A genetic timer through noise-induced stabilization of an unstable state

Multimodal distribution of transient time of predator extinction in a three-species food chain

The transient dynamics capture the time history in the behavior of a system before reaching an attractor. This paper deals with the statistics of transient dynamics in a classic tri-trophic food chain with bistability. The species of the food chain model either coexist or undergo a partial extinction with predator death after a transient time depending upon the initial population density. The distribution of transient time to predator extinction shows interesting patterns of inhomogeneity and anisotropy in the basin of the predator-free state. More precisely, the distribution shows a multimodal character when the initial points are located near a basin boundary and a unimodal character when chosen from a location far away from the boundary. The distribution is also anisotropic because the number of modes depends on the direction of the local of initial points. We define two new metrics, viz., homogeneity index and local isotropic index, to characterize the distinctive features of the distribution. We explain the origin of such multimodal distributions and try to present their ecological implications.

Noise-induced quasiperiod and period switching

We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.

Independent control of mean and noise by convolution of gene expression distributions

Gene expression noise can reduce cellular fitness or facilitate processes such as alternative metabolism, antibiotic resistance, and differentiation. Unfortunately, efforts to study the impacts of noise have been hampered by a scaling relationship between noise and expression level from individual promoters. Here, we use theory to demonstrate that mean and noise can be controlled independently by expressing two copies of a gene from separate inducible promoters in the same cell. We engineer low and high noise inducible promoters to validate this result in Escherichia coli, and develop a model that predicts the experimental distributions. Finally, we use our method to reveal that the response of a promoter to a repressor is less sensitive with higher repressor noise and explain this result using a law from probability theory. Our approach can be applied to investigate the effects of noise on diverse biological pathways or program cellular heterogeneity for synthetic biology applications.

A tunable population timer in multicellular consortia

Processing time-dependent information requires cells to quantify the duration of past regulatory events and program the time span of future signals. At the single-cell level, timer mechanisms can be implemented with genetic circuits. However, such systems are difficult to implement in single cells due to saturation in molecular components and stochasticity in the limited intracellular space. In contrast, multicellular implementations outsource some of the components of information-processing circuits to the extracellular space, potentially escaping these constraints. Here, we develop a theoretical framework, based on trilinear coordinate representation, to study the collective behavior of populations composed of three cell types under stationary conditions. This framework reveals that distributing different processes (in our case the production, detection and degradation of a time-encoding signal) across distinct strains enables the implementation of a multicellular timer. Our analysis also shows that the circuit can be easily tunable by varying the cellular composition of the consortium.

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We propose a chemical wire architecture for distributed biological computation

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Our model predicts how input signals can be restored or modulated in the output

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Chemical wires can store temporal information and the system can act as a timer

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Digital periodic input signals can be filtered by altering the strain ratios

Theoretical study of the impact of adaptation on cell-fate heterogeneity and fractional killing

Fractional killing illustrates the cell propensity to display a heterogeneous fate response over a wide range of stimuli. The interplay between the nonlinear and stochastic dynamics of biochemical networks plays a fundamental role in shaping this probabilistic response and in reconciling requirements for heterogeneity and controllability of cell-fate decisions. The stress-induced fate choice between life and death depends on an early adaptation response which may contribute to fractional killing by amplifying small differences between cells. To test this hypothesis, we consider a stochastic modeling framework suited for comprehensive sensitivity analysis of dose response curve through the computation of a fractionality index. Combining bifurcation analysis and Langevin simulation, we show that adaptation dynamics enhances noise-induced cell-fate heterogeneity by shifting from a saddle-node to a saddle-collision transition scenario. The generality of this result is further assessed by a computational analysis of a detailed regulatory network model of apoptosis initiation and by a theoretical analysis of stochastic bifurcation mechanisms. Overall, the present study identifies a cooperative interplay between stochastic, adaptation and decision intracellular processes that could promote cell-fate heterogeneity in many contexts.

Synaptic mechanisms for motor variability in a feedforward network

Behavioral variability often arises from variable activity in the behavior-generating neural network. The synaptic mechanisms underlying this variability are poorly understood. We show that synaptic noise, in conjunction with weak feedforward excitation, generates variable motor output in the Aplysia feeding system. A command-like neuron (CBI-10) triggers rhythmic motor programs more variable than programs triggered by CBI-2. CBI-10 weakly excites a pivotal pattern-generating interneuron (B34) strongly activated by CBI-2. The activation properties of B34 substantially account for the degree of program variability. CBI-10- and CBI-2-induced EPSPs in B34 vary in amplitude across trials, suggesting that there is synaptic noise. Computational studies show that synaptic noise is required for program variability. Further, at network state transition points when synaptic conductance is low, maximum program variability is promoted by moderate noise levels. Thus, synaptic strength and noise act together in a nonlinear manner to determine the degree of variability within a feedforward network.

Role of noise and parametric variation in the dynamics of gene regulatory circuits

Stochasticity in gene expression impacts the dynamics and functions of gene regulatory circuits. Intrinsic noises, including those that are caused by low copy number of molecules and transcriptional bursting, are usually studied by stochastic simulations. However, the role of extrinsic factors, such as cell-to-cell variability and heterogeneity in the microenvironment, is still elusive. To evaluate the effects of both the intrinsic and extrinsic noises, we develop a method, named sRACIPE, by integrating stochastic analysis with random circuit perturbation (RACIPE) method. RACIPE uniquely generates and analyzes an ensemble of models with random kinetic parameters. Previously, we have shown that the gene expression from random models form robust and functionally related clusters. In sRACIPE we further develop two stochastic simulation schemes, aiming to reduce the computational cost without sacrificing the convergence of statistics. One scheme uses constant noise to capture the basins of attraction, and the other one uses simulated annealing to detect the stability of states. By testing the methods on several synthetic gene regulatory circuits and an epithelial-mesenchymal transition network in squamous cell carcinoma, we demonstrate that sRACIPE can interpret the experimental observations from single-cell gene expression data. We observe that parametric variation (the spread of parameters around a median value) increases the spread of the gene expression clusters, whereas high noise merges the states. Our approach quantifies the robustness of a gene circuit in the presence of noise and sheds light on a new mechanism of noise-induced hybrid states. We have implemented sRACIPE as an R package.

Stabilizing effect of volatility in financial markets

In financial markets, greater volatility is usually considered to be synonymous with greater risk and instability. However, large market downturns and upturns are often preceded by long periods where price returns exhibit only small fluctuations. To investigate this surprising feature, here we propose using the mean first hitting time, i.e., the average time a stock return takes to undergo for the first time a large negative (crashes) or positive variation (rallies), as an indicator of price stability, and relate this to a standard measure of volatility. In an empirical analysis of daily returns for 1071 stocks traded in the New York Stock Exchange, we find that this measure of stability displays nonmonotonic behavior, with a maximum, as a function of volatility. Also, we show that the statistical properties of the empirical data can be reproduced by a nonlinear Heston model. This analysis implies that, contrary to conventional wisdom, not only high, but also low volatility values can be associated with higher instability in financial markets. This proposed measure of stability can be extremely useful in risk control.

Computing with biological switches and clocks

The complex dynamics of biological systems is primarily driven by molecular interactions that underpin the regulatory networks of cells. These networks typically contain positive and negative feedback loops, which are responsible for switch-like and oscillatory dynamics, respectively. Many computing systems rely on switches and clocks as computational modules. While the combination of such modules in biological systems leads to a variety of dynamical behaviours, it is also driving development of new computing algorithms. Here we present a historical perspective on computation by biological systems, with a focus on switches and clocks, and discuss parallels between biology and computing. We also outline our vision for the future of biological computing.

Robust stochastic Turing patterns in the development of a one-dimensional cyanobacterial organism

Under nitrogen deprivation, the one-dimensional cyanobacterial organism Anabaena sp. PCC 7120 develops patterns of single, nitrogen-fixing cells separated by nearly regular intervals of photosynthetic vegetative cells. We study a minimal, stochastic model of developmental patterns in Anabaena that includes a nondiffusing activator, two diffusing inhibitor morphogens, demographic fluctuations in the number of morphogen molecules, and filament growth. By tracking developing filaments, we provide experimental evidence for different spatiotemporal roles of the two inhibitors during pattern maintenance and for small molecular copy numbers, justifying a stochastic approach. In the deterministic limit, the model yields Turing patterns within a region of parameter space that shrinks markedly as the inhibitor diffusivities become equal. Transient, noise-driven, stochastic Turing patterns are produced outside this region, which can then be fixed by downstream genetic commitment pathways, dramatically enhancing the robustness of pattern formation, also in the biologically relevant situation in which the inhibitors' diffusivities may be comparable.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Genetic Redundancies Enhance Information Transfer in Noisy Regulatory Circuits

Cellular decision making is based on regulatory circuits that associate signal thresholds to specific physiological actions. This transmission of information is subjected to molecular noise what can decrease its fidelity. Here, we show instead how such intrinsic noise enhances information transfer in the presence of multiple circuit copies. The result is due to the contribution of noise to the generation of autonomous responses by each copy, which are altogether associated with a common decision. Moreover, factors that correlate the responses of the redundant units (extrinsic noise or regulatory cross-talk) contribute to reduce fidelity, while those that further uncouple them (heterogeneity within the copies) can lead to stronger information gain. Overall, our study emphasizes how the interplay of signal thresholding, redundancy, and noise influences the accuracy of cellular decision making. Understanding this interplay provides a basis to explain collective cell signaling mechanisms, and to engineer robust decisions with noisy genetic circuits.

There is increasing evidence that the presence of molecular noise greatly influences function in biological systems. This could imply, for instance, that genetic circuits adopt particular architectures in order to reduce noise. On the other hand, noise can be beneficial. Here, we show that this could be the case for the functioning of analog to digital genetic devices, which are commonly found in cellular decision making situations. We use the framework of information theory to illustrate first how noise can enhance information transfer in these devices. In those regimes in which noise is detrimental, we discuss how genetic redundancies allow information to be maximized, and how this effect depends on the specifics of the devices, and the interdependence among them. These results provide overall an additional rationale for genetic redundancies in genomic systems.

Dissecting Embryonic Stem Cell Self-Renewal and Differentiation Commitment from Quantitative Models

To model quantitatively embryonic stem cell (ESC) self-renewal and differentiation by computational approaches, we developed a unified mathematical model for gene expression involved in cell fate choices. Our quantitative model comprised ESC master regulators and lineage-specific pivotal genes. It took the factors of multiple pathways as input and computed expression as a function of intrinsic transcription factors, extrinsic cues, epigenetic modifications, and antagonism between ESC master regulators and lineage-specific pivotal genes. In the model, the differential equations of expression of genes involved in cell fate choices from regulation relationship were established according to the transcription and degradation rates. We applied this model to the Murine ESC self-renewal and differentiation commitment and found that it modeled the expression patterns with good accuracy. Our model analysis revealed that Murine ESC was an attractor state in culture and differentiation was predominantly caused by antagonism between ESC master regulators and lineage-specific pivotal genes. Moreover, antagonism among lineages played a critical role in lineage reprogramming. Our results also uncovered that the ordered expression alteration of ESC master regulators over time had a central role in ESC differentiation fates. Our computational framework was generally applicable to most cell-type maintenance and lineage reprogramming.

Transitions in a genetic transcriptional regulatory system under Lévy motion

Based on a stochastic differential equation model for a single genetic regulatory system, we examine the dynamical effects of noisy fluctuations, arising in the synthesis reaction, on the evolution of the transcription factor activator in terms of its concentration. The fluctuations are modeled by Brownian motion and α-stable Lévy motion. Two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), are used to analyse the transitions from the low to high concentration states. A shorter MFET or higher FEP in the low concentration region facilitates such a transition. We have observed that higher noise intensities and larger jumps of the Lévy motion shortens the MFET and thus benefits transitions. The Lévy motion activates a transition from the low concentration region to the non-adjacent high concentration region, while Brownian motion can not induce this phenomenon. There are optimal proportions of Gaussian and non-Gaussian noises, which maximise the quantities MFET and FEP for each concentration, when the total sum of noise intensities are kept constant. Because a weaker stability indicates a higher transition probability, a new geometric concept is introduced to quantify the basin stability of the low concentration region, characterised by the escaping behaviour.

A Statistical Approach Reveals Designs for the Most Robust Stochastic Gene Oscillators

The engineering of transcriptional networks presents many challenges due to the inherent uncertainty in the system structure, changing cellular context, and stochasticity in the governing dynamics. One approach to address these problems is to design and build systems that can function across a range of conditions; that is they are robust to uncertainty in their constituent components. Here we examine the parametric robustness landscape of transcriptional oscillators, which underlie many important processes such as circadian rhythms and the cell cycle, plus also serve as a model for the engineering of complex and emergent phenomena. The central questions that we address are: Can we build genetic oscillators that are more robust than those already constructed? Can we make genetic oscillators arbitrarily robust? These questions are technically challenging due to the large model and parameter spaces that must be efficiently explored. Here we use a measure of robustness that coincides with the Bayesian model evidence, combined with an efficient Monte Carlo method to traverse model space and concentrate on regions of high robustness, which enables the accurate evaluation of the relative robustness of gene network models governed by stochastic dynamics. We report the most robust two and three gene oscillator systems, plus examine how the number of interactions, the presence of autoregulation, and degradation of mRNA and protein affects the frequency, amplitude, and robustness of transcriptional oscillators. We also find that there is a limit to parametric robustness, beyond which there is nothing to be gained by adding additional feedback. Importantly, we provide predictions on new oscillator systems that can be constructed to verify the theory and advance design and modeling approaches to systems and synthetic biology.

Intrinsic noise induces critical behavior in leaky Markovian networks leading to avalanching

The role intrinsic statistical fluctuations play in creating avalanches – patterns of complex bursting activity with scale-free properties – is examined in leaky Markovian networks. Using this broad class of models, we develop a probabilistic approach that employs a potential energy landscape perspective coupled with a macroscopic description based on statistical thermodynamics. We identify six important thermodynamic quantities essential for characterizing system behavior as a function of network size: the internal potential energy, entropy, free potential energy, internal pressure, pressure, and bulk modulus. In agreement with classical phase transitions, these quantities evolve smoothly as a function of the network size until a critical value is reached. At that value, a discontinuity in pressure is observed that leads to a spike in the bulk modulus demarcating loss of thermodynamic robustness. We attribute this novel result to a reallocation of the ground states (global minima) of the system's stationary potential energy landscape caused by a noise-induced deformation of its topographic surface. Further analysis demonstrates that appreciable levels of intrinsic noise can cause avalanching, a complex mode of operation that dominates system dynamics at near-critical or subcritical network sizes. Illustrative examples are provided using an epidemiological model of bacterial infection, where avalanching has not been characterized before, and a previously studied model of computational neuroscience, where avalanching was erroneously attributed to specific neural architectures. The general methods developed here can be used to study the emergence of avalanching (and other complex phenomena) in many biological, physical and man-made interaction networks.

Networks of noisy interacting components arise in diverse scientific disciplines. Here, we develop a mathematical framework to study the underlying causes of a bursting phenomenon in network activity known as avalanching. As prototypical examples, we study a model of disease spreading in a population of individuals and a model of brain activity in a neural network. Although avalanching is well-documented in neural networks, thought to be crucial for learning, information processing, and memory, it has not been studied before in disease spreading. We employ tools originally used to analyze thermodynamic systems to argue that randomness in the actions of individual network components plays a fundamental role in avalanche formation. We show that avalanching is a spontaneous behavior, brought about by a phenomenon reminiscent to a phase transition in statistical mechanics, caused by increasing randomness as the network size decreases. Our work demonstrates that a previously suggested balanced feed-forward network structure is not necessary for neuronal avalanching. Instead, we attribute avalanching to a reallocation of the global minima of the network's stationary potential energy landscape, caused by a noise-induced deformation of its topographic surface.

Demethylation and re-expression of epigenetically silenced tumor suppressor genes: sensitization of cancer cells by combination therapy

Epigenetic regulation in eukaryotic and mammalian systems is a complex and emerging field of study. While histone modifications create an open chromatin conformation allowing for gene transcription, CpG methylation adds a further dimension to the expression of specific genes in developmental pathways and carcinogenesis. In this review, we will highlight DNA methylation as one of the distinct mechanisms for gene silencing and try to provide insight into the role of epigenetics in cancer progenitor cell formation and carcinogenesis. We will also introduce the concept of a dynamic methylation-demethylation system and the potential for the existence of a demethylating enzyme in this process. Finally, we will explain how re-expression of epigenetically silenced tumor suppressor genes could be exploited to develop effective drug therapies. In particular, we will consider how a combination therapy that includes epigenetic drugs could possibly kill cancer progenitor cells and reduce the chance of relapse following chemotherapy.

Modeling gene expression in time and space

Cell populations rarely exhibit gene-expression profiles that are homogeneous in time and space. In the temporal domain, dynamical behaviors such as oscillations and pulses of protein production pervade cell biology, underlying phenomena as diverse as circadian rhythmicity, cell cycle control, stress and damage responses, and stem-cell pluripotency. In multicellular populations, spatial heterogeneities are crucial for decision making and development, among many other functions. Cells need to exquisitely coordinate this temporal and spatial variation to survive. Although the spatiotemporal character of gene expression is challenging to quantify experimentally at the level of individual cells, it is beneficial from the modeling viewpoint, because it provides strong constraints that can be probed by theoretically analyzing mathematical models of candidate gene and protein circuits. Here, we review recent examples of temporal dynamics and spatial patterning in gene expression to show how modeling such phenomenology can help us unravel the molecular mechanisms of cellular function.

The energy pump and the origin of the non-equilibrium flux of the dynamical systems and the networks

The global stability of dynamical systems and networks is still challenging to study. We developed a landscape and flux framework to explore the global stability. The potential landscape is directly linked to the steady state probability distribution of the non-equilibrium dynamical systems which can be used to study the global stability. The steady state probability flux together with the landscape gradient determines the dynamics of the system. The non-zero probability flux implies the breaking down of the detailed balance which is a quantitative signature of the systems being in non-equilibrium states. We investigated the dynamics of several systems from monostability to limit cycle and explored the microscopic origin of the probability flux. We discovered that the origin of the probability flux is due to the non-equilibrium conditions on the concentrations resulting energy input acting like non-equilibrium pump or battery to the system. Another interesting behavior we uncovered is that the probabilistic flux is closely related to the steady state deterministic chemical flux. For the monostable model of the kinetic cycle, the analytical expression of the probabilistic flux is directly related to the deterministic flux, and the later is directly generated by the chemical potential difference from the adenosine triphosphate (ATP) hydrolysis. For the limit cycle of the reversible Schnakenberg model, we also show that the probabilistic flux is correlated to the chemical driving force, as well as the deterministic effective flux. Furthermore, we study the phase coherence of the stochastic oscillation against the energy pump, and argue that larger non-equilibrium pump results faster flux and higher coherence. This leads to higher robustness of the biological oscillations. We also uncovered how fluctuations influence the coherence of the oscillations in two steps: (1) The mild fluctuations influence the coherence of the system mainly through the probability flux while maintaining the regular landscape topography. (2) The larger fluctuations lead to flat landscape and the complete loss of the stability of the whole system.

The role of feedback control mechanisms on the establishment of oscillatory regimes in the Ras/cAMP/PKA pathway in S. cerevisiae

In the yeast Saccharomyces cerevisiae, the Ras/cAMP/PKA pathway is involved in the regulation of cell growth and proliferation in response to nutritional sensing and stress conditions. The pathway is tightly regulated by multiple feedback loops, exerted by the protein kinase A (PKA) on a few pivotal components of the pathway. In this article, we investigate the dynamics of the second messenger cAMP by performing stochastic simulations and parameter sweep analysis of a mechanistic model of the Ras/cAMP/PKA pathway, to determine the effects that the modulation of these feedback mechanisms has on the establishment of stable oscillatory regimes. In particular, we start by studying the role of phosphodiesterases, the enzymes that catalyze the degradation of cAMP, which represent the major negative feedback in this pathway. Then, we show the results on cAMP oscillations when perturbing the amount of protein Cdc25 coupled with the alteration of the intracellular ratio of the guanine nucleotides (GTP/GDP), which are known to regulate the switch of the GTPase Ras protein. This multi-level regulation of the amplitude and frequency of oscillations in the Ras/cAMP/PKA pathway might act as a fine tuning mechanism for the downstream targets of PKA, as also recently evidenced by some experimental investigations on the nucleocytoplasmic shuttling of the transcription factor Msn2 in yeast cells.

The transition between stochastic and deterministic behavior in an excitable gene circuit

We explore the connection between a stochastic simulation model and an ordinary differential equations (ODEs) model of the dynamics of an excitable gene circuit that exhibits noise-induced oscillations. Near a bifurcation point in the ODE model, the stochastic simulation model yields behavior dramatically different from that predicted by the ODE model. We analyze how that behavior depends on the gene copy number and find very slow convergence to the large number limit near the bifurcation point. The implications for understanding the dynamics of gene circuits and other birth-death dynamical systems with small numbers of constituents are discussed.

Asymmetric stochastic switching driven by intrinsic molecular noise

Low-copy-number molecules are involved in many functions in cells. The intrinsic fluctuations of these numbers can enable stochastic switching between multiple steady states, inducing phenotypic variability. Herein we present a theoretical and computational study based on Master Equations and Fokker-Planck and Langevin descriptions of stochastic switching for a genetic circuit of autoactivation. We show that in this circuit the intrinsic fluctuations arising from low-copy numbers, which are inherently state-dependent, drive asymmetric switching. These theoretical results are consistent with experimental data that have been reported for the bistable system of the gallactose signaling network in yeast. Our study unravels that intrinsic fluctuations, while not required to describe bistability, are fundamental to understand stochastic switching and the dynamical relative stability of multiple states.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in this paper we develop a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. First, we develop a model that takes into account fluctuations of relative concentrations of proteins during special events of cell cycles. Such fluctuations are induced by varying rates of relative concentrations of proteins and/or by relative concentrations of proteins themselves. As such fluctuations may be responsible for qualitative changes in the cell, we develop a new model that accounts for the effect of cellular dynamics on the cell cycle. Finally, we analyze numerically nonlinear effects in the cell cycle by constructing phase portraits based on the newly developed model and carry out a parametric sensitivity analysis in order to identify parameters for an efficient cell cycle control. The results of computational experiments demonstrate that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

Optimizing periodicity and polymodality in noise-induced genetic oscillators

Many cellular functions are based on the rhythmic organization of biological processes into self-repeating cascades of events. Some of these periodic processes, such as the cell cycles of several species, exhibit conspicuous irregularities in the form of period skippings, which lead to polymodal distributions of cycle lengths. A recently proposed mechanism that accounts for this quantized behavior is the stabilization of a Hopf-unstable state by molecular noise. Here we investigate the effect of varying noise in a model system, namely an excitable activator-repressor genetic circuit, that displays this noise-induced stabilization effect. Our results show that an optimal noise level enhances the regularity (coherence) of the cycles, in a form of coherence resonance. Similar noise levels also optimize the multimodal nature of the cycle lengths. Together, these results illustrate how molecular noise within a minimal gene regulatory motif confers robust generation of polymodal patterns of periodicity.

Stochastic approaches in systems biology

The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches--the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker-Planck equation and the two-moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts.

Biological role of noise encoded in a genetic network motif

Genetic circuits that regulate distinct cellular processes can differ in their wiring pattern of interactions (architecture) and susceptibility to stochastic fluctuations (noise). Whether the link between circuit architecture and noise is of biological importance remains, however, poorly understood. To investigate this problem, we performed a computational study of gene expression noise for all possible circuit architectures of feed-forward loop (FFL) motifs. Results revealed that FFL architectures fall into two categories depending on whether their ON (stimulated) or OFF (unstimulated) steady states exhibit noise. To explore the biological importance of this difference in noise behavior, we analyzed 858 documented FFLs in Escherichia coli that were divided into 39 functional categories. The majority of FFLs were found to regulate two subsets of functional categories. Interestingly, these two functional categories associated with FFLs of opposite noise behaviors. This opposite noise preference revealed two noise-based strategies to cope with environmental constraints where cellular responses are either initiated or terminated stochastically to allow probabilistic sampling of alternative states. FFLs may thus be selected for their architecture-dependent noise behavior, revealing a biological role for noise that is encoded in gene circuit architectures.

Avoiding transcription factor competition at promoter level increases the chances of obtaining oscillation

Background

The ultimate goal of synthetic biology is the conception and construction of genetic circuits that are reliable with respect to their designed function (e.g. oscillators, switches). This task remains still to be attained due to the inherent synergy of the biological building blocks and to an insufficient feedback between experiments and mathematical models. Nevertheless, the progress in these directions has been substantial.

Results

It has been emphasized in the literature that the architecture of a genetic oscillator must include positive (activating) and negative (inhibiting) genetic interactions in order to yield robust oscillations. Our results point out that the oscillatory capacity is not only affected by the interaction polarity but by how it is implemented at promoter level. For a chosen oscillator architecture, we show by means of numerical simulations that the existence or lack of competition between activator and inhibitor at promoter level affects the probability of producing oscillations and also leaves characteristic fingerprints on the associated period/amplitude features.

Conclusions

In comparison with non-competitive binding at promoters, competition drastically reduces the region of the parameters space characterized by oscillatory solutions. Moreover, while competition leads to pulse-like oscillations with long-tail distribution in period and amplitude for various parameters or noisy conditions, the non-competitive scenario shows a characteristic frequency and confined amplitude values. Our study also situates the competition mechanism in the context of existing genetic oscillators, with emphasis on the Atkinson oscillator.

Switchable genetic oscillator operating in quasi-stable mode

Ring topologies of repressing genes have qualitatively different long-term dynamics if the number of genes is odd (they oscillate) or even (they exhibit bistability). However, these attractors may not fully explain the observed behaviour in transient and stochastic environments such as the cell. We show here that even repressilators possess quasi-stable, travelling wave periodic solutions that are reachable, long-lived and robust to parameter changes. These solutions underlie the sustained oscillations observed in even rings in the stochastic regime, even if these circuits are expected to behave as switches. The existence of such solutions can also be exploited for control purposes: operation of the system around the quasi-stable orbit allows us to turn on and off the oscillations reliably and on demand. We illustrate these ideas with a simple protocol based on optical interference that can induce oscillations robustly both in the stochastic and deterministic regimes.