Evaluation of the performance of mechanisms for noise attenuation in a single-gene expression

Mutational robustness of gene regulatory networks

Mutational robustness of gene regulatory networks refers to their ability to generate constant biological output upon mutations that change network structure. Such networks contain regulatory interactions (transcription factor – target gene interactions) but often also protein-protein interactions between transcription factors. Using computational modeling, we study factors that influence robustness and we infer several network properties governing it. These include the type of mutation, i.e. whether a regulatory interaction or a protein-protein interaction is mutated, and in the case of mutation of a regulatory interaction, the sign of the interaction (activating vs. repressive). In addition, we analyze the effect of combinations of mutations and we compare networks containing monomeric with those containing dimeric transcription factors. Our results are consistent with available data on biological networks, for example based on evolutionary conservation of network features. As a novel and remarkable property, we predict that networks are more robust against mutations in monomer than in dimer transcription factors, a prediction for which analysis of conservation of DNA binding residues in monomeric vs. dimeric transcription factors provides indirect evidence.

Effective stochastic resonance under noise of heterogeneous amplitude

Stochastic resonance is numerically and analytically studied using a model wherein two coupled particles are exposed to noise of heterogeneous (i.e., particle-dependent) amplitude. In a certain range of coupling constant and under noise of heterogeneous amplitude, a particle is observed to resonate intensively to the external field. In other words, one particle under noise of very small or zero amplitude exhibits intensive resonance with the assistance of a highly fluctuating second particle under noise of large amplitude. This intensive resonance is interpreted as a product of the unique combinatory dynamics between one particle that stochastically fluctuates and another particle that classically resonates to the external field.

Counter-intuitive stochastic behavior of simple gene circuits with negative feedback

It has often been taken for granted that negative feedback loops in gene regulation work as homeostatic control mechanisms. If one increases the regulation strength a less noisy signal is to be expected. However, recent theoretical studies have reported the exact contrary, counter-intuitive observation, which has left a question mark over the relationship between negative feedback loops and noise. We explore and systematically analyze several minimal models of gene regulation, where a transcriptional repressor negatively regulates its own expression. For models including a quasi-steady-state assumption, we identify processes that buffer noise change (RNA polymerase binding) or accentuate it (repressor dimerization) alongside increasing feedback strength. Moreover, we show that lumping together transcription and translation in simplified models clearly underestimates the impact of negative feedback strength on the system's noise. In contrast, in systems without a quasi-steady-state assumption, noise always increases with negative feedback strength. Hence, subtle mathematical properties and model assumptions yield different types of noise profiles and, by consequence, previous studies have simultaneously reported decrease, increase or persistence of noise levels with increasing feedback. We discuss our findings in terms of separation of timescales and time correlations between molecular species distributions, extending current theoretical findings on the topic and allowing us to propose what we believe new ways to better characterize noise.

Multiple feedback loop design in the tryptophan regulatory network of Escherichia coli suggests a paradigm for robust regulation of processes in series

Biological networks have evolved through adaptation in uncertain environments. Of the different possible design paradigms, some may offer functional advantages over others. These designs can be quantified by the structure of the network resulting from molecular interactions and the parameter values. One may, therefore, like to identify the design motif present in the evolved network that makes it preferable over other alternatives. In this work, we focus on the regulatory networks characterized by serially arranged processes, which are regulated by multiple feedback loops. Specifically, we consider the tryptophan system present in Escherichia coli, which may be conceptualized as three processes in series, namely transcription, translation and tryptophan synthesis. The multiple feedback loop motif results from three distinct negative feedback loops, namely genetic repression, mRNA attenuation and enzyme inhibition. A framework is introduced to identify the key design components of this network responsible for its physiological performance. We demonstrate that the multiple feedback loop motif, as seen in the tryptophan system, enables robust performance to variations in system parameters while maintaining a rapid response to achieve homeostasis. Superior performance, if arising from a design principle, is intrinsic and, therefore, inherent to any similarly designed system, either natural or engineered. An experimental engineering implementation of the multiple feedback loop design on a two-tank system supports the generality of the robust attributes offered by the design.

Accuracy of positional information provided by multiple morphogen gradients with correlated noise

Normal development of multicellular organisms requires cells to respond properly according to their positions. Positional information is often provided to cells as concentrations of diffusive chemicals called morphogens with spatial gradients. However, the spatial profiles of their concentrations include various kinds of noises, making positional information unreliable. In many developmental systems, multiple morphogen gradients are adopted to specify the spatial position along a single axis, presumably to achieve a sufficiently high precision of information on the location of each cell. In this paper, we ask how the precision of positional information depends on the number of morphogens. We derive a formula for the limit of precision when each cell adopts the maximum-likelihood estimation of the "true" position from noisy inputs. The precision increases with the number of morphogens and interestingly it also depends on the correlation of noises. The positional specification can be made more precisely if their gradients are of the opposite (same) direction when noises of the two morphogens are positively (negatively) correlated. The formula also tells us a minimum number of morphogens needed to achieve a given precision of positional information. We illustrate the theory by analyzing experimental data for the gradients of two diffusive chemicals, Bicoid and Caudal, in the early development of Drosophila embryo. The analysis suggests that combined information provided by the two chemicals is able to give accurate positional information in the middle part of the embryo, where the embryo segmentation occurs in later stages, much more than near both ends.

Estimation of the source-by-source effect of autorepression on genetic noise

Transcriptional autorepression has been thought to be one of the simplest control circuits to attenuate fluctuations in gene expression. Here, we explored the effect of autorepression on fluctuations from different noise sources. We theoretically represent the fluctuations in the copy number of proteins as the sum of several terms, each of which is related to a specific noise source and expressed as the product of the source-specific fluctuations under no autorepression (path gain) and the effect of autorepression on them (loop gain). Inspection of each term demonstrates the source-independent noise-attenuating effect of autorepression as well as its source-dependent efficiency. Our experiments using a synthetic autorepression module revealed that autorepression attenuates fluctuations of various noise compositions. These findings indicate that the noise-attenuating effect of autorepression is robust against variation in noise compositions. We also experimentally estimated the loop gain for mRNA noise, demonstrating that loop gains are measurable parameters. Decomposition of fluctuations followed by experimental estimation of path and loop gains would help us to understand the noise-related feature of design principles underlying loop-containing biological networks.

Optimal placement of multiple morphogen sources

During development, cells grow, differentiate, divide, and die according to their spatial positions, yet the positional information given to cells by morphogens (diffusive chemicals) includes considerable noises from various origins. In this paper, we examine a relationship between fluctuations in morphogen concentrations that the cells receive and the precision of positional specification by the morphogens in multidimensional space. As a method to quantify the precision, we introduce a measure of "ambiguity of positional information," based on the information entropy. We discover that the location of morphogen sources crucially affects the ambiguity, and that the ambiguity becomes minimum when the angle made by gradient vectors of different morphogens cross at a right angle in a target region under a given organ geometry (orthogonality principle). We conjecture that morphogen sources in development might be placed at the nearly optimal position that minimizes the ambiguity of positional information. This is supported by experimental data on the configurations of two major sources of spatial patterning, the apical ectodermal ridge (AER) and the zone of polarizing activity (ZPA), in vertebrate limb development. Indeed, their predicted configuration agrees very well with the one observed in experiments. We believe that the placement of morphogen sources to minimize the ambiguity of positional information is a basic principle in development of multicellular organisms beyond this particular example.

Large-scale on-off switching of genetic activity mediated by the folding-unfolding transition in a giant DNA molecule: an hypothesis

We present a model to describe the on-off switching of transcriptional activity in a genetic assembly by considering the intrinsic characteristics of a giant genomic DNA molecule which can undergo a discrete structural transition between coiled and compact states. We propose a model in which the transition in the higher-order structure of DNA plays an essential role in regulating stable on-off switching and/or the oscillation of a large number of genes under the fluctuations in a living cell, where such a structural transition is caused by environmental factors. This model explains the rapid and broad transcriptional response in a genetic assembly as well as its robustness against fluctuations.

Critical condition for the occurrence of a noise-reduction effect

Based on a copy number control model of bacterial plasmid, the internal noise near a given steady state is investigated by using the linear noise approximation. All the parameters are restricted to a certain region so that the time spent near the steady state is long enough and the absorbing state can be neglected. For the noise in the plasmid molecules, a transition occurs with increasing the noise in the signal molecules under certain conditions. A noise-reduction mechanism, noise suppression by noise, is found. More importantly, the critical condition for the occurrence of the noise-reduction effect is given in our theoretical treatment.

Dose response relationship in anti-stress gene regulatory networks

To maintain a stable intracellular environment, cells utilize complex and specialized defense systems against a variety of external perturbations, such as electrophilic stress, heat shock, and hypoxia, etc. Irrespective of the type of stress, many adaptive mechanisms contributing to cellular homeostasis appear to operate through gene regulatory networks that are organized into negative feedback loops. In general, the degree of deviation of the controlled variables, such as electrophiles, misfolded proteins, and O₂, is first detected by specialized sensor molecules, then the signal is transduced to specific transcription factors. Transcription factors can regulate the expression of a suite of anti-stress genes, many of which encode enzymes functioning to counteract the perturbed variables. The objective of this study was to explore, using control theory and computational approaches, the theoretical basis that underlies the steady-state dose response relationship between cellular stressors and intracellular biochemical species (controlled variables, transcription factors, and gene products) in these gene regulatory networks. Our work indicated that the shape of dose response curves (linear, superlinear, or sublinear) depends on changes in the specific values of local response coefficients (gains) distributed in the feedback loop. Multimerization of anti-stress enzymes and transcription factors into homodimers, homotrimers, or even higher-order multimers, play a significant role in maintaining robust homeostasis. Moreover, our simulation noted that dose response curves for the controlled variables can transition sequentially through four distinct phases as stressor level increases: initial superlinear with lesser control, superlinear more highly controlled, linear uncontrolled, and sublinear catastrophic. Each phase relies on specific gain-changing events that come into play as stressor level increases. The low-dose region is intrinsically nonlinear, and depending on the level of local gains, presence of gain-changing events, and degree of feedforward gene activation, this region can appear as superlinear, sublinear, or even J-shaped. The general dose response transition proposed here was further examined in a complex anti-electrophilic stress pathway, which involves multiple genes, enzymes, and metabolic reactions. This work would help biologists and especially toxicologists to better assess and predict the cellular impact brought about by biological stressors.

To maintain a stable intracellular environment, cells are equipped with multiple specialized defense programs that are launched in response to various external chemical and physical stressors. These anti-stress mechanisms comprise primarily gene regulatory networks, and like many manmade control devices, such as thermostats and automobile cruise controls, they are often organized into negative feedback circuits. A quantitative understanding of how these control circuits operate in the cell can help us to assess and predict more accurately the cellular impacts brought about by perturbing stressors, such as environmental toxicants. Using control theory and computer simulations, we explored nature's design principle for anti-stress gene regulatory networks, and the manner in which cells respond and adapt to perturbations. We showed that cells can exploit multiple mechanisms, such as protein homodimerization, cooperative binding, and auto-regulation, to enhance the feedback loop gain, which, according to control theory, is a basic principle for effective perturbation resistance. We also illustrated that the steady-state dose response curve is likely to transition through multiple phases as stressor level increases, and that the low-dose region is inherently nonlinear. Our results challenge the common practice of linear extrapolation for evaluating the low-dose effect, and would lead to improved human health risk assessment for exposures to environmental toxicants.

An optimal number of molecules for signal amplification and discrimination in a chemical cascade

Understanding the information processing ability of signal transduction pathways is of great importance because of their crucial roles in triggering various cellular responses. Despite continuing theoretical investigation, some important aspects of signal transduction such as a transient response and its connection to stochasticity originating from a small number of molecules have not yet been well understood. It is, however, through these aspects that unexpected and nontrivial properties of the information processing emerge. In this article, we analyze the transient behavior of a simple signaling cascade by taking into account the stochasticity originating from the small number of molecules. We identify several properties of the signaling cascade that emerge as a result of the interplay between the stochasticity and transient dynamics of the cascade. We specifically demonstrate that each step of the cascade has an optimal number of signaling molecules at which the average signal amplitude becomes maximal. We further investigate the connection between a finite number of molecules and the ability of the cascade to discriminate between true and error signals, which cannot be inferred from deterministic descriptions. The implications of our results are discussed from both biological and mathematical viewpoints.