Superstability of the yeast cell-cycle dynamics: ensuring causality in the presence of biochemical stochasticity

Gene regulatory dynamics are governed by molecular processes and therefore exhibits an inherent stochasticity. However, for the survival of an organism it is a strict necessity that this intrinsic noise does not prevent robust functioning of the system. It is still an open question how dynamical stability is achieved in biological systems despite the omnipresent fluctuations. In this paper we investigate the cell cycle of the budding yeast Saccharomyces cerevisiae as an example of a well-studied organism. We study a genetic network model of 11 genes that coordinate the cell-cycle dynamics using a modeling framework which generalizes the concept of discrete threshold dynamics. By allowing for fluctuations in the process times, we introduce noise into the model, accounting for the effects of biochemical stochasticity. We study the dynamical attractor of the cell cycle and find a remarkable robustness against fluctuations of this kind. We identify mechanisms that ensure reliability in spite of fluctuations: 'Catcher states' and persistence of activity levels contribute significantly to the stability of the yeast cell cycle despite the inherent stochasticity.     

A maximum entropy criterion of filtering and coding for stationary autoregressive signals: Its physical interpretations and suggestions for its application to neural information transmission

The operations of encoding and decoding in communication agree with filtering operations of convolution and deconvolution for Gaussian signal processing. In an analogy with power transmission in thermodynamics, an autoregressive model of information transmission is proposed for representing a continuous communication system which requires a pair of an internal noise source and a signal source to encode or decode a message. In this model transinformation (informational entropy) equals the increase in stationary nonequilibrium organization formed through the amplification of white noise by a positive feedback system. The channel capacity is finite due to the existence of inherent noise in the system. The maximum entropy criterion in information dynamics corresponds to the 2^nd law of thermodynamics. If the process is stationary, the communication system is invertible, and has the maximum efficiency of transformation. The total variation in informational entropy is zero in the cycle of the invertible system, while in the noninvertible system the entropy of decoding is less than that of encoding. A noisy autoregressive coding which maximizes transinformation is optimum, but is also ideal.     

Colored extrinsic fluctuations and stochastic gene expression

Stochasticity is both exploited and controlled by cells. Although the intrinsic stochasticity inherent in biochemistry is relatively well understood, cellular variation, or ‘noise’, is predominantly generated by interactions of the system of interest with other stochastic systems in the cell or its environment. Such extrinsic fluctuations are nonspecific, affecting many system components, and have a substantial lifetime, comparable to the cell cycle (they are ‘colored’). Here, we extend the standard stochastic simulation algorithm to include extrinsic fluctuations. We show that these fluctuations affect mean protein numbers and intrinsic noise, can speed up typical network response times, and can explain trends in high‐throughput measurements of variation. If extrinsic fluctuations in two components of the network are correlated, they may combine constructively (amplifying each other) or destructively (attenuating each other). Consequently, we predict that incoherent feedforward loops attenuate stochasticity, while coherent feedforwards amplify it. Our results demonstrate that both the timescales of extrinsic fluctuations and their nonspecificity substantially affect the function and performance of biochemical networks.     

Computational design of synthetic regulatory networks from a genetic library to characterize the designability of dynamical behaviors

The engineering of synthetic gene networks has mostly relied on the assembly of few characterized regulatory elements using rational design principles. It is of outmost importance to analyze the scalability and limits of such a design workflow. To analyze the design capabilities of libraries of regulatory elements, we have developed the first automated design approach that combines such elements to search the genotype space associated to a given phenotypic behavior. Herein, we calculated the designability of dynamical functions obtained from circuits assembled with a given genetic library. By designing circuits working as amplitude filters, pulse counters and oscillators, we could infer new mechanisms for such behaviors. We also highlighted the hierarchical design and the optimization of the interface between devices. We dissected the functional diversity of a constrained library and we found that even such libraries can provide a rich variety of behaviors. We also found that intrinsic noise slightly reduces the designability of digital circuits, but it increases the designability of oscillators. Finally, we analyzed the robust design as a strategy to counteract the evolvability and noise in gene expression of the engineered circuits within a cellular background, obtaining mechanisms for robustness through non-linear negative feedback loops.     

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.     

Dynamical coding of sensory information with competitive networks.

Based on experiments with the locust olfactory system, we demonstrate that model sensory neural networks with lateral inhibition can generate stimulus specific identity-temporal patterns in the form of stimulus-dependent switching among small and dynamically changing neural ensembles (each ensemble being a group of synchronized projection neurons). Networks produce this switching mode of dynamical activity when lateral inhibitory connections are strongly non-symmetric. Such coding uses 'winner-less competitive' (WLC) dynamics. In contrast to the well known winner-take-all competitive (WTA) networks and Hopfield nets, winner-less competition represents sensory information dynamically. Such dynamics are reproducible, robust against intrinsic noise and sensitive to changes in the sensory input. We demonstrate the validity of sensory coding with WLC networks using two different formulations of the dynamics, namely the average and spiking dynamics of projection neurons (PN).     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

‘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.     

In-phase and anti-phase synchronization in noisy Hodgkin–Huxley neurons

We numerically investigate the influence of intrinsic channel noise on the dynamical response of delay-coupling in neuronal systems. The stochastic dynamics of the spiking is modeled within a stochastic modification of the standard Hodgkin–Huxley model wherein the delay-coupling accounts for the finite propagation time of an action potential along the neuronal axon. We quantify this delay-coupling of the Pyragas-type in terms of the difference between corresponding presynaptic and postsynaptic membrane potentials. For an elementary neuronal network consisting of two coupled neurons we detect characteristic stochastic synchronization patterns which exhibit multiple phase-flip bifurcations: The phase-flip bifurcations occur in form of alternate transitions from an in-phase spiking activity towards an anti-phase spiking activity. Interestingly, these phase-flips remain robust for strong channel noise and in turn cause a striking stabilization of the spiking frequency.     

Coupled feedback loops form dynamic motifs of cellular networks

Cellular networks are composed of complicated interconnections among components, and some subnetworks of particular functioning are often identified as network motifs. Among such network motifs, feedback loops are thought to play important dynamical roles. Intriguingly, such feedback loops are very often found as a coupled structure in cellular circuits. Therefore, we integrated all the scattered information regarding the coupled feedbacks in various cellular circuits and investigated the dynamical role of each coupled structure. Finally, we discovered that coupled positive feedbacks enhance signal amplification and bistable characteristics; coupled negative feedbacks realize enhanced homeostasis; coupled positive and negative feedbacks enable reliable decision-making by properly modulating signal responses and effectively dealing with noise.     

Hysteresis and cell cycle transitions: how crucial is it?

Recently, experiments have shown that cyclin-dependent kinase (CDK) activity exhibits hysteresis in its response to total cyclin when cyclin is made nondegradable and controlled externally. This observation was taken to support mathematical modeling predictions regarding the underlying dynamics of the cell cycle. However, cell cycle dynamics can also be generated by other nonhysteretic mechanisms. To examine the robustness of the hysteretic response of CDK activity to total cyclin, we simulated various cell cycle signal transduction networks, and correlated the dynamics to the response function of CDK activity versus total cyclin. By randomly searching the parameter space, we assessed robustness by estimating the frequency of hysteretic versus nonhysteretic dynamical mechanisms. When the dynamical instabilities were caused by feedback loops in CDK phosphorylation and dephosphorylation or by feedback between cyclin and the CDK inhibitor, the response function of CDK activity versus total cyclin correlated well with the dynamical instabilities. However, when the dynamical instabilities originated from feedback between cyclin and APC-CDH1 or RB-E2F, the response function did not correlate with dynamical instabilities. Thus, although a hysteretic response is neither necessary nor sufficient, it is in general a much more robust mechanism for generating cell cycle dynamics than nonhysteretic mechanisms.     

Dynamics and mechanical stability of the developing dorsoventral organizer of the wing imaginal disc

Shaping the primordia during development relies on forces and mechanisms able to control cell segregation. In the imaginal discs of Drosophila the cellular populations that will give rise to the dorsal and ventral parts on the wing blade are segregated and do not intermingle. A cellular population that becomes specified by the boundary of the dorsal and ventral cellular domains, the so-called organizer, controls this process. In this paper we study the dynamics and stability of the dorsal-ventral organizer of the wing imaginal disc of Drosophila as cell proliferation advances. Our approach is based on a vertex model to perform in silico experiments that are fully dynamical and take into account the available experimental data such as: cell packing properties, orientation of the cellular divisions, response upon membrane ablation, and robustness to mechanical perturbations induced by fast growing clones. Our results shed light on the complex interplay between the cytoskeleton mechanics, the cell cycle, the cell growth, and the cellular interactions in order to shape the dorsal-ventral organizer as a robust source of positional information and a lineage controller. Specifically, we elucidate the necessary and sufficient ingredients that enforce its functionality: distinctive mechanical properties, including increased tension, longer cell cycle duration, and a cleavage criterion that satisfies the Hertwig rule. Our results provide novel insights into the developmental mechanisms that drive the dynamics of the DV organizer and set a definition of the so-called Notch fence model in quantitative terms.     

Robustness and topology of the yeast cell cycle Boolean network

Yeast cell cycle Boolean network was used as a case study of robustness to protein noise. Robustness was interpreted as involving stability of G1 steady state and sequence of gene expression from cell cycle START to stationary G1. A robustness measure to evaluate robustness strength of a network was proposed. Robust putative networks corresponding to the same steady state and sequence of gene expression of wild-type network were sampled. Architecture of wild-type yeast cell cycle network can be revealed by average topology profile of sampled robust putative networks.     

Quantifying robustness and dissipation cost of yeast cell cycle network: the funneled energy landscape perspectives

We study the origin of robustness of yeast cell cycle cellular network through uncovering its underlying energy landscape. This is realized from the information of the steady-state probabilities by solving a discrete set of kinetic master equations for the network. We discovered that the potential landscape of yeast cell cycle network is funneled toward the global minimum, G1 state. The ratio of the energy gap between G1 and average versus roughness of the landscape termed as robustness ratio (RR) becomes a quantitative measure of the robustness and stability for the network. The funneled landscape is quite robust against random perturbations from the inherent wiring or connections of the network. There exists a global phase transition between the more sensitive response or less self-degradation phase leading to underlying funneled global landscape with large RR, and insensitive response or more self-degradation phase leading to shallower underlying landscape of the network with small RR. Furthermore, we show that the more robust landscape also leads to less dissipation cost of the network. Least dissipation and robust landscape might be a realization of Darwinian principle of natural selection at cellular network level. It may provide an optimal criterion for network wiring connections and design.     

Computational models of signalling networks for non-linear control

Artificial signalling networks (ASNs) are a computational approach inspired by the signalling processes inside cells that decode outside environmental information. Using evolutionary algorithms to induce complex behaviours, we show how chaotic dynamics in a conservative dynamical system can be controlled. Such dynamics are of particular interest as they mimic the inherent complexity of non-linear physical systems in the real world. Considering the main biological interpretations of cellular signalling, in which complex behaviours and robust cellular responses emerge from the interaction of multiple pathways, we introduce two ASN representations: a stand-alone ASN and a coupled ASN. In particular we note how sophisticated cellular communication mechanisms can lead to effective controllers, where complicated problems can be divided into smaller and independent tasks.