Simple Gene Regulatory Network Research Articles

On the evolutionary origin of discrete phenotypic plasticity.

Phenotypic plasticity provides an attractive strategy for adapting to various environments, but the evolutionary mechanism of the underlying genetic system is poorly understood. We use a simple gene regulatory network model to explore how a species acquires phenotypic plasticity, particularly focusing on discrete phenotypic plasticity, which has been difficult to explain by quantitative genetic models. Our approach employs a population genetic framework that integrates the developmental process, where each individual undergoes growth to develop its phenotype, which subsequently becomes subject to selection pressures. Our model considers two alternative types of environments, with the gene regulatory network including a sensor gene that turns on and off depending on the type of environment. With this assumption, we demonstrate that the system gradually adapts by acquiring the ability to produce two distinct optimum phenotypes under two types of environments without changing genotype, resulting in phenotypic plasticity. We find that the resulting plasticity is often discrete after a lengthy period of evolution. Our results suggest that gene regulatory networks have a notable capacity to flexibly produce various phenotypes in response to environmental changes. This study also shows that the evolutionary dynamics of phenotype may differ significantly between mechanistic-based developmental models and quantitative genetics models, suggesting the utility of incorporating gene regulatory networks into evolutionary models.

The distribution of fitness effects during adaptive walks using a simple genetic network.

The tempo and mode of adaptation depends on the availability of beneficial alleles. Genetic interactions arising from gene networks can restrict this availability. However, the extent to which networks affect adaptation remains largely unknown. Current models of evolution consider additive genotype-phenotype relationships while often ignoring the contribution of gene interactions to phenotypic variance. In this study, we model a quantitative trait as the product of a simple gene regulatory network, the negative autoregulation motif. Using forward-time genetic simulations, we measure adaptive walks towards a phenotypic optimum in both additive and network models. A key expectation from adaptive walk theory is that the distribution of fitness effects of new beneficial mutations is exponential. We found that both models instead harbored distributions with fewer large-effect beneficial alleles than expected. The network model also had a complex and bimodal distribution of fitness effects among all mutations, with a considerable density at deleterious selection coefficients. This behavior is reminiscent of the cost of complexity, where correlations among traits constrain adaptation. Our results suggest that the interactions emerging from genetic networks can generate complex and multimodal distributions of fitness effects.

Deterministic and probabilistic fate decisions co-exist in a single retinal lineage.

Correct nervous system development depends on the timely differentiation of progenitor cells into neurons. While the output of progenitor differentiation is well investigated at the population and clonal level, how stereotypic or variable fate decisions are during development is still more elusive. To fill this gap, we here follow the fate outcome of single neurogenic progenitors in the zebrafish retina over time using live imaging. We find that neurogenic progenitor divisions produce two daughter cells, one of deterministic and one of probabilistic fate. Interference with the deterministic branch of the lineage affects lineage progression. In contrast, interference with fate probabilities of the probabilistic branch results in a broader range of fate possibilities than in wild-type and involves the production of any neuronal cell type even at non-canonical developmental stages. Combining the interference data with stochastic modelling of fate probabilities revealed that a simple gene regulatory network is able to predict the observed fate decision probabilities during wild-type development. These findings unveil unexpected lineage flexibility that could ensure robust development of the retina and other tissues.

Time-optimal control of piecewise affine bistable gene-regulatory networks: preliminary results

In this preliminary work we give a geometric characterization of the time-optimal trajectories for a piecewise affine bistable switch. Such hybrid models play a major role in systems biology, as they can expressively account for the behaviors of simple gene-regulatory networks. A key example of these biological circuits is the genetic toggle switch, for which the optimality of a state switch has not been studied to the present day. The main tool of this work is an adaptation of the Hybrid Pontryagin's Maximum Principle to our setting, which allows to prove that the optimal control strategy is indeed a simple bang-bang feedback control law. In opposition to previous works, we show that a time-optimal transition between states should pass by an undifferentiated state, well known in cell biology for playing a major role in fate differentiation of cells. We conclude by showing numerical simulations of optimal trajectories illustrating the structure of the bang-bang optimal control for different scenarios.

Stability of Systems with Stochastic Delays and Applications to Genetic Regulatory Networks.

The dynamics of systems with stochastically varying time delays are investigated in this paper. It is shown that the mean dynamics can be used to derive necessary conditions for the stability of equilibria of the stochastic system. Moreover, the second moment dynamics can be used to derive sufficient conditions for almost sure stability of equilibria. The results are summarized using stability charts that are obtained via semi-discretization. The theoretical methods are applied to simple gene regulatory networks where it is demonstrated that stochasticity in the delay can improve the stability of steady protein production.

Pluripotency, Differentiation, and Reprogramming: A Gene Expression Dynamics Model with Epigenetic Feedback Regulation.

Embryonic stem cells exhibit pluripotency: they can differentiate into all types of somatic cells. Pluripotent genes such as Oct4 and Nanog are activated in the pluripotent state, and their expression decreases during cell differentiation. Inversely, expression of differentiation genes such as Gata6 and Gata4 is promoted during differentiation. The gene regulatory network controlling the expression of these genes has been described, and slower-scale epigenetic modifications have been uncovered. Although the differentiation of pluripotent stem cells is normally irreversible, reprogramming of cells can be experimentally manipulated to regain pluripotency via overexpression of certain genes. Despite these experimental advances, the dynamics and mechanisms of differentiation and reprogramming are not yet fully understood. Based on recent experimental findings, we constructed a simple gene regulatory network including pluripotent and differentiation genes, and we demonstrated the existence of pluripotent and differentiated states from the resultant dynamical-systems model. Two differentiation mechanisms, interaction-induced switching from an expression oscillatory state and noise-assisted transition between bistable stationary states, were tested in the model. The former was found to be relevant to the differentiation process. We also introduced variables representing epigenetic modifications, which controlled the threshold for gene expression. By assuming positive feedback between expression levels and the epigenetic variables, we observed differentiation in expression dynamics. Additionally, with numerical reprogramming experiments for differentiated cells, we showed that pluripotency was recovered in cells by imposing overexpression of two pluripotent genes and external factors to control expression of differentiation genes. Interestingly, these factors were consistent with the four Yamanaka factors, Oct4, Sox2, Klf4, and Myc, which were necessary for the establishment of induced pluripotent stem cells. These results, based on a gene regulatory network and expression dynamics, contribute to our wider understanding of pluripotency, differentiation, and reprogramming of cells, and they provide a fresh viewpoint on robustness and control during development.

Mathematical modeling of the intracellular protein dynamics: The importance of active transport along microtubules

In this paper we propose a mathematical model of protein and mRNA transport inside a cell. The spatio-temporal model takes into account the active transport along microtubules in the cytoplasm as well as diffusion and is able to reproduce the oscillatory changes in protein concentration observed in many experimental data. In the model the protein and the mRNA interact with each other that allows us to classify the model as a simple gene regulatory network. The proposed model is generic and may be adapted to specific signaling pathways. On the basis of numerical simulations, we formulate a new hypothesis that the oscillatory dynamics is allowed by the mRNA active transport along microtubules from the nucleus to distant locations.

EVOLUTION OF TRANSCRIPTION NETWORKS IN RESPONSE TO TEMPORAL FLUCTUATIONS

Organisms respond to changes in their environment over a wide range of biological and temporal scales. Such phenotypic plasticity can involve developmental, behavioral, physiological, and genetic shifts. The adaptive value of a plastic response is known to depend on the nature of the information that is available to the organism as well as the direct and indirect costs of the plastic response. We modeled the dynamic process of simple gene regulatory networks as they responded to temporal fluctuations in environmental conditions. We simulated the evolution of networks to determine when genes that function solely as transcription factors, with no direct function of their own, are beneficial to the function of the network. When there is perfect information about the environment and there is no timing information to be extracted then there is no advantage to adding pure transcription factor genes to the network. In contrast, when there is either timing information that can be extracted or only indirect information about the current state of the environment then additional transcription factor genes improve the evolved network fitness.

Robustification as a tool in modeling biochemical reaction networks

Biological functions have evolved to become robust against a multitude of perturbations such as gene mutations, intracellular noise and changes in the physical and chemical environment. This robustness should be reflected in models of the underlying biochemical networks, and robustness analysis has consequently been established as an important tool for model validation in systems biology. However, while robustness analysis can be used to invalidate a given model, it does not support the postulation of model modifications that can serve to improve the robustness. Herein we propose a method for this purpose, based on computing the sensitivity of the robustness with respect to generic dynamic perturbations applied to the individual network edges. To quantify robustness we compute the smallest simultaneous change in the activity of the network nodes that induces a bifurcation in the network, resulting in a qualitative change in the network behavior. The proposed method can be used to identify network interactions with the most significant impact on the overall robustness of a given function. By considering the impact of adding new nodes and edges, the method can also be used to postulate the presence of important unmodeled components and interactions. The focus here is on biological functions related to bistable switches and sustained oscillations, and the proposed methodology is demonstrated through application to metabolic oscillations in white blood cells and bistable switching in MAPK signal transduction. The classical Goodwin model of oscillations in a simple gene regulatory network is used for illustration throughout the paper.

Effects of nonlinear degradation of protein on the oscillatory dynamics in a simple gene regulatory network

Cooperative stability of protein is here defined as the tendency for the oligomers to be more stable than their monomeric components and to perform their physiological functions. In this paper, we incorporate nonlinear degradation of protein induced by cooperative stability into a simple model which has been previously presented in the biological literature. Linear analysis gives a critical time delay beyond which a periodic solution is born in a Hopf bifurcation. Lindstedt’s method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. Our findings indicate that cooperative stability can cause periodic dynamics through reducing the critical time delay. In addition, we show that cooperative stability may increase the amplitude of oscillation. Our study contributes to the theoretical demonstration of the effect of cooperative stability in the simple gene regulatory network.

Why are cellular switches Boolean? General conditions for multistable genetic circuits

The logic of cellular decision-making is largely controlled by regulatory circuits defining molecular switches. Such switching elements allow to turn a graded input signal into an all-or-nothing output. Traditional studies have focused on this bistable picture of regulation, but higher-order scenarios involving tristable and tetrastable states are possible too. Are these multiswitches allowed in simple gene regulatory networks? Are they likely to be observed? If not, why not? In this paper we present the examination of this question by means of a simple but powerful geometric approach. We examine the relation between multistability, the degree of multimerization of the regulators and the role of autoloops within a deterministic setting, finding that N-stable circuits are possible, although their likelihood to occur rapidly decays with the order of the switch. Our work indicates that, despite two-component circuits are able to implement multistability, they are optimal for Boolean switches. The evolutionary implications are outlined.

Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs

More and more experiments show that small RNAs regulate gene expression by repressing translation of messenger RNAs (mRNAs) or degrading mRNAs. In this paper, we incorporate the small RNAs into a simple gene regulatory network and investigate its dynamical behaviors. In addition, we also derive the theoretical results of globally asymptotic stability and provide the sufficient conditions for the oscillation of the simple gene regulatory network, and further demonstrate that the amplitudes against the change of delay in the gene regulatory network are robust.

An improved dynamic Finite State Projection algorithm for the numerical solution of the chemical master equation with applications

Recently, Munsky and Khammash suggested the Finite State Projection (FSP) algorithm for the numerical solution of the Chemical Master Equation, which provides a discrete and stochastic modelling framework for chemical kinetics. The important question of whether or not the algorithm is guaranteed to terminate is not addressed in the original work. We show that the well-known explosive birth process provides a counter example. We also give sufficient criteria for a model to be suitable for the FSP technique. We demonstrate the FSP technique on three novel applications. Results are presented for: (i) the Schlogl reactions; (ii) another example from Gillespie's celebrated book; and (iii) models for the role that dimerization plays in reducing noise in simple gene regulatory networks. Finally, we augment the dimerization model to include tetramers and show that this enhances the noise reduction properties of the network.

Dynamics and Intrinsic Statistical Fluctuations of a Gene Switch

We studied a simple gene regulatory network, the toggle switch. Specifically, we examined the means and statistical fluctuations in numbers of proteins. We found that when omega, the ratio of rates of protein-gene unbinding to protein degradation, was between approximately 10(-3) and approximately 10, the fluctuations were much larger than those we would have expected from Poisson statistics. In addition, we examined characteristic time values for system relaxation and found both that they increased with omega and that they have significant phase transition effects, with a secondary time scale appearing near the boundary between bistable and other phases. Last, we discuss the bistability of the toggle switch.

Potential energy landscape and robustness of a gene regulatory network: toggle switch.

Finding a multidimensional potential landscape is the key for addressing important global issues, such as the robustness of cellular networks. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This was realized by explicitly constructing the steady state probability of the gene switch in the protein concentration space in the presence of the intrinsic statistical fluctuations due to the small number of proteins in the cell. We explored the global phase space for the system. We found that the protein synthesis rate and the unbinding rate of proteins to the gene were small relative to the protein degradation rate; the gene switch is monostable with only one stable basin of attraction. When both the protein synthesis rate and the unbinding rate of proteins to the gene are large compared with the protein degradation rate, two global basins of attraction emerge for a toggle switch. These basins correspond to the biologically stable functional states. The potential energy barrier between the two basins determines the time scale of conversion from one to the other. We found as the protein synthesis rate and protein unbinding rate to the gene relative to the protein degradation rate became larger, the potential energy barrier became larger. This also corresponded to systems with less noise or the fluctuations on the protein numbers. It leads to the robustness of the biological basins of the gene switches. The technique used here is general and can be applied to explore the potential energy landscape of the gene networks.

Single molecule dynamics and statistical fluctuations of gene regulatory networks: A repressilator

The authors developed a time dependent method to study the single molecule dynamics of a simple gene regulatory network: a repressilator with three genes mutually repressing each other. They quantitatively characterize the time evolution dynamics of the repressilator. Furthermore, they study purely dynamical issues such as statistical fluctuations and noise evolution. They illustrated some important features of the biological network such as monostability, spirals, and limit cycle oscillation. Explicit time dependent Fano factors which describe noise evolution and show statistical fluctuations out of equilibrium can be significant and far from the Poisson distribution. They explore the phase space and the interrelationships among fluctuations, order, amplitude, and period of oscillations of the repressilators. The authors found that repressilators follow ordered limit cycle orbits and are more likely to appear in the lower fluctuating regions. The amplitude of the repressilators increases as the suppressing of the genes decreases and production of proteins increases. The oscillation period of the repressilators decreases as the suppressing of the genes decreases and production of proteins increases.