Genetic Feedback Loops Research Articles

Transcription factor competition facilitates self-sustained oscillations in single gene genetic circuits.

Genetic feedback loops can be used by cells to regulate internal processes or to keep track of time. It is often thought that, for a genetic circuit to display self-sustained oscillations, a degree of cooperativity is needed in the binding and unbinding of actor species. This cooperativity is usually modeled using a Hill function, regardless of the actual promoter architecture. Furthermore, genetic circuits do not operate in isolation and often transcription factors are shared between different promoters. In this work we show how mathematical modelling of genetic feedback loops can be facilitated with a mechanistic fold-change function that takes into account the titration effect caused by competing binding sites for transcription factors. The model shows how the titration effect facilitates self-sustained oscillations in a minimal genetic feedback loop: a gene that produces its own repressor directly without cooperative transcription factor binding. The use of delay-differential equations leads to a stability contour that predicts whether a genetic feedback loop will show self-sustained oscillations, even when taking the bursty nature of transcription into account.

Hopf bifurcation analysis for models on genetic negative feedback loops

Negative feedback loops have been identified as genetic generation of oscillations in cells. The Goodwin model, a prototypical negative feedback oscillator, has been applied largely in the studies of biological rhythms. By applying the degenerate Routh-Hurwitz criterion, we perform a thorough Hopf bifurcation analysis on the Goodwin model and two other models on negative feedback loops. We not only derive the precise conditions for the Hopf bifurcation in these models, but also obtain the frequencies of the bifurcating periodic solutions. Such frequencies are acquired by factorizing the characteristic polynomials of the linearized systems at the bifurcation points and the bifurcation values. We also compute the higher-order terms which determine the stability of the bifurcating periodic solution and other properties of the Hopf bifurcation for the Goodwin model. Our result indicates that the minimal Hill coefficient does not increase and no further condition is required for the bifurcating periodic solution to be stable. We allocate the parameter values according to the present theories to illustrate limit-cycle oscillations in several numerical examples for these models.

Collective Oscillations in Coupled-Cell Systems

We investigate oscillations in coupled systems. The methodology is based on the Hopf bifurcation theorem and a condition extended from the Routh–Hurwitz criterion. Such a condition leads to locating the bifurcation values of the parameters. With such an approach, we analyze a single-cell system modeling the minimal genetic negative feedback loop and the coupled-cell system composed by these single-cell systems. We study the oscillatory properties for these systems and compare these properties between the model with Hill-type repression and the one with protein-sequestration-based repression. As the parameters move from the Hopf bifurcation value for single cells to the one for coupled cells, we compute the eigenvalues of the linearized systems to obtain the magnitude of the collective frequency when the periodic solution of the coupled-cell system is generated. Extending from this information on the parameter values, we further compute and compare the collective frequency for the coupled-cell system and the average frequency of the decoupled individual cells. To compare these scenarios with other biological oscillators, we perform parallel analysis and computations on a segmentation clock model.

Steady-state fluctuations of a genetic feedback loop with fluctuating rate parameters using the unified colored noise approximation

A common model of stochastic auto-regulatory gene expression describes promoter switching via cooperative protein binding, effective protein production in the active state and dilution of proteins. Here we consider an extension of this model whereby colored noise with a short correlation time is added to the reaction rate parameters—we show that when the size and timescale of the noise is appropriately chosen it accounts for fast reactions that are not explicitly modeled, e.g., in models with no mRNA description, fluctuations in the protein production rate can account for rapid multiple stages of nuclear mRNA processing which precede translation in eukaryotes. We show how the unified colored noise approximation can be used to derive expressions for the protein number distribution that is in good agreement with stochastic simulations. We find that even when the noise in the rate parameters is small, the protein distributions predicted by our model can be significantly different than models assuming constant reaction rates.

Control strategies for sustained oscillations in a disrupted biological clock

As many disorders have been correlated with dysfunctional biophysical rhythms, new therapies based on the control of clock functions are emerging for the slowdown of disease progression. In this context, a general disrupted biological clock is modeled by the canonical form of a genetic negative feedback loop. When the unique fixed point of this N-dimensional non-linear differential system is stable, the model reproduces accurately the damped oscillations observed in a damaged oscillator. First, a synthetic modification of the network is proved to generate sustained oscillations and allow to recover a functional clock. The desired periodic trajectories are obtained by destabilizing the fixed point of the model and monotone properties are applied for global results. In a limit case, this modification of the loop is shown to be equivalent to an external piecewise constant control law, supporting the conjecture that simple qualitative control strategies may be able to guarantee sustained oscillations. From the perspective of a biological implementation, this result is promising as these types of control are well adapted to experimental constraints. To support this theoretical work, the methods are applied to the disrupted circadian clock observed in human cancer cells.

Parameter estimation for biochemical reaction networks using Wasserstein distances

We present a method for estimating parameters in stochastic models of biochemical reaction networks by fitting steady-state distributions using Wasserstein distances. We simulate a reaction network at different parameter settings and train a Gaussian process to learn the Wasserstein distance between observations and the simulator output for all parameters. We then use Bayesian optimization to find parameters minimizing this distance based on the trained Gaussian process. The effectiveness of our method is demonstrated on the three-stage model of gene expression and a genetic feedback loop for which moment-based methods are known to perform poorly. Our method is applicable to any simulator model of stochastic reaction networks, including Brownian dynamics.

Qualitative control of undesired oscillations in a genetic negative feedback loop with uncertain measurements

In the context of gene regulatory networks, a negative feedback loop is modeled by N-coupled ordinary differential equations. The resulting system is highly non-linear due to the use of smooth Hill functions. This classical dynamical system properly captures the two main biological behaviors arising from this type of recurrent network motif: homeostasis under global stability of the unique fixed point, and biochemical oscillations otherwise. When homeostatic conditions are disrupted, undesired sustained oscillations can appear. In this context, a biologically relevant control strategy is designed in order to suppress these undesirable oscillations. As biological measurement techniques do not provide a quantitative knowledge of the system, the control law is chosen piecewise constant and dependent on specific regions of the state space. Moreover, due to biological devices inaccuracies, the measurements are considered uncertain leading to regions in which the control law is undefined. Under appropriate conditions on the control inputs, successive repelling regions of the state space are determined in order to prove the global convergence of the system towards an adjustable zone around the fixed point. These results are illustrated with the well-known p53-Mdm2 genetic feedback loop.

Revisiting the Reduction of Stochastic Models of Genetic Feedback Loops with Fast Promoter Switching

Propensity functions of the Hill type are commonly used to model transcriptional regulation in stochastic models of gene expression. This leads to an effective reduced master equation for the mRNA and protein dynamics only. Based on deterministic considerations, it is often stated or tacitly assumed that such models are valid in the limit of rapid promoter switching. Here, starting from the chemical master equation describing promoter-protein interactions, mRNA transcription, protein translation, and decay, we prove that in the limit of fast promoter switching, the distribution of protein numbers is different than that given by standard stochastic models with Hill-type propensities. We show the differences are pronounced whenever the protein-DNA binding rate is much larger than the unbinding rate, a special case of fast promoter switching. Furthermore, we show using both theory and simulations that use of the standard stochastic models leads to drastically incorrect predictions for the switching properties of positive feedback loops and that these differences decrease with increasing mean protein burst size. Our results confirm that commonly used stochastic models of gene regulatory networks are only accurate in a subset of the parameter space consistent with rapid promoter switching.

A new qualitative control strategy for the genetic Toggle Switch

Genetic positive feedback loops are essential for cell decision making and cell differentiation. They are traditionally described by a two-dimensional smooth non-linear differential system composed of Hill functions. The bistability of this classic model properly captures the decision properties of these biological motifs. This paper designs a new control strategy based on the measurement and control of a unique gene within the loop, in order to stabilize the system around its unstable fixed point. The quantized nature of genetic measurements and the new synthetic control approaches available in biology encourage the use of a piecewise constant control law. A specific partitioning of the state space and the study of successive repelling regions allow to prove global convergence and global stability for the resulting system. The same strategy is shown to be efficient as well in the more realistic context in which measurements are considered possibly uncertain. This new control strategy is compared with a real biological experiment that was implemented for the Toggle Switch with the same stabilization objective, but for which both genes were measured and controlled.

Global stabilization of a genetic positive feedback loop via the design of a synthetic auto-repression

Genetic positive feedback loops are essential for cell differentiation processes. They are accurately modeled with N-dimensional non-linear monotone dynamical systems that display bi-stability: the two stable fixed points represent two distinct cell differentiated states, whereas the unstable fixed point is interpreted as a cell undifferentiated state. This paper shows that the synthetic design of a simple self-inhibition of one gene in the loop is able to globally stabilize the unstable fixed point of the network. This modification may lead to a promising cell dedifferentiation process during which cells regress from a specialized state to an earlier developmental state. Compared to a similar experiment designed for the Toggle Switch, this new synthetic circuit prevents the use of any input and measurement devices, reducing greatly the complexity of the biological set-up. In order to take into account inherent biological uncertainties, the cell undifferentiated state is later considered as a region of the state space around the unstable fixed point and is shown to be globally attractive with the same simple synthetic modification of the loop. Some conditions are given such that all the possible fixed points of the circuit are confined in the undifferentiated region and the global results are proved with the theory of monotone dynamical systems.

How likely are oscillations in a genetic feedback loop with delay?

Some genetic control networks display temporal oscillations as a result of delays in their homeostatic control. A relevant question about these systems is whether the oscillating regime is a rare feature, or it corresponds to a sizeable volume of the space of parameters. The answer is not trivial mainly due to the large number of parameters controlling the rate equations which describe the network. We have developed an efficient sampling scheme of the parameter space, based on a Monte Carlo algorithm, and applied it to a two-node system with delay, characterised by a 8-dimension parameter space. The result is that the volume fraction of the parameter space associated with oscillations is small but not negligible, and it is weakly dependent on the duration of the delay. The most critical parameter to control oscillations is the coupling production rates, which must have opposite sign, giving rise to a negative feedback loop. The oscillating regions are connected except along the equilibrium constants between the two species, not allowing neutral evolution along this parameter.

Elucidating the Functional Roles of Spatial Organization in Cross-Membrane Signal Transduction by a Hybrid Simulation Method.

The ligand-binding of membrane receptors on cell surfaces initiates the dynamic process of cross-membrane signal transduction. It is an indispensable part of the signaling network for cells to communicate with external environments. Recent experiments revealed that molecular components in signal transduction are not randomly mixed, but spatially organized into distinctive patterns. These patterns, such as receptor clustering and ligand oligomerization, lead to very different gene expression profiles. However, little is understood about the molecular mechanisms and functional impacts of this spatial-temporal regulation in cross-membrane signal transduction. In order to tackle this problem, we developed a hybrid computational method that decomposes a model of signaling network into two simulation modules. The physical process of binding between receptors and ligands on cell surfaces are simulated by a diffusion-reaction algorithm, while the downstream biochemical reactions are modeled by stochastic simulation of Gillespie algorithm. These two processes are coupled together by a synchronization framework. Using this method, we tested the dynamics of a simple signaling network in which the ligand binding of cell surface receptors triggers the phosphorylation of protein kinases, and in turn regulates the expression of target genes. We found that spatial aggregation of membrane receptors at cellular interfaces is able to either amplify or inhibit downstream signaling outputs, depending on the details of clustering mechanism. Moreover, by providing higher binding avidity, the co-localization of ligands into multi-valence complex modulates signaling in very different ways that are closely related to the binding affinity between ligand and receptor. We also found that the temporal oscillation of the signaling pathway that is derived from genetic feedback loops can be modified by the spatial clustering of membrane receptors. In summary, our method demonstrates the functional importance of spatial organization in cross-membrane signal transduction. The method can be applied to any specific signaling pathway in cells.

Elucidating the Functional Roles of Spatial Organization in Cross-Membrane Signal Transduction by a Hybrid Simulation Method

The ligand-binding of membrane receptors on cell surfaces initiates the dynamic process of cross-membrane signal transduction. Recent experiments revealed that molecular components in signal transduction are not randomly mixed, but spatially organized into distinctive patterns that lead to very different gene-expression profiles. However, little is understood about the molecular mechanisms and functional impacts of this spatial-temporal regulation. Here we developed a hybrid computational method that decomposes signaling networks into two simulation modules. The physical process of binding between receptors and ligands are simulated by a diffusion-reaction algorithm, while the downstream biochemical reactions are modeled by stochastic simulation of Gillespie algorithm. Using this method, we tested the dynamics of a simple signaling system. We found that spatial aggregation of membrane receptors is able to either amplify or inhibit downstream signaling outputs, depending on the clustering mechanism. Moreover, by providing higher binding avidity, the co-localization of ligands into multi-valence complex modulates signaling in very different ways that are closely related to the binding affinity between ligand and receptor. We also found that the temporal oscillation of the signaling pathway from genetic feedback loops can be modified by receptors clustering. Our method demonstrates the functional importance of spatial organization in cross-membrane signal transduction.

Systems Perspective on Mechanobiology: Producing the Right Proteins in the Right Place at the Right Time

Systems Perspective on Mechanobiology: Producing the Right Proteins in the Right Place at the Right Time

Stochasticity in plant cellular growth and patterning

Plants, along with other multicellular organisms, have evolved specialized regulatory mechanisms to achieve proper tissue growth and morphogenesis. During development, growing tissues generate specialized cell types and complex patterns necessary for establishing the function of the organ. Tissue growth is a tightly regulated process that yields highly reproducible outcomes. Nevertheless, the underlying cellular and molecular behaviors are often stochastic. Thus, how does stochasticity, together with strict genetic regulation, give rise to reproducible tissue development? This review draws examples from plants as well as other systems to explore stochasticity in plant cell division, growth, and patterning. We conclude that stochasticity is often needed to create small differences between identical cells, which are amplified and stabilized by genetic and mechanical feedback loops to begin cell differentiation. These first few differentiating cells initiate traditional patterning mechanisms to ensure regular development.

Computational Modeling of the Interplay between Cadherin-Mediated Cell Adhesion and Wnt Signaling Pathway

Wnt signaling and cadherin-mediated adhesion have been implicated in both processes of embryonic development and the progression of carcinomas. Recent experimental studies revealed that Wnt signaling and cadherin-mediated cell adhesion have close crosstalk with each other. A comprehensive model that investigates the dynamic balance of β-catenins in Wnt signaling and cell adhesion will improve our understanding to embryonic development and carcinomas. We constructed a network model to evaluate the dynamic interplay between adhesion and Wnt signaling. The network is decomposed into three interdependent modules: the cell adhesion, the degradation circle and the transcriptional regulation. In the cell adhesion module, we consider the effect of cadherin’s lateral clustering. We found adhesion negatively contributes to Wnt signaling through competition for cytoplasmic β-catenins. In the network of degradation circle, we incorporated features from various existing models. Our simulations reproduced the most recent experimental phenomena with semi-quantitative accuracy. Finally, in the transcriptional regulation module, we developed a function selection strategy to analyze the outcomes of genetic feedback loops in modulating the gene expression of Wnt targets. The specific cellular phenomena such as cadherin switch and Axin oscillation were archived and their biological insights were discussed. Our model provides the theoretical basis of how spatial organization regulates the dynamics of cellular signaling pathways. We suggest that cell adhesion affects Wnt signaling in both negative and positive ways. Cadherins can inhibit Wnt signaling not only in a way as a stoichiometric binding partner of β-catenins that sequesters them from signaling, but also in a way through their clustering to impacts the rate at which β-catenins are involved in the destruction loop. Additionally, cadherin clustering increases the phosphorylation rate of β-catenins and promotes its signaling in nucleus.

A novel protein, CHRONO, functions as a core component of the mammalian circadian clock.

Author SummaryThe circadian clock has a fundamental role in regulating biological temporal rhythms in organisms, and it is tightly controlled by a molecular circuit consisting of positive and negative regulatory feedback loops. Although many of the clock genes comprising this circuit have been identified, there are still some critical components missing. Here, we characterize a circadian gene renamed Chrono (Gm129) and show that it functions as a transcriptional repressor of the negative feedback loop in the mammalian clock. Chrono binds to the regulatory region of clock genes and its occupancy oscillates in a circadian manner. Chrono knockout and Avp-neuron-specific knockout mice display longer circadian periods and altered expression of core clock genes. We show that Chrono-mediated repression involves the suppression of BMAL1–CLOCK activity via an epigenetic mechanism and that it regulates metabolic pathways triggered by behavioral stress. Our study suggests that Chrono functions as a clock repressor and reveals the molecular mechanisms underlying its function.

Effects of bursty protein production on the noisy oscillatory properties of downstream pathways

Experiments show that proteins are translated in sharp bursts; similar bursty phenomena have been observed for protein import into compartments. Here we investigate the effect of burstiness in protein expression and import on the stochastic properties of downstream pathways. We consider two identical pathways with equal mean input rates, except in one pathway proteins are input one at a time and in the other proteins are input in bursts. Deterministically the dynamics of these two pathways are indistinguishable. However the stochastic behavior falls in three categories: (i) both pathways display or do not display noise-induced oscillations; (ii) the non-bursty input pathway displays noise-induced oscillations whereas the bursty one does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify systems involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production.

Preconditioned Bayesian Regression for Stochastic Chemical Kinetics

We develop a preconditioned Bayesian regression method that enables sparse polynomial chaos representations of noisy outputs for stochastic chemical systems with uncertain reaction rates. The approach is based on the definition of an appropriate multiscale transformation of the state variables coupled with a Bayesian regression formalism. This enables efficient and robust recovery of both the transient dynamics and the corresponding noise levels. Implementation of the present approach is illustrated through applications to a stochastic Michaelis---Menten dynamics and a higher dimensional example involving a genetic positive feedback loop. In all cases, a stochastic simulation algorithm (SSA) is used to compute the system dynamics. Numerical experiments show that Bayesian preconditioning algorithms can simultaneously accommodate large noise levels and large variability with uncertain parameters, and that robust estimates can be obtained with a small number of SSA realizations.

Steady-state fluctuations of a genetic feedback loop: An exact solution

Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.