Breaking an Epigenetic Chromatin Switch: Curious Features of Hysteresis in Saccharomyces cerevisiae Telomeric Silencing

In addition to gene network switches, local epigenetic modifications to DNA and histones play an important role in all-or-none cellular decision-making. Here, we study the dynamical design of a well-characterized epigenetic chromatin switch: the yeast SIR system, in order to understand the origin of the stability of epigenetic states. We study hysteresis in this system by perturbing it with a histone deacetylase inhibitor. We find that SIR silencing has many characteristics of a non-linear bistable system, as observed in conventional genetic switches, which are based on activities of a few promoters affecting each other through the abundance of their gene products. Quite remarkably, our experiments in yeast telomeric silencing show a very distinctive pattern when it comes to the transition from bistability to monostability. In particular, the loss of the stable silenced state, upon increasing the inhibitor concentration, does not seem to show the expected saddle node behavior, instead looking like a supercritical pitchfork bifurcation. In other words, the 'off' state merges with the 'on' state at a threshold concentration leading to a single state, as opposed to the two states remaining distinct up to the threshold and exhibiting a discontinuous jump from the 'off' to the 'on' state. We argue that this is an inevitable consequence of silenced and active regions coexisting with dynamic domain boundaries. The experimental observations in our study therefore have broad implications for the understanding of chromatin silencing in yeast and beyond

Predicting Phenotypic Diversity and the Underlying Quantitative Molecular Transitions

During development, signaling networks control the formation of multicellular patterns. To what extent quantitative fluctuations in these complex networks may affect multicellular phenotype remains unclear. Here, we describe a computational approach to predict and analyze the phenotypic diversity that is accessible to a developmental signaling network. Applying this framework to vulval development in C. elegans, we demonstrate that quantitative changes in the regulatory network can render ~500 multicellular phenotypes. This phenotypic capacity is an order-of-magnitude below the theoretical upper limit for this system but yet is large enough to demonstrate that the system is not restricted to a select few outcomes. Using metrics to gauge the robustness of these phenotypes to parameter perturbations, we identify a select subset of novel phenotypes that are the most promising for experimental validation. In addition, our model calculations provide a layout of these phenotypes in network parameter space. Analyzing this landscape of multicellular phenotypes yielded two significant insights. First, we show that experimentally well-established mutant phenotypes may be rendered using non-canonical network perturbations. Second, we show that the predicted multicellular patterns include not only those observed in C. elegans, but also those occurring exclusively in other species of the Caenorhabditis genus. This result demonstrates that quantitative diversification of a common regulatory network is indeed demonstrably sufficient to generate the phenotypic differences observed across three major species within the Caenorhabditis genus. Using our computational framework, we systematically identify the quantitative changes that may have occurred in the regulatory network during the evolution of these species. Our model predictions show that significant phenotypic diversity may be sampled through quantitative variations in the regulatory network without overhauling the core network architecture. Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentally observed across multiple species, we systematically trace the quantitative regulatory changes that may have occurred during the evolution of the Caenorhabditis genus

The role of multiple marks in epigenetic silencing and the emergence of a stable bivalent chromatin state

We introduce and analyze a minimal model of epigenetic silencing in budding yeast, built upon known biomolecular interactions in the system. Doing so, we identify the epigenetic marks essential for the bistability of epigenetic states. The model explicitly incorporates two key chromatin marks, namely H4K16 acetylation and H3K79 methylation, and explores whether the presence of multiple marks lead to a qualitatively different systems behavior. We find that having both modifications is important for the robustness of epigenetic silencing. Besides the silenced and transcriptionally active fate of chromatin, our model leads to a novel state with bivalent (i.e., both active and silencing) marks under certain perturbations (knock-out mutations, inhibition or enhancement of enzymatic activity). The bivalent state appears under several perturbations and is shown to result in patchy silencing. We also show that the titration effect, owing to a limited supply of silencing proteins, can result in counter-intuitive responses. The design principles of the silencing system is systematically investigated and disparate experimental observations are assessed within a single theoretical framework. Specifically, we discuss the behavior of Sir protein recruitment, spreading and stability of silenced regions in commonly-studied mutants (e.g., sas2, dot1) illuminating the controversial role of Dot1 in the systems biology of yeast silencing.Comment: Supplementary Material, 14 page

Robustness Analysis and Behavior Discrimination in Enzymatic Reaction Networks

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics

Shape, Size, and Robustness: Feasible Regions in the Parameter Space of Biochemical Networks

The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks

A Theoretical Exploration of Birhythmicity in the p53-Mdm2 Network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities

‘Glocal’ Robustness Analysis and Model Discrimination for Circadian Oscillators

To characterize the behavior and robustness of cellular circuits with many unknown parameters is a major challenge for systems biology. Its difficulty rises exponentially with the number of circuit components. We here propose a novel analysis method to meet this challenge. Our method identifies the region of a high-dimensional parameter space where a circuit displays an experimentally observed behavior. It does so via a Monte Carlo approach guided by principal component analysis, in order to allow efficient sampling of this space. This ‘global’ analysis is then supplemented by a ‘local’ analysis, in which circuit robustness is determined for each of the thousands of parameter sets sampled in the global analysis. We apply this method to two prominent, recent models of the cyanobacterial circadian oscillator, an autocatalytic model, and a model centered on consecutive phosphorylation at two sites of the KaiC protein, a key circadian regulator. For these models, we find that the two-sites architecture is much more robust than the autocatalytic one, both globally and locally, based on five different quantifiers of robustness, including robustness to parameter perturbations and to molecular noise. Our ‘glocal’ combination of global and local analyses can also identify key causes of high or low robustness. In doing so, our approach helps to unravel the architectural origin of robust circuit behavior. Complementarily, identifying fragile aspects of system behavior can aid in designing perturbation experiments that may discriminate between competing mechanisms and different parameter sets

A Developmental Systems Perspective on Epistasis: Computational Exploration of Mutational Interactions in Model Developmental Regulatory Networks

The way in which the information contained in genotypes is translated into complex phenotypic traits (i.e. embryonic expression patterns) depends on its decoding by a multilayered hierarchy of biomolecular systems (regulatory networks). Each layer of this hierarchy displays its own regulatory schemes (i.e. operational rules such as +/− feedback) and associated control parameters, resulting in characteristic variational constraints. This process can be conceptualized as a mapping issue, and in the context of highly-dimensional genotype-phenotype mappings (GPMs) epistatic events have been shown to be ubiquitous, manifested in non-linear correspondences between changes in the genotype and their phenotypic effects. In this study I concentrate on epistatic phenomena pervading levels of biological organization above the genetic material, more specifically the realm of molecular networks. At this level, systems approaches to studying GPMs are specially suitable to shed light on the mechanistic basis of epistatic phenomena. To this aim, I constructed and analyzed ensembles of highly-modular (fully interconnected) networks with distinctive topologies, each displaying dynamic behaviors that were categorized as either arbitrary or functional according to early patterning processes in the Drosophila embryo. Spatio-temporal expression trajectories in virtual syncytial embryos were simulated via reaction-diffusion models. My in silico mutational experiments show that: 1) the average fitness decay tendency to successively accumulated mutations in ensembles of functional networks indicates the prevalence of positive epistasis, whereas in ensembles of arbitrary networks negative epistasis is the dominant tendency; and 2) the evaluation of epistatic coefficients of diverse interaction orders indicates that, both positive and negative epistasis are more prevalent in functional networks than in arbitrary ones. Overall, I conclude that the phenotypic and fitness effects of multiple perturbations are strongly conditioned by both the regulatory architecture (i.e. pattern of coupled feedback structures) and the dynamic nature of the spatio-temporal expression trajectories displayed by the simulated networks

A hyper-heuristic with two guidance indicators for bi-objective mixed-shift vehicle routing problem with time windows

In this paper, a Mixed-Shift Vehicle Routing Problem is proposed based on a real-life container transportation problem. In a long planning horizon of multiple shifts, transport tasks are completed satisfying the time constraints. Due to the different travel distances and time of tasks, there are two types of shifts (long shift and short shift) in this problem. The unit driver cost for long shifts is higher than that of short shifts. A mathematical model of this Mixed-Shift Vehicle Routing Problem with Time Windows (MS-VRPTW) is established in this paper, with two objectives of minimizing the total driver payment and the total travel distance. Due to the large scale and nonlinear constraints, the exact search showed is not suitable to MS-VRPTW. An initial solution construction heuristic (EBIH) and a selective perturbation Hyper-Heuristic (GIHH) are thus developed. In GIHH, five heuristics with different extents of perturbation at the low level are adaptively selected by a high level selection scheme with the Hill Climbing acceptance criterion. Two guidance indicators are devised at the high level to adaptively adjust the selection of the low level heuristics for this bi-objective problem. The two indicators estimate the objective value improvement and the improvement direction over the Pareto Front, respectively. To evaluate the generality of the proposed algorithms, a set of benchmark instances with various features is extracted from real-life historical datasets. The experiment results show that GIHH significantly improves the quality of the final Pareto Solution Set, outperforming the state-of-the-art algorithms for similar problems. Its application on VRPTW also obtains promising results