Nucleation of stem cell domains in a bistable activator-inhibitor model of the shoot apical meristem.

Shoot apical meristems (SAMs) give rise to all above-ground tissues of a plant. Expansion of meristematic tissue is derived from the growth and division of stem cells that reside in a central zone of the SAM. This reservoir of stem cells is maintained by expression of a transcription factor WUSCHEL that is responsible for the development of stem cells in the central zone. WUSCHEL expression is self-activating and downregulated by a signaling pathway initiated by CLAVATA proteins, which are upregulated by WUSCHEL. This classic activator-inhibitor network can generate localized patterns of WUSCHEL activity by a Turing instability provided certain constraints on reaction rates and diffusion constants of WUSCHEL and CLAVATA are satisfied, and most existing mathematical models of nucleation and confinement of stem cells in the SAM rely on Turing's mechanism. However, Turing patterns have certain properties that are inconsistent with observed patterns of stem cell differentiation in the SAM. As an alternative mechanism, we propose a model for stem cell confinement based on a bistable-switch in WUSCHEL-CLAVATA interactions. We study the bistable-switch mechanism for pattern formation in a spatially continuous domain and in a discrete cellularized tissue in the presence of a non-uniform field of a rapidly diffusing hormone. By comparing domain formation by Turing and bistable-switch mechanisms in these contexts, we show that bistable switching provides a superior account of nucleation and confinement of the stem cell domain under reasonable assumptions on reaction rates and diffusion constants.

Deciphering the Dynamics of Interlocked Feedback Loops in a Model of the Mammalian Circadian Clock.

Mathematical models of fundamental biological processes play an important role in consolidating theory and experiments, especially if they are systematically developed, thoroughly characterized, and well tested by experimental data. In this work, we report a detailed bifurcation analysis of a mathematical model of the mammalian circadian clock network developed by Relógio et al., noteworthy for its consistency with available data. Using one- and two-parameter bifurcation diagrams, we explore how oscillations in the model depend on the expression levels of its constituent genes and the activities of their encoded proteins. These bifurcation diagrams allow us to decipher the dynamics of interlocked feedback loops by parametric variation of genes and proteins in the model. Among other results, we find that REV-ERB, a member of a subfamily of orphan nuclear receptors, plays a critical role in the intertwined dynamics of Relógio's model. The bifurcation diagrams reported here can be used for predicting how the core clock network responds-in terms of period, amplitude and phases of oscillations-to different perturbations.

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network.

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences.

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Domain nucleation and confinement in agent-controlled bistable systems.

We report a mechanism of pattern formation in growing bistable systems coupled indirectly. A modified Fujita et al. model is studied as an example of a reaction-diffusion system of nondiffusive activator and inhibitor molecules immersed in the medium of a fast diffusive agent. Here we show that, as the system grows, a new domain nucleates spontaneously in the area where the local level of the agent becomes critical. Newly nucleated domains are stable and the pattern formation is different from Turing's mechanism in monostable systems. Domains are spatially confined by the agent even if the activator and inhibitor molecules diffuse. With the spatial extension of the system, a larger domain may undergo a wave number instability, and the concentrations of active molecules within the neighboring elements of a domain can become sharply different. The mechanism reported in this work could be generic for pattern formation systems involving multistability, growth, and indirect coupling.

Comparative characterization of the Arabidopsis subfamily a1 beta-galactosidases.

The Arabidopsis genome contains 17 predicted beta-galactosidase genes, all of which belong to glycosyl hydrolase (GH) Family 35. These genes have been further grouped into seven subfamilies based on sequence similarity. The largest of these, subfamily a1, consists of six genes, Gal-1 (At3g13750), Gal-2 (At3g52840), Gal-3 (At4g36360), Gal-4 (At5g56870), Gal-5 (At1g45130), and Gal-12 (At4g26140), some of which were characterized in previous studies. We report here the purification and biochemical characterization of recombinant Gal-1, Gal-3, Gal-4 and Gal-12 from Pichiapastoris, completing the analysis of all six recombinant proteins, as well as the isolation and characterization of the native Gal-2 protein from Arabidopsis leaves. Comparison of the relative expression levels of the subfamily a1 beta-galactosidases at the mRNA and protein levels uncovered evidence of differential regulation, which may involve post-transcriptional and post-translational processes. In addition, this study provides further support for the proposed function of the subfamily a1 beta-galactosidases in cell wall modification based on analysis of the organ-specific expression and subcellular localization of Gal-1 and Gal-12. Our study suggests that, despite some differences in individual biochemical characteristics and expression patterns, each member of the family has the potential to contribute to the dynamics of the Arabidopsis plant cell wall.

Synchronization of eukaryotic cells by periodic forcing.

We study a cell population described by a minimal mathematical model of the eukaryotic cell cycle subject to periodic forcing that simultaneously perturbs the dynamics of the cell cycle engine and cell growth, and we show that the population can be synchronized in a mode-locked regime. By simplifying the model to two variables, for the phase of cell cycle progression and the mass of the cell, we calculate the Lyapunov exponents to obtain the parameter window for synchronization. We also discuss the effects of intrinsic mitotic fluctuations, asymmetric division, and weak mutual coupling on the pace of synchronization.

Analysis of a generic model of eukaryotic cell-cycle regulation.

We propose a protein interaction network for the regulation of DNA synthesis and mitosis that emphasizes the universality of the regulatory system among eukaryotic cells. The idiosyncrasies of cell cycle regulation in particular organisms can be attributed, we claim, to specific settings of rate constants in the dynamic network of chemical reactions. The values of these rate constants are determined ultimately by the genetic makeup of an organism. To support these claims, we convert the reaction mechanism into a set of governing kinetic equations and provide parameter values (specific to budding yeast, fission yeast, frog eggs, and mammalian cells) that account for many curious features of cell cycle regulation in these organisms. Using one-parameter bifurcation diagrams, we show how overall cell growth drives progression through the cell cycle, how cell-size homeostasis can be achieved by two different strategies, and how mutations remodel bifurcation diagrams and create unusual cell-division phenotypes. The relation between gene dosage and phenotype can be summarized compactly in two-parameter bifurcation diagrams. Our approach provides a theoretical framework in which to understand both the universality and particularity of cell cycle regulation, and to construct, in modular fashion, increasingly complex models of the networks controlling cell growth and division.

Periodic forcing of a mathematical model of the eukaryotic cell cycle.

In a differential equation model of the molecular network governing cell growth and division, cell cycle phases and transitions through checkpoints are associated with certain bifurcations of the underlying vector field. If the cell cycle is driven by another rhythmic process, interactions between forcing and bifurcations lead to emergent orbits and oscillations. In this paper, by varying the amplitude and frequency of forcing of the synthesis rates of regulatory proteins and the mass growth rate in a minimal model of the eukaryotic cell cycle, we study changes of the probability distributions of interdivision time and mass at division. By computing numerically the Lyapunov exponent of the model, we show that the splitting of probability distributions is associated with mode-locked solutions. We also introduce a simple, integrate-and-fire model to analyze mode locking in the cell cycle.

Bifurcation analysis of a model of the budding yeast cell cycle.

We study the bifurcations of a set of nine nonlinear ordinary differential equations that describe regulation of the cyclin-dependent kinase that triggers DNA synthesis and mitosis in the budding yeast, Saccharomyces cerevisiae. We show that Clb2-dependent kinase exhibits bistability (stable steady states of high or low kinase activity). The transition from low to high Clb2-dependent kinase activity is driven by transient activation of Cln2-dependent kinase, and the reverse transition is driven by transient activation of the Clb2 degradation machinery. We show that a four-variable model retains the main features of the nine-variable model. In a three-variable model exhibiting birhythmicity (two stable oscillatory states), we explore possible effects of extrinsic fluctuations on cell cycle progression.

Turbulence near cyclic fold bifurcations in birhythmic media.

We show that at the onset of a cyclic fold bifurcation, a birhythmic medium composed of glycolytic oscillators displays turbulent dynamics. By computing the largest Lyapunov exponent, the spatial correlation function, and the average transient lifetime, we classify it as weak turbulence of a transient nature. Virtual heterogeneities generating unstable fast oscillations account for the transient turbulence. In the presence of a wave number instability, unstable oscillations can be reinjected, leading to stationary turbulence. We also find similar turbulence in a cell cycle model. These findings suggest that weak turbulence may be universal in biochemical birhythmic media exhibiting cyclic fold bifurcations.