing out historical milestones, then describe existing models for the Notch pathway in the synchronization of noisy neighbouring oscillators, and a new role in the modulation of gene expression wave patterns. We ask what functions Notch signalling may have in arthropod segmentation and explore the relationship between Notch-mediated lateral inhibition and synchronization. Finally, we propose open questions and technical challenges to guide future investigations into Notch signalling in segmentation.

Modeling delayed processes in biological systems

Delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations (DDEs) and their extension to include stochastic effects. DDEs do not explicitly incorporate intermediate states associated with a delayed process but instead use an estimated average delay time. In an effort to examine the validity of this approach, we study systems with significant delays by explicitly incorporating intermediate steps. We show that such explicit models often yield significantly different equilibrium distributions and transition times as compared to DDEs with deterministic delay values. Additionally, different explicit models with qualitatively different dynamics can give rise to the same DDEs revealing important ambiguities. We also show that DDE-based predictions of oscillatory behavior may fail for the corresponding explicit model.