Landscape reveals critical network structures for sharpening gene expression boundaries

Beyond homogeneity: Assessing the validity of the Michaelis-Menten rate law in spatially heterogeneous environments

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations.

Comparison of cell state models derived from single-cell RNA sequencing data: graph versus multi-dimensional space

Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two cell state geometries for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; (i) by using partial differential equations on a graph representing intermediate cell states between known cell types, and (ii) by using the equations on a multi-dimensional continuous cell state-space. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell states during the pathogenesis of acute myeloid leukemia. We particularly focus on comparing the strength and weakness of the graph model and multi-dimensional model.

A stochastic model of homeostasis: The roles of noise and nuclear positioning in deciding cell fate

We study a population-based cellular model that starts from a single stem cell that divides stochastically to give rise to either daughter stem cells or differentiated daughter cells. There are three main components in the model: nucleus position, the underlying gene-regulatory network, and stochastic segregation of transcription factors in the daughter cells. The proportion of self-renewal and differentiated cell lines as a function of the nucleus position which in turn decides the plane of cleavage is studied. Both nuclear position and noise play an important role in determining the stem cell genealogies. We have observed both long and short genealogies in model simulation, and these compare well with experimental results from neuroblast and B-cell division. Symmetric divisions are observed in apical nuclei, while asymmetric division occurs when the nucleus is toward the base. In this model, the number of clones decreases over time, although the average clone size increases.

Landscape and kinetic path quantify critical transitions in epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT), a basic developmental process that might promote cancer metastasis, has been studied from various perspectives. Recently, the early warning theory has been used to anticipate critical transitions in EMT from mathematical modeling. However, the underlying mechanisms of EMT involving complex molecular networks remain to be clarified. Especially, how to quantify the global stability and stochastic transition dynamics of EMT and what the underlying mechanism for early warning theory in EMT is remain to be fully clarified. To address these issues, we constructed a comprehensive gene regulatory network model for EMT and quantified the corresponding potential landscape. The landscape for EMT displays multiple stable attractors, which correspond to E, M, and some other intermediate states. Based on the path-integral approach, we identified the most probable transition paths of EMT, which are supported by experimental data. Correspondingly, the results of transition actions demonstrated that intermediate states can accelerate EMT, consistent with recent studies. By integrating the landscape and path with early warning concept, we identified the potential barrier height from the landscape as a global and more accurate measure for early warning signals to predict critical transitions in EMT. The landscape results also provide an intuitive and quantitative explanation for the early warning theory. Overall, the landscape and path results advance our mechanistic understanding of dynamical transitions and roles of intermediate states in EMT, and the potential barrier height provides a new, to our knowledge, measure for critical transitions and quantitative explanations for the early warning theory.

FGF Gradient Controls Boundary Position Between Proliferating and Differentiating Cells and Regulates Lacrimal Gland Growth Dynamics

Fibroblast growth factor (FGF) signaling plays an important role in controlling cell proliferation, survival, and cell movements during branching morphogenesis of many organs. In mammals branching morphogenesis is primarily regulated by members of the FGF7-subfamily (FGF7 and FGF10), which are expressed in the mesenchyme, and signal to the epithelial cells through the “b” isoform of fibroblast growth factor receptor-2 (FGFR2). Our previous work demonstrated that FGF7 and FGF10 form different gradients in the extracellular matrix (ECM) and induce distinct cellular responses and gene expression profiles in the lacrimal and submandibular glands. The last finding was the most surprising since both FGF7 and FGF10 bind signal most strongly through the same fibroblast growth factor receptor-2b isoform (FGFR2b). Here we revisit this question to gain an explanation of how the different FGFs regulate gene expression. For this purpose, we employed our ex vivo epithelial explant migration assay in which isolated epithelial explants are grown near the FGF loaded beads. We demonstrate that the graded distribution of FGF induces activation of ERK1/2 MAP kinases that define the position of the boundary between proliferating “bud” and differentiating “stalk” cells of growing lacrimal gland epithelium. Moreover, we showed that gene expression profiles of the epithelial explants exposed to distinct FGFs strictly depend on the ratio between “bud” and “stalk” area. Our data also suggests that differentiation of “stalk” and “bud” regions within the epithelial explants is necessary for directional and persistent epithelial migration. Gaining a better understanding of FGF functions is important for development of new approaches to enhance tissue regeneration.