Hopf bifurcation in the presomitic mesoderm during the mouse segmentation

Vertebrae and ribs arise from embryonic tissues called somites. Somites arise sequentially from the unsegmented embryo tail, called presomitic mesoderm (PSM). The pace of somite formation is controlled by gene products such as hairy and enhancer of split 7 (Hes7) whose expression oscillates in the PSM. In addition to the cyclic genes, there is a gradient of fibroblast growth factor 8 (Fgf8) mRNA from posterior to anterior PSM. Recent experiments have shown that in the absence of Fgf signaling, Hes7 oscillations in the anterior and posterior PSM are lost. On the other hand, Notch mutants reduce the amplitude of posterior Hes7 oscillations and abolish anterior Hes7 oscillations. To understand these phenotypes, we delineated and simulated a logical and a delay differential equation (DDE) model with similar network topology in wild-type and mutant situations. Both models reproduced most wild-type and mutant phenotypes suggesting that the chosen topology is robust to explain these phenotypes. Numerical continuation of the model showed that even in the wild-type situation, the system changed from sustained to damped, i.e. a Hopf bifurcation occurred, when the Fgf concentration decreased in the PSM. This numerical continuation analysis further indicated that the most sensitive parameters for the oscillations are the parameters of Hes7 followed by those of Lunatic fringe (Lfng) and Notch1. In the wild-type, the damping of Hes7 oscillations was not so strong so that cells reached the new somites before they lose Hes7 oscillations. By contrast, in the fibroblast growth factor receptor 1 (Fgfr1) conditional knock-out (cKO) mutant simulation, Notch signaling was not able to maintain sustained Hes7 oscillations. Our analysis suggests that Fgf signaling makes cells enter an oscillatory state of Hes7 expression. After moving to the anterior PSM, where Fgf signaling is missing, Notch signaling compensates the damping of Hes7 oscillations in the anterior PSM.     

Stochastic Oscillations Induced by Intrinsic Fluctuations in a Self-Repressing Gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.     

Neural fate decisions mediated by combinatorial regulation of Hes1 and miR-9

In the nervous system, Hes1 shows an oscillatory manner in neural progenitors but a persistent one in neurons. Many models involving Hes1 have been provided for the study of neural differentiation but few of them take the role of microRNA into account. It is known that a microRNA, miR-9, plays crucial roles in modulating Hes1 oscillations. However, the roles of miR-9 in controlling Hes1 oscillations and inducing transition between different cell fates still need to be further explored. Here we provide a mathematical model to show the interaction between miR-9 and Hes1, with the aim of understanding how the Hes1 oscillations are produced, how they are controlled, and further, how they are terminated. Based on the experimental findings, the model demonstrates the essential roles of Hes1 and miR-9 in regulating the dynamics of the system. In particular, the model suggests that the balance between miR-9 and Hes1 plays important roles in the choice between progenitor maintenance and neural differentiation. In addition, the synergistic (or antagonistic) effects of several important regulations are investigated so as to elucidate the effects of combinatorial regulation in neural decision-making. Our model provides a qualitative mechanism for understanding the process in neural fate decisions regulated by Hes1 and miR-9.     

Simulations of propagated mouse ventricular action potentials: effects of molecular heterogeneity

The molecular heterogeneity of repolarizing currents produces significant spatial heterogeneity and/or dispersion of repolarization in many mammalian cardiac tissues. Transgenic mice are prominent experimental models for the study of the molecular basis of repolarization and arrhythmias. However, it is debated whether the small mouse heart can sustain physiologically relevant heterogeneity of repolarization. We used a comprehensive model of the mouse action potential (AP) to predict how small a region of the cardiac tissue can maintain spatial gradients of repolarization due to differential expression of channels. Our simulations of a one-dimensional multicellular ring or cable predict that substantial gradients in repolarization and intracellular Ca(2+) concentration transients can be maintained through heterogeneity of expression of K(+) channels in distances of approximately 10 cells that are sufficient to block propagation. The abruptness of expression gradients and the site of stimulation can cause Ca(2+) transient oscillations and affect the stability of Ca(2+) dynamics and AP propagation. Two different mechanisms of instability of AP propagation in one-dimensional cable occur at fast pacing rates. Transitions from periodic activity to alternans or to irregular behavior were observed. Abrupt gradients of channel expression can cause alternans at slower pacing rates than gradual changes. Our simulations demonstrate the importance of incorporating realistic Ca(2+) dynamics and current densities into models of propagated AP. They also emphasize that microscopic aspects of tissue organization are important for predicting large-scale propagation phenomena. Finally, our results predict that the mouse heart should be able to sustain substantial molecularly based heterogeneity of repolarization.     

Identification of oscillatory genes in somitogenesis from functional genomic analysis of a human mesenchymal stem cell model

During somitogenesis, oscillatory expression of genes in the notch and wnt signaling pathways plays a key role in regulating segmentation. These oscillations in expression levels are elements of a species-specific developmental mechanism. To date, the periodicity and components of the human clock remain unstudied. Here we show that a human mesenchymal stem/stromal cell (MSC) model can be induced to display oscillatory gene expression. We observed that the known cycling gene HES1 oscillated with a 5 h period consistent with available data on the rate of somitogenesis in humans. We also observed cycling of Hes1 expression in mouse C2C12 myoblasts with a period of 2 h, consistent with previous in vitro and embryonic studies. Furthermore, we used microarray and quantitative PCR (Q-PCR) analysis to identify additional genes that display oscillatory expression both in vitro and in mouse embryos. We confirmed oscillatory expression of the notch pathway gene Maml3 and the wnt pathway gene Nkd2 by whole mount in situ hybridization analysis and Q-PCR. Expression patterns of these genes were disrupted in Wnt3a(tm1Amc) mutants but not in Dll3(pu) mutants. Our results demonstrate that human and mouse in vitro models can recapitulate oscillatory expression observed in embryo and that a number of genes in multiple developmental pathways display dynamic expression in vitro.     

Mechanical and biochemical modeling of cortical oscillations in spreading cells

Actomyosin-based cortical contractility is a common feature of eukaryotic cells and is involved in cell motility, cell division, and apoptosis. In nonmuscle cells, oscillations in contractility are induced by microtubule depolymerization during cell spreading. We developed an ordinary differential equation model to describe this behavior. The computational model includes 36 parameters. The values for all but two of the model parameters were taken from experimental measurements found in the literature. Using these values, we demonstrate that the model generates oscillatory behavior consistent with current experimental observations. The rhythmic behavior occurs because of the antagonistic effects of calcium-induced contractility and stretch-activated calcium channels. The model makes several experimentally testable predictions: 1), buffering intracellular calcium increases the period and decreases the amplitude of cortical oscillations; 2), increasing the number or activity of stretch activated channels leads to an increase in period and amplitude of cortical oscillations; 3), inhibiting Ca²⁺ pump activity increases the period and amplitude of oscillations; and 4), a threshold exists for the calcium concentration below which oscillations cease.     

Lineage tracing reveals the dynamic contribution of Hes1+ cells to the developing and adult pancreas

Notch signaling regulates numerous developmental processes, often acting either to promote one cell fate over another or else to inhibit differentiation altogether. In the embryonic pancreas, Notch and its target gene Hes1 are thought to inhibit endocrine and exocrine specification. Although differentiated cells appear to downregulate Hes1, it is unknown whether Hes1 expression marks multipotent progenitors, or else lineage-restricted precursors. Moreover, although rare cells of the adult pancreas express Hes1, it is unknown whether these represent a specialized progenitor-like population. To address these issues, we developed a mouse Hes1(CreERT2) knock-in allele to inducibly mark Hes1(+) cells and their descendants. We find that Hes1 expression in the early embryonic pancreas identifies multipotent, Notch-responsive progenitors, differentiation of which is blocked by activated Notch. In later embryogenesis, Hes1 marks exocrine-restricted progenitors, in which activated Notch promotes ductal differentiation. In the adult pancreas, Hes1 expression persists in rare differentiated cells, particularly terminal duct or centroacinar cells. Although we find that Hes1(+) cells in the resting or injured pancreas do not behave as adult stem cells for insulin-producing beta (β)-cells, Hes1 expression does identify stem cells throughout the small and large intestine. Together, these studies clarify the roles of Notch and Hes1 in the developing and adult pancreas, and open new avenues to study Notch signaling in this and other tissues.     

Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study

MOTIVATION: In this study, we address the problem of estimating the parameters of regulatory networks and provide the first application of Markov chain Monte Carlo (MCMC) methods to experimental data. As a case study, we consider a stochastic model of the Hes1 system expressed in terms of stochastic differential equations (SDEs) to which rigorous likelihood methods of inference can be applied. When fitting continuous-time stochastic models to discretely observed time series the lengths of the sampling intervals are important, and much of our study addresses the problem when the data are sparse. RESULTS: We estimate the parameters of an autoregulatory network providing results both for simulated and real experimental data from the Hes1 system. We develop an estimation algorithm using MCMC techniques which are flexible enough to allow for the imputation of latent data on a finer time scale and the presence of prior information about parameters which may be informed from other experiments as well as additional measurement error.     

Regulation of active and quiescent somatic stem cells by Notch signaling

Somatic stem/progenitor cells actively proliferate and give rise to different types of mature cells (active state) in embryonic tissues while they are mostly dormant (quiescent state) in many adult tissues. Notch signaling is known to regulate both active and quiescent states of somatic stem cells, but how it regulates these different states is unknown. Recent studies revealed that the Notch effector Hes1 is expressed differently during the active and quiescent states during neurogenesis and myogenesis: high in the quiescent state and oscillatory in the active state. When the Hes1 expression level is high, both Ascl1 and MyoD expression are continuously suppressed. By contrast, when Hes1 expression oscillates, it periodically represses expression of the neurogenic factor Ascl1 and the myogenic factor MyoD, thereby driving Ascl1 and MyoD oscillations. High levels of Hes1 and the resultant Ascl1 suppression promote the quiescent state of neural stem cells, while Hes1 oscillation-dependent Ascl1 oscillations regulate their active state. Similarly, in satellite cells of muscles, known adult muscle stem cells, high levels of Hes1 and the resultant MyoD suppression seem to promote their quiescent state, while Hes1 oscillation-dependent MyoD oscillations activate their proliferation and differentiation. Therefore, the expression dynamics of Hes1 is a key regulatory mechanism of generating and maintaining active/quiescent stem cell states.