Dorjsuren Battogtokh

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.

Comparative characterization of the Arabidopsis subfamily a1 β-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.

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.

Multi-scaled turbulence in large populations of oscillators in a diffusive medium

Large populations of oscillatory units distributed densely and uniformly in a diffusive medium exhibit turbulence with multi-scaling properties similar to fully developed fluid turbulence. Unlike the latter, however, the scaling exponents are parameter-dependent, and there exists a critical condition for the onset of individual motion of the oscillators. A three-component reaction-diffusion model is numerically analyzed to demonstrate our assertion. The universality of this type of turbulence which extends far beyond the reaction-diffusion systems is also discussed.

Mean-Field Theory Revives in Self-Oscillatory Fields with Non-Local Coupling

A simple mean-field idea is applicable to the pattern dynamics of large assemblies of limit-cycle oscillators with non-local coupling. This is demonstrated by developing a mathematical theory for the following two specific examples of pattern dynamics. Firstly, we discuss propagation of phase waves in noisy oscillatory media, with particular concern with the existence of a critical condition for persistent propagation of the waves throughout the medium, and also with the possibility of noise-induced turbulence. Secondly, we discuss the existence of an exotic class of patterns peculiar to non-local coupling called chimera where the system is composed of two distinct domains, one coherent and the other incoherent, separated from each other with sharp boundaries.

Pattern Formation in Nonlocally Coupled Oscillators

A Turing-Hopf mixed mode solution is found in Kuramoto’s reaction-diffusion model, 3) which can be considered a nonlocally coupled system. Analytic and numerical analysis shows that the nonlocally coupled complex Ginzburg-Landau equation possesses the mixed mode solution.

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.

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 others 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.

A Bistable Switch Mechanism for Stem Cell Domain Nucleation in the Shoot Apical Meristem

In plants, the stem cells residing in shoot apical meristems (SAM) give rise to above-ground tissues (Aichinger et al., 2012). Hence, the maintenance of stem cell niches is of central importance to a plants continued growth and development (Fletcher and Meyerowitz, 2000; Gordon et al., 2009). For the flowering plant Arabidopsis thaliana, the genetic determinants of stem cell growth, division, and localization have been identified, and negative feedback between a homeodomain transcription factor, WUSCHEL (WUS), and a receptor kinase, CLAVATA (CLV), is known to play a crucial role in controlling the reservoir of stem cells in the central domain of a SAM. The morphology of plant stems and floral organs is controlled in large part by the size and stability of SAMs, which is controlled, in turn, by spatiotemporal patterns of WUS and CLV expression in meristems. For example, loss of restrictive signals in clv mutants of Arabidopsis leads to enlargement of shoot and floral meristems, resulting in extra floral organs and club-shaped siliques (Jonsson et al., 2005). The size, localization and stability of stem cell domains should be determined, in principle, by the interactions of WUS and CLV proteins, especially by their propensities to diffuse through the domain and by the rates of the molecular reactions that control their activities. Within this paradigm, reaction-diffusion (RD) models of WUS-CLV interactions have been popular mathematical models of SAM dynamics (Jonsson et al., 2005; Hohm et al., 2010; Fujita et al., 2011). In RD models, the spontaneous generation of inhomogeneous distributions of WUS and CLV in SAM domains is usually attributed to mechanisms based on a “Turing” instability (Turing, 1952; Segel and Jackson, 1972). The generic RD equations for spatiotemporal changes in the concentrations, u(x,t) and v(x,t), of two interacting proteins are  ∂u∂t=f(u,v)+Du∂2u∂x2,      ∂v∂t=g(u,v)+Dv∂2v∂x2, where f(u,v) and g(u,v) are nonlinear functions describing their local chemical interactions. A unique, uniform, steady-state solution, u(x,t) = u0 = constant and v(x,t) = v0, of these equations can become unstable with respect to small, non-uniform perturbations, u(x,t) = u0 + eλt·δu·cos(qx) and v(x,t) = v0 + eλt·δv·cos(qx), δu > Du, generating standing waves of wavelength l ≈ 2π/qcrit in the simulations of the RD system (Gierer and Meinhardt, 1972; Murray, 2003). At present, the diffusive lengths of CLV and WUS in SAMs have not been determined, and there is no evidence to suggest that the Turing condition (diffusivity of CLV >> diffusivity of WUS) is satisfied in the central zone of a SAM.

Nonlinear Effects in the Site Blocking Induced Oscillations in the Catalytic Reaction of CO Oxidation

Higher order nonlinear effects of the site blocking induced oscillations in the Monte Carlo simulations of CO oxidation are outlined. It is shown that the rate equations accounting for these effects display a supercritical Hopf bifurcation.