Stochastic control of spontaneous signal generation for gradient sensing in chemotaxis

Somite boundary determination in normal and clock-less vertebrate embryos

Vertebrate segments called somites are generated by periodic segmentation of the presomitic mesoderm (PSM). In the most accepted theoretical model for somite segmentation, the clock and wavefront (CW) model, a clock that ticks to determine particular timings and a wavefront that moves posteriorly are presented in the PSM, and somite positions are determined when the clock meets the posteriorly moving wavefront somewhere in the PSM. Over the last two decades, it has been revealed that the molecular mechanism of the clock and wavefront in vertebrates is based on clock genes including Hes family transcription factors and Notch effectors that oscillate within the PSM to determine particular timings and fibroblast growth factor (FGF) gradients, acting as the posteriorly moving wavefront to determine the position of somite segmentation. A clock-less condition in the CW model was predicted to form no somites; however, irregularly sized somites were still formed in mice and zebrafish, suggesting that this was one of the limitations of the CW model. Recently, we performed interdisciplinary research of experimental and theoretical biological studies and revealed the mechanisms of somite boundary determination in normal and clock-less conditions by characterization of the FGF/extracellular signal-regulated kinase (ERK) activity dynamics. Since features of the molecular clock have already been described in-depth in several reviews, we summarized recent findings regarding the role of FGF/ERK signaling in somite boundary formation and described our current understanding of how FGF/ERK signaling contributes to somitogenesis in normal and clock-less conditions in this review.

Noise-resistant developmental reproducibility in vertebrate somite formation

The reproducibility of embryonic development is remarkable, although molecular processes are intrinsically stochastic at the single-cell level. How the multicellular system resists the inevitable noise to acquire developmental reproducibility constitutes a fundamental question in developmental biology. Toward this end, we focused on vertebrate somitogenesis as a representative system, because somites are repeatedly reproduced within a single embryo whereas such reproducibility is lost in segmentation clock gene-deficient embryos. However, the effect of noise on developmental reproducibility has not been fully investigated, because of the technical difficulty in manipulating the noise intensity in experiments. In this study, we developed a computational model of ERK-mediated somitogenesis, in which bistable ERK activity is regulated by an FGF gradient, cell-cell communication, and the segmentation clock, subject to the intrinsic noise. The model simulation generated our previous in vivo observation that the ERK activity was distributed in a step-like gradient in the presomitic mesoderm, and its boundary was posteriorly shifted by the clock in a stepwise manner, leading to regular somite formation. Here, we showed that this somite regularity was robustly maintained against the noise. Removing the clock from the model predicted that the stepwise shift of the ERK activity occurs at irregular timing with irregular distance owing to the noise, resulting in somite size variation. This model prediction was recently confirmed by live imaging of ERK activity in zebrafish embryos. Through theoretical analysis, we presented a mechanism by which the clock reduces the inherent somite irregularity observed in clock-deficient embryos. Therefore, this study indicates a novel role of the segmentation clock in noise-resistant developmental reproducibility.

The segmentation clock has been widely considered vital for somite formation, because clock-deficient embryos display severe segmental defects. However, irregular somites are still formed, suggesting that the clock is not required for somite formation itself but rather endows it with developmental reproducibility. Thus, the following questions arose: How do irregular somites emerge in a clock-independent manner? How is the irregularity reduced in the presence of the clock? To address these questions, we developed a computational model of somitogenesis. We then clarified that the intrinsic noise induces spontaneous formation of irregular-sized somites in the absence of the clock, and that the clock plays an important role in suppressing the noise effect for reproducible somite formation.

The threshold of an excitable system serves as a control mechanism for noise filtering during chemotaxis

Chemotaxis, the migration of cells in the direction of a chemical gradient, is of utmost importance in various biological processes. In recent years, research has demonstrated that the underlying mechanism that controls cell migration is an excitable network. One of the properties that characterizes excitability is the presence of a threshold for activation. Here, we show that excitable systems possess noise filtering capabilities that enable faster and more efficient directed migration compared to other systems that also include a threshold, such as ultrasensitive switches. We demonstrate that this filtering ability is a consequence of the varying responses of excitable systems to step and pulse stimuli. Whereas the response to step inputs is determined solely by the magnitude of the stimulus, for pulse stimuli, the response depends on both the magnitude and duration of the stimulus. We then show that these two forms of threshold behavior can be decoupled from one another, allowing finer control in chemotaxis. Finally, we use a simple model of chemotaxis to demonstrate that cells that rely on an excitable system display faster and more effective directed migration that a hypothetical cell guided by an ultra-sensitive switch.

Revisiting chemoaffinity theory: Chemotactic implementation of topographic axonal projection

Neural circuits are wired by chemotactic migration of growth cones guided by extracellular guidance cue gradients. How growth cone chemotaxis builds the macroscopic structure of the neural circuit is a fundamental question in neuroscience. I addressed this issue in the case of the ordered axonal projections called topographic maps in the retinotectal system. In the retina and tectum, the erythropoietin-producing hepatocellular (Eph) receptors and their ligands, the ephrins, are expressed in gradients. According to Sperry's chemoaffinity theory, gradients in both the source and target areas enable projecting axons to recognize their proper terminals, but how axons chemotactically decode their destinations is largely unknown. To identify the chemotactic mechanism of topographic mapping, I developed a mathematical model of intracellular signaling in the growth cone that focuses on the growth cone's unique chemotactic property of being attracted or repelled by the same guidance cues in different biological situations. The model presented mechanism by which the retinal growth cone reaches the correct terminal zone in the tectum through alternating chemotactic response between attraction and repulsion around a preferred concentration. The model also provided a unified understanding of the contrasting relationships between receptor expression levels and preferred ligand concentrations in EphA/ephrinA- and EphB/ephrinB-encoded topographic mappings. Thus, this study redefines the chemoaffinity theory in chemotactic terms.

Multi-phasic bi-directional chemotactic responses of the growth cone

The nerve growth cone is bi-directionally attracted and repelled by the same cue molecules depending on the situations, while other non-neural chemotactic cells usually show uni-directional attraction or repulsion toward their specific cue molecules. However, how the growth cone differs from other non-neural cells remains unclear. Toward this question, we developed a theory for describing chemotactic response based on a mathematical model of intracellular signaling of activator and inhibitor. Our theory was first able to clarify the conditions of attraction and repulsion, which are determined by balance between activator and inhibitor, and the conditions of uni- and bi-directional responses, which are determined by dose-response profiles of activator and inhibitor to the guidance cue. With biologically realistic sigmoidal dose-responses, our model predicted tri-phasic turning response depending on intracellular Ca²⁺ level, which was then experimentally confirmed by growth cone turning assays and Ca²⁺ imaging. Furthermore, we took a reverse-engineering analysis to identify balanced regulation between CaMKII (activator) and PP1 (inhibitor) and then the model performance was validated by reproducing turning assays with inhibitions of CaMKII and PP1. Thus, our study implies that the balance between activator and inhibitor underlies the multi-phasic bi-directional turning response of the growth cone.

Modeling Excitable Dynamics of Chemotactic Networks

The study of chemotaxis has benefited greatly from computational models that describe the response of cells to chemoattractant stimuli. These models must keep track of spatially and temporally varying distributions of numerous intracellular species. Moreover, recent evidence suggests that these are not deterministic interactions, but also include the effect of stochastic variations that trigger an excitable network. In this chapter we illustrate how to create simulations of excitable networks using the Virtual Cell modeling environment.

Interaction of motility, directional sensing, and polarity modules recreates the behaviors of chemotaxing cells

Chemotaxis involves the coordinated action of separable but interrelated processes: motility, gradient sensing, and polarization. We have hypothesized that these are mediated by separate modules that account for these processes individually and that, when combined, recreate most of the behaviors of chemotactic cells. Here, we describe a mathematical model where the modules are implemented in terms of reaction-diffusion equations. Migration and the accompanying changes in cellular morphology are demonstrated in simulations using a mechanical model of the cell cortex implemented in the level set framework. The central module is an excitable network that accounts for random migration. The response to combinations of uniform stimuli and gradients is mediated by a local excitation, global inhibition module that biases the direction in which excitability is directed. A polarization module linked to the excitable network through the cytoskeleton allows unstimulated cells to move persistently and, for cells in gradients, to gradually acquire distinct sensitivity between front and back. Finally, by varying the strengths of various feedback loops in the model we obtain cellular behaviors that mirror those of genetically altered cell lines.

Chemotaxis is the movement of cells in response to spatial gradients of chemical cues. While single-celled organisms rely on sensing and responding to chemical gradients to search for nutrients, chemotaxis is also an essential component of the mammalian immune system. However, chemotaxis can also be deleterious, since chemotactic tumor cells can lead to metastasis. Due to its importance, understanding the process by which cells sense and respond to chemical gradients has attracted considerable interest. Moreover, because of the complexity of chemotactic signaling, which includes multiple feedback loops and redundant pathways, this has been a research area in which computational models have had a significant impact in understanding experimental findings. Here, we propose a modular description of the signaling network that regulates chemotaxis. The modules describe different processes that are observed in chemotactic cells. In addition to accounting for these behaviors individually, we show that the overall system recreates many features of the directed motion of migrating cells. The signaling described by our modules is implemented as a series of equations, whereas movement and the accompanying cellular deformations are simulated using a mechanical model of the cell and implemented using level set methods, a method that allows simulations of cells as they change morphology.

Excitable behavior in amoeboid chemotaxis

Chemotaxis, the directed motion of cells in response to chemical gradients, is a fundamental process. Eukaryotic cells detect spatial differences in chemoattractant receptor occupancy with high precision and use these differences to bias the location of actin-rich protrusions to guide their movement. Research into chemotaxis has benefitted greatly from a systems biology approach that combines novel experimental and computational tools to pose and test hypotheses. Recently, one such hypothesis has been postulated proposing that chemotaxis in eukaryotic cells is mediated by locally biasing the activity of an underlying excitable system. The excitable system hypothesis can account for a number of cellular behaviors related to chemotaxis, including the stochastic nature of the movement of unstimulated cells, the directional bias imposed by chemoattractant gradients, and the observed spatial and temporal distribution of signaling and cytoskeleton proteins.

Levels of germination proteins in dormant and superdormant spores of Bacillus subtilis

Bacillus subtilis spores that germinated poorly with saturating levels of nutrient germinants, termed superdormant spores, were separated from the great majority of dormant spore populations that germinated more rapidly. These purified superdormant spores (1.5 to 3% of spore populations) germinated extremely poorly with the germinants used to isolate them but better with germinants targeting germinant receptors not activated in superdormant spore isolation although not as well as the initial dormant spores. The level of β-galactosidase from a gerA-lacZ fusion in superdormant spores isolated by germination via the GerA germinant receptor was identical to that in the initial dormant spores. Levels of the germination proteins GerD and SpoVAD were also identical in dormant and superdormant spores. However, levels of subunits of a germinant receptor or germinant receptors activated in superdormant spore isolation were 6- to 10-fold lower than those in dormant spores, while levels of subunits of germinant receptors not activated in superdormant spore isolation were only ≤ 2-fold lower. These results indicate that (i) levels of β-galactosidase from lacZ fusions to operons encoding germinant receptors may not be an accurate reflection of actual germinant receptor levels in spores and (ii) a low level of a specific germinant receptor or germinant receptors is a major cause of spore superdormancy.

Multi-cellular logistics of collective cell migration

During development, the formation of biological networks (such as organs and neuronal networks) is controlled by multicellular transportation phenomena based on cell migration. In multi-cellular systems, cellular locomotion is restricted by physical interactions with other cells in a crowded space, similar to passengers pushing others out of their way on a packed train. The motion of individual cells is intrinsically stochastic and may be viewed as a type of random walk. However, this walk takes place in a noisy environment because the cell interacts with its randomly moving neighbors. Despite this randomness and complexity, development is highly orchestrated and precisely regulated, following genetic (and even epigenetic) blueprints. Although individual cell migration has long been studied, the manner in which stochasticity affects multi-cellular transportation within the precisely controlled process of development remains largely unknown. To explore the general principles underlying multicellular migration, we focus on the migration of neural crest cells, which migrate collectively and form streams. We introduce a mechanical model of multi-cellular migration. Simulations based on the model show that the migration mode depends on the relative strengths of the noise from migratory and non-migratory cells. Strong noise from migratory cells and weak noise from surrounding cells causes “collective migration,” whereas strong noise from non-migratory cells causes “dispersive migration.” Moreover, our theoretical analyses reveal that migratory cells attract each other over long distances, even without direct mechanical contacts. This effective interaction depends on the stochasticity of the migratory and non-migratory cells. On the basis of these findings, we propose that stochastic behavior at the single-cell level works effectively and precisely to achieve collective migration in multi-cellular systems.

Biased excitable networks: how cells direct motion in response to gradients

The actin cytoskeleton in motile cells has many of the hallmarks of an excitable medium, including the presence of propagating waves. This excitable behavior can account for the spontaneous migration of cells. A number of reports have suggested that the chemoattractant-mediated signaling can bias excitability, thus providing a means by which cell motility can be directed. In this review, we discuss some of these observations and theories proposed to explain them. We also suggest a mechanism for cell polarity that can be incorporated into the existing framework.

A multiphysical model of cell migration integrating reaction-diffusion, membrane and cytoskeleton

Cellular motility is a complicated phenomenon that involves multiphysics, including the cytoskeleton, the plasma membrane and intracellular signal transduction. In this study, a hybrid computational model was developed for the simulation of whole-cell migration behaviors. The model integrates sub-models of reaction-diffusion, actin filaments (F-actin) and the plasma membrane. Reaction-diffusion was calculated as if enclosed by a moving membrane. Individual F-actins were reorganized on the basis of stochastic kinetic events, such as polymerization, capping, branching and severing. Membrane dynamics were modeled using an optimization of energy function that depends on cell volume, surface area, smoothness and the elasticity of F-actin against the membrane. Simulations of this model demonstrated self-organization of F-actin networks, as in lamellipodia, and chemotactic migration. Furthermore, this method was extended to address external obstacles to simulate the dynamic cellular morphological changes seen during invasive migration.

Bayes-optimal chemotaxis

Chemotaxis plays a crucial role in many biological processes, including nervous system development. However, fundamental physical constraints limit the ability of a small sensing device such as a cell or growth cone to detect an external chemical gradient. One of these is the stochastic nature of receptor binding, leading to a constantly fluctuating binding pattern across the cell's array of receptors. This is analogous to the uncertainty in sensory information often encountered by the brain at the systems level. Here we derive analytically the Bayes-optimal strategy for combining information from a spatial array of receptors in both one and two dimensions to determine gradient direction. We also show how information from more than one receptor species can be optimally integrated, derive the gradient shapes that are optimal for guiding cells or growth cones over the longest possible distances, and illustrate that polarized cell behavior might arise as an adaptation to slowly varying environments. Together our results provide closed-form predictions for variations in chemotactic performance over a wide range of gradient conditions.

Implementation of dynamic Bayesian decision making by intracellular kinetics

Decision making in a noisy and dynamically changing environment is a fundamental task for a cell. To choose appropriate decisions over time, a cell must be equipped with intracellular kinetics that can conduct dynamic and efficient decision making. By using the theory of sequential inference, I demonstrate that dynamic Bayesian decision making can be implemented by an intracellular kinetics with a dual positive feedback structure. I also show that the combination of linear instantaneous and nonlinear stationary sensitivities to the input dominantly contributes to decision making efficiency, and that the state-dependent sensitivity change further suppresses noisy response. The statistical principles underlying these two factors are further clarified to be a log-likelihood-dependent quantification of the input information and uncertainty-dependent sensitivity control.