External Noise and External Signal Induced Transition of Gene Switch and Coherence Resonance in the Genetic Regulatory System

The transition of gene switch induced by external noises (multiplicative external noise and additive external noise) and external signals is investigated in the genetic regulatory system. Results show that the state-to-state transition of gene switch as well as resonant behaviors, such as the explicit coherence resonance (ECR), implicit coherence resonance (ICR) and control parameter coherence biresonance (CPCBR), can appear when noises are injected into the genetic regulatory system. The ECR is increased with the increase of the control parameter value when starting from the supercritical Hopf bifurcation parameter point, and there exists a critical control parameter value for the occurrence of ECR. However, the ICR is decreased as the control parameter value is increased when starting from the subcritical Hopf bifurcation point. In particular, the coherence of ECR is higher and more sensitive to noise than that of ICR. When an external signal is introduced into the system, the enhancement or suppression of the CPCBR and the number of peaks strongly depend on the frequency and amplitude of the external signal. Furthermore, the gene regulation system can selectively enhance or decrease the noise-induced oscillation signals at preferred frequency and amplitude of an external signal.     

Signal transduction and amplification in a circadian oscillator: Interaction between two colored noises

The signal transduction and amplification in a Neurospora circadian clock system is studied by using the mechanism of internal signal stochastic resonance (ISSR). Two cases have been investigated: the case of no correlations between multiplicative and additive colored noises and the case of correlations between two noises. The results show that, in both cases, the noise-induced circadian oscillations can be transduced with the phenomenon of internal signal stochastic resonance (ISSR). However, the correlation time and intensity of an additive colored noise play different roles for the ISSR, driven by multiplicative colored noise, while the correlation time and intensity of multiplicative colored noise hardly influence the ISSR driven by additive colored noise. In addition, the ISSR can be amplified or suppressed at an appropriate range of the correlation intensity between two colored noises. The fundamental frequency of noise-induced circadian oscillations is hardly shifted with the increment of the intensity and correlation time of colored noises, which implies that the Neurospora system could be resistant to colored noises, exhibit strong vitality and sustain intrinsic circadian rhythms.     

Multiplicative background correction for spotted microarrays to improve reproducibility

We propose a simple approach, the multiplicative background correction, to solve a perplexing problem in spotted microarray data analysis: correcting the foreground intensities for the background noise, especially for spots with genes that are weakly expressed or not at all. The conventional approach, the additive background correction, directly subtracts the background intensities from foreground intensities. When the foreground intensities marginally dominate the background intensities, the additive background correction provides unreliable estimates of the differential gene expression levels and usually presents M-A plots with fishtails or fans. Unreliable additive background correction makes it preferable to ignore the background noise, which may increase the number of false positives. Based on the more realistic multiplicative assumption instead of the conventional additive assumption, we propose to logarithmically transform the intensity readings before the background correction, with the logarithmic transformation symmetrizing the skewed intensity readings. This approach not only precludes the fishtails and fans in the M-A plots, but provides highly reproducible background-corrected intensities for both strongly and weakly expressed genes. The superiority of the multiplicative background correction to the additive one as well as the no background correction is justified by publicly available self-hybridization datasets.     

Epistasis Can Facilitate the Evolution of Reproductive Isolation by Peak Shifts: A Two-Locus Two-Allele Model

The influence of epistasis on the evolution of reproductive isolation by peak shifts is studied in a two-locus two-allele model of a quantitative genetic character under stabilizing selection. Epistasis is introduced by a simple multiplicative term in the function that maps gene effects onto genotypic values. In the model with only additive effects on the trait, the probability of a peak shift and the amount of reproductive isolation are always inversely related, i.e., the higher the peak shift rate, the lower the amount of reproductive isolation caused by the peak shift. With epistatic characters there is no consistent relationship between these two values. Interestingly, there are causes where both transition rates as well as the amount of reproductive isolation are increased relative to the additive model. This effect has two main causes: a shift in the location of the transition point, and the hybrids between the two alternative optimal genotypes have lower average fitness in the epistatic case. A review of the empirical literature shows that the fitness relations resulting in higher peak shift rates and more reproductive isolation are qualitatively the same as those observed for genes causing hybrid inferiority.     

Propagation effects of current and conductance noise in a model neuron with subthreshold oscillations

We have examined the effects of current and conductance noise in a single-neuron model which can generate a variety of physiologically important impulse patterns. Current noise enters the membrane equation directly while conductance noise is propagated through the activation variables. Additive Gaussian white noise which is implemented as conductance noise appears in the voltage equations as an additive and a multiplicative term. Moreover, the originally white noise is turned into colored noise. The noise correlation time is a function of the system's control parameters which may explain the different effects of current and conductance noise in different dynamic states. We have found the most significant, qualitative differences between different noise implementations in a pacemaker-like, tonic firing regime at the transition to chaotic burst discharges. This reflects a dynamic state of high physiological relevance.     

Theoretical analysis of epigenetic cell memory by nucleosome modification

Chromosomal regions can adopt stable and heritable alternative states resulting in bistable gene expression without changes to the DNA sequence. Such epigenetic control is often associated with alternative covalent modifications of histones. The stability and heritability of the states are thought to involve positive feedback where modified nucleosomes recruit enzymes that similarly modify nearby nucleosomes. We developed a simplified stochastic model for dynamic nucleosome modification based on the silent mating-type region of the yeast Schizosaccharomyces pombe. We show that the mechanism can give strong bistability that is resistant both to high noise due to random gain or loss of nucleosome modifications and to random partitioning upon DNA replication. However, robust bistability required: (1) cooperativity, the activity of more than one modified nucleosome, in the modification reactions and (2) that nucleosomes occasionally stimulate modification beyond their neighbor nucleosomes, arguing against a simple continuous spreading of nucleosome modification.     

Bistability and phase variation in Salmonella enterica

Cell-to-cell differences in bacterial gene expression can merely reflect the occurrence of noise. In certain cases, however, heterogeneous gene expression is a programmed event that results in bistable expression. If bistability is heritable, bacterial lineages are formed. When programmed bistability is reversible, the phenomenon is known as phase variation. In certain cases, bistability is controlled by genetic mechanisms (e. g., DNA rearrangement). In other cases, bistability has epigenetic origin. A robust epigenetic mechanism for the formation of bacterial lineages is the formation of heritable DNA methylation patterns. However, bistability can also arise upon propagation of gene expression patterns by feedback loops that are stable upon cell division. This review describes examples of bistability and phase variation in Salmonella enterica and discusses their adaptive value, sometimes in a speculative manner.     

Spatial pattern formation induced by Gaussian white noise

The ability of Gaussian noise to induce ordered states in dynamical systems is here presented in an overview of the main stochastic mechanisms able to generate spatial patterns. These mechanisms involve: (i) a deterministic local dynamics term, accounting for the local rate of variation of the field variable, (ii) a noise component (additive or multiplicative) accounting for the unavoidable environmental disturbances, and (iii) a linear spatial coupling component, which provides spatial coherence and takes into account diffusion mechanisms. We investigate these dynamics using analytical tools, such as mean-field theory, linear stability analysis and structure function analysis, and use numerical simulations to confirm these analytical results.     

Gene perturbation and intervention in probabilistic Boolean networks

MOTIVATION: A major objective of gene regulatory network modeling, in addition to gaining a deeper understanding of genetic regulation and control, is the development of computational tools for the identification and discovery of potential targets for therapeutic intervention in diseases such as cancer. We consider the general question of the potential effect of individual genes on the global dynamical network behavior, both from the view of random gene perturbation as well as intervention in order to elicit desired network behavior. RESULTS: Using a recently introduced class of models, called Probabilistic Boolean Networks (PBNs), this paper develops a model for random gene perturbations and derives an explicit formula for the transition probabilities in the new PBN model. This result provides a building block for performing simulations and deriving other results concerning network dynamics. An example is provided to show how the gene perturbation model can be used to compute long-term influences of genes on other genes. Following this, the problem of intervention is addressed via the development of several computational tools based on first-passage times in Markov chains. The consequence is a methodology for finding the best gene with which to intervene in order to most likely achieve desirable network behavior. The ideas are illustrated with several examples in which the goal is to induce the network to transition into a desired state, or set of states. The corresponding issue of avoiding undesirable states is also addressed. Finally, the paper turns to the important problem of assessing the effect of gene perturbations on long-run network behavior. A bound on the steady-state probabilities is derived in terms of the perturbation probability. The result demonstrates that states of the network that are more 'easily reachable' from other states are more stable in the presence of gene perturbations. Consequently, these are hypothesized to correspond to cellular functional states. AVAILABILITY: A library of functions written in MATLAB for simulating PBNs, constructing state-transition matrices, computing steady-state distributions, computing influences, modeling random gene perturbations, and finding optimal intervention targets, as described in this paper, is available on request from is@ieee.org.     

External noise and feedback regulation: steady-state statistics of auto-regulatory genetic network

The steady-state statistics of a single gene auto-regulatory genetic network with the additive external Gaussian white noises is investigated. The main result shows that the negative feedback will result in that the mRNA noise has a positive contribution to the protein noise, but the positive feedback will result in that the mRNA noise has a negative contribution to the protein noise. If there is no feed back, then the contribution of mRNA noise to protein noise is always positive. On the other hand, the analysis and numerical simulations of linear and nonlinear feedback show that it is possible that the negative feedback increases, but the positive feedback decreases, the protein noise.     

Signaling in small subcellular volumes. II. Stochastic and diffusion effects on synaptic network properties

The synaptic signaling network is capable of sophisticated cellular computations. These include the ability to respond selectively to different patterns of input, and to sustain changes in response over long periods. The small volume of the synapse complicates the analysis of signaling because the chemical environment is strongly affected by diffusion and stochasticity. This study is based on an updated version of a previously proposed synaptic signaling circuit (Bhalla and Iyengar, 1999) and analyzes three network computation properties in small volumes: bistability, thresholding, and pattern selectivity. Simulations show that although there are diffusive regimes in which bistability may persist, chemical noise at small volumes overwhelms bistability. In the deterministic situation, the network exhibits a sharp threshold for transition between lower and upper stable states. This transition is broadened and individual runs partition between lower and upper states, when stochasticity is considered. The third network property, pattern selectivity, is severely degraded at synaptic volumes. However, there are regimes in which a process similar to stochastic resonance operates and amplifies pattern selectivity. These results imply that simple scaling of signaling conditions to femtoliter volumes is unlikely, and microenvironments, such as reaction complex formation, may be essential for reliable small-volume signaling.     

Topology and robustness in the Drosophila segment polarity network

A complex hierarchy of genetic interactions converts a single-celled Drosophila melanogaster egg into a multicellular embryo with 14 segments. Previously, von Dassow et al. reported that a mathematical model of the genetic interactions that defined the polarity of segments (the segment polarity network) was robust (von Dassow et al. 2000). As quantitative information about the system was unavailable, parameters were sampled randomly. A surprisingly large fraction of these parameter sets allowed the model to maintain and elaborate on the segment polarity pattern. This robustness is due to the positive feedback of gene products on their own expression, which induces individual cells in a model segment to adopt different stable expression states (bistability) corresponding to different cell types in the segment polarity pattern. A positive feedback loop will only yield multiple stable states when the parameters that describe it satisfy a particular inequality. By testing which random parameter sets satisfy these inequalities, I show that bistability is necessary to form the segment polarity pattern and serves as a strong predictor of which parameter sets will succeed in forming the pattern. Although the original model was robust to parameter variation, it could not reproduce the observed effects of cell division on the pattern of gene expression. I present a modified version that incorporates recent experimental evidence and does successfully mimic the consequences of cell division. The behavior of this modified model can also be understood in terms of bistability in positive feedback of gene expression. I discuss how this topological property of networks provides robust pattern formation and how large changes in parameters can change the specific pattern produced by a network.     

Duration time of a one-dimensional random walk as a function of the energies of the intermediate states: application for dissociation and relaxation processes in DNA hybrids.

Kinetic parameters of macromolecular systems are important for their function in vitro and in vivo. These parameters describe how fast the system dissociates (the characteristic dissociation time), and how fast the system reaches equilibrium (characteristic relaxation time). For many macromolecular systems, the transitions within the systems are described as a random walk through a number of states with various free energies. The rate of transition between two given states within the system is characterized by the average time which passes between starting the movement from one state, and reaching the other state. This time is referred to as the mean first-passage time between two given states. The characteristic dissociation and relaxation times of the system depend on the first-passages times between the states within the system. Here, for a one-dimensional random walk we derived an equation, which connects the mean first-passage time between two states with the free energies of the states within the system. We also derived the general equation, which is not restricted to one-dimensional systems, connecting the relaxation time of the system with the first-passage times between states. The application of these equations to DNA branch migration, DNA structural transitions and other processes is discussed.     

Noise and Information Transmission in Promoters with Multiple Internal States

Based on the measurements of noise in gene expression performed during the past decade, it has become customary to think of gene regulation in terms of a two-state model, where the promoter of a gene can stochastically switch between an ON and an OFF state. As experiments are becoming increasingly precise and the deviations from the two-state model start to be observable, we ask about the experimental signatures of complex multistate promoters, as well as the functional consequences of this additional complexity. In detail, we i), extend the calculations for noise in gene expression to promoters described by state transition diagrams with multiple states, ii), systematically compute the experimentally accessible noise characteristics for these complex promoters, and iii), use information theory to evaluate the channel capacities of complex promoter architectures and compare them with the baseline provided by the two-state model. We find that adding internal states to the promoter generically decreases channel capacity, except in certain cases, three of which (cooperativity, dual-role regulation, promoter cycling) we analyze in detail.     

Positive-feedback loops as a flexible biological module

BACKGROUND: Bistability in genetic networks allows cells to remember past events and to make discrete decisions in response to graded signals. Bistable behavior can result from positive feedback, but feedback loops can have other roles in signal transduction as well. RESULTS: We introduced positive feedback into the budding-yeast pheromone response to convert it into a bistable system. In the presence of feedback, transient induction with high pheromone levels caused persistent pathway activation, whereas at lower levels a fraction of cells became persistently active but the rest inactivated completely. We also generated mutations that quantitatively tuned the basal and induced expression levels of the feedback promoter and showed that they qualitatively changed the behavior of the system. Finally, we developed a simple stochastic model of our positive-feedback system and showed the agreement between our simulations and experimental results. CONCLUSIONS: The positive-feedback loop can display several different behaviors, including bistability, and can switch between them as a result of simple mutations.