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Stochastic fluctuations in reaction-diffusion processes often have substantial effect on spatial and temporal dynamics of signal transductions in complex biological systems. One popular approach for simulating these processes is to divide the system into small spatial compartments assuming that molecules react only within the same compartment and jump between adjacent compartments driven by the diffusion. While the approach is convenient in terms of its implementation, its computational cost may become prohibitive when diffusive jumps occur significantly more frequently than reactions, as in the case of rapid diffusion. Here, we present a hybrid continuous-discrete method in which diffusion is simulated using continuous approximation while reactions are based on the Gillespie algorithm. Specifically, the diffusive jumps are approximated as continuous Gaussian random vectors with time-dependent means and covariances, allowing use of a large time step, even for rapid diffusion. By considering the correlation among diffusive jumps, the approximation is accurate for the second moment of the diffusion process. In addition, a criterion is obtained for identifying the region in which such diffusion approximation is required to enable adaptive calculations for better accuracy. Applications to a linear diffusion system and two nonlinear systems of morphogens demonstrate the effectiveness and benefits of the new hybrid method.

The patterning of many developing tissues is organized by morphogens. Genetic and environmental perturbations of gene expression, protein synthesis and ligand binding are among the sources of unreliability that limit the accuracy and precision of morphogen-mediated patterning. While it has been found that the robustness of morphogen gradients to the perturbation of morphogen synthesis can be enhanced by particular mechanisms, how such mechanisms affect robustness to other perturbations, such as to receptor synthesis for the same morphogen, has been little explored. Here, we investigate the interplay between the robustness of patterning to the changes in receptor synthesis and morphogen synthesis and to the effects of cell-to-cell variability. Our analysis elucidates the trade-offs and constraints that arise as a result of achieving these three performance objectives simultaneously in the context of simple, steady-state morphogen gradients formed by diffusion and receptor-mediated uptake. Analysis of the interdependence between length scales of patterning and these performance objectives reveals several potential mechanisms for mitigating such trade-offs and constraints. One involves downregulation of receptor synthesis in the morphogen source, while another involves the presence of non-signalling cell-surface morphogen-binding molecules. Both of these mechanisms occur in Drosophila wing discs during their patterning. We computationally elucidate how these mechanisms improve the robustness and precision of morphogen-mediated patterning.

BackgroundHow morphogen gradients form has long been a subject of controversy. The strongest support for the view that morphogens do not simply spread by free diffusion has come from a variety of studies of the Decapentaplegic (Dpp) gradient of the Drosophila larval wing disc.ResultsIn the present study, we initially show how the failure, in such studies, to consider the coupling of transport to receptor-mediated uptake and degradation has led to estimates of transport rates that are orders of magnitude too low, lending unwarranted support to a variety of hypothetical mechanisms, such as "planar transcytosis" and "restricted extracellular diffusion." Using several independent dynamic methods, we obtain data that are inconsistent with such models and show directly that Dpp transport occurs by simple, rapid diffusion in the extracellular space. We discuss the implications of these findings for other morphogen systems in which complex transport mechanisms have been proposed.ConclusionsWe believe that these findings resolve a major, longstanding question about morphogen gradient formation and provide a solid framework for interpreting experimental observations of morphogen gradient dynamics.

In developing and self-renewing tissues, terminally differentiated (TD) cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit-amplifying) cell stages, which ultimately give rise to TD cells. As the tissue reaches a tightly controlled steady-state size, cells at different lineage stages assume distinct spatial locations within the tissue. Although tissue stratification appears to be genetically specified, the underlying mechanisms that direct tissue lamination are not yet completely understood. Herein, we use modeling and simulations to explore several potential mechanisms that can be utilized to create stratification during developmental or regenerative growth in general systems and in the model system, the olfactory epithelium of mouse. Our results show that tissue stratification can be generated and maintained through controlling spatial distribution of diffusive signaling molecules that regulate the proliferation of each cell type within the lineage. The ability of feedback molecules to stratify a tissue is dependent on a low TD death rate: high death rates decrease tissue lamination. Regulation of the cell cycle lengths of stem cells by feedback signals can lead to transient accumulation of stem cells near the base and apex of tissue.

A large, diverse, and growing number of strategies have been proposed to explain how morphogen gradients achieve robustness and precision. We argue that, to be useful, the evaluation of such strategies must take into account the constraints imposed by competing objectives and performance tradeoffs. This point is illustrated through a mathematical and computational analysis of the strategy of self-enhanced morphogen clearance. The results suggest that the usefulness of this strategy comes less from its ability to increase robustness to morphogen source fluctuations per se, than from its ability to overcome specific kinds of noise, and to increase the fraction of a morphogen gradient within which robust threshold positions may be established. This work also provides new insights into the longstanding question of why morphogen gradients show a maximum range in vivo.

Studies of developing and self-renewing tissues have shown that differentiated cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit amplifying; TA) cell stages, which ultimately give rise to terminally differentiated (TD) cells. In several cases, self-renewal and differentiation of stem and progenitor cells within such lineages have been shown to be under feedback regulation. Together, the existence of multiple cell stages within a lineage and complex feedback regulation are thought to confer upon a tissue the ability to autoregulate development and regeneration, in terms of both cell number (total tissue volume) and cell identity (the proportions of different cell types, especially TD cells, within the tissue). In this paper, we model neurogenesis in the olfactory epithelium (OE) of the mouse, a system in which the lineage stages and mediators of feedback regulation that govern the generation of terminally differentiated olfactory receptor neurons (ORNs) have been the subject of much experimental work. Here we report on the existence and uniqueness of steady states in this system, as well as local and global stability of these steady states. In particular, we identify parameter conditions for the stability of the system when negative feedback loops are represented either as Hill functions, or in more general terms. Our results suggest that two factors -- autoregulation of the proliferation of transit amplifying (TA) progenitor cells, and a low death rate of TD cells -- enhance the stability of this system.

COVID-19 vaccine-induced protection declines over time. This waning of immunity has been described in modelling as a lower level of protection. This study incorporated fine-scale vaccine waning into modelling to predict the next surge of the Omicron variant of the SARS-CoV-2 virus. In Hong Kong, the Omicron subvariant BA.2 caused a significant epidemic wave between February and April 2022, which triggered high vaccination rates. About half a year later, a second outbreak, dominated by a combination of BA.2, BA.4 and BA.5 subvariants, began to spread. We developed mathematical equations to formulate continuous changes in vaccine boosting and waning based on empirical serological data. These equations were incorporated into a multi-strain discrete-time Susceptible-Exposed-Infectious-Removed model. The daily number of reported cases during the first Omicron outbreak, with daily vaccination rates, the population mobility index and daily average temperature, were used to train the model. The model successfully predicted the size and timing of the second surge and the variant replacement by BA.4/5. It estimated 655,893 cumulative reported cases from June 1, 2022 to 31 October 2022, which was only 2.69% fewer than the observed cumulative number of 674,008. The model projected that increased vaccine protection (by larger vaccine coverage or no vaccine waning) would reduce the size of the second surge of BA.2 infections substantially but would allow more subsequent BA.4/5 infections. Increased vaccine coverage or greater vaccine protection can reduce the infection rate during certain periods when the immune-escape variants co-circulate; however, new immune-escape variants spread more by out-competing the previous strain., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

Cell polarity refers to spatial di erences in the shape and structure of cells, which leads to the generation of diverse cell types playing di erent roles in biological processes. Cell polarization usually involves the localization of some specific signaling molecules to a proper location of the cell membrane. Recent studies proposed that delayed negative feedback may be important for maintaining the robustness of cell polarization and the observed oscillating behavior of signaling cluster. However, the fundamental mechanisms for achieving cell polarization under negative feedback remain controversial. In this paper, we formulate the cell polarization system as a non-local reaction di usion equation with positive and delayed negative feedback loops. Through the Turing stability analysis, we identify the parameter conditions, including the range of the time delay constant, for achieving cell polarization without any inhomogeneous spatial cues. Also, our numerical results support that by controlling the length of the time delay in negative feedback and the magnitude of positive feedback, the oscillating behavior of signaling cluster can be observed in our simulations.

Cell polarization occurs along a single axis that is generally determined by a spatial cue. Cells of the budding yeast exhibit a characteristic pattern of budding, which depends on cell-type-specific cortical markers, reflecting a genetic programming for the site of cell polarization. The Cdc42 GTPase plays a key role in cell polarization in various cell types. Although previous studies in budding yeast suggested positive feedback loops whereby Cdc42 becomes polarized, these mechanisms do not include spatial cues, neglecting the normal patterns of budding. Here we combine live-cell imaging and mathematical modeling to understand how diploid daughter cells establish polarity preferentially at the pole distal to the previous division site. Live-cell imaging shows that daughter cells of diploids exhibit dynamic polarization of Cdc42-GTP, which localizes to the bud tip until the M phase, to the division site at cytokinesis, and then to the distal pole in the next G1 phase. The strong bias toward distal budding of daughter cells requires the distal-pole tag Bud8 and Rga1, a GTPase activating protein for Cdc42, which inhibits budding at the cytokinesis site. Unexpectedly, we also find that over 50% of daughter cells lacking Rga1 exhibit persistent Cdc42-GTP polarization at the bud tip and the distal pole, revealing an additional role of Rga1 in spatiotemporal regulation of Cdc42 and thus in the pattern of polarized growth. Mathematical modeling indeed reveals robust Cdc42-GTP clustering at the distal pole in diploid daughter cells despite random perturbation of the landmark cues. Moreover, modeling predicts different dynamics of Cdc42-GTP polarization when the landmark level and the initial level of Cdc42-GTP at the division site are perturbed by noise added in the model.