Modeling of negative autoregulated genetic networks in single cells

Damped oscillations in a multiple delayed feedback NF-κB signaling module

The NF-[Formula: see text]B signaling system regulates a number of cellular processes. Recent studies with simplified models found a damped function of the dual delayed feedback NF-κB signaling module. We use a computational model to investigate how multiple delayed feedback aids achieving damping oscillation in the system and how internal noise can influence the damping function. A curve-fitting method (CFM) is introduced to quantify the damped oscillation. Our results show that (1) the structure of multiple delayed feedback, containing double or triple significantly delayed feedback, determines achieving damped oscillation. (2) Internal noise could aid the system to achieve damped oscillation under almost all circumstances.

Dual delayed feedback provides sensitivity and robustness to the NF-κB signaling module

Many cellular stress-responsive signaling systems exhibit highly dynamic behavior with oscillatory features mediated by delayed negative feedback loops. What remains unclear is whether oscillatory behavior is the basis for a signaling code based on frequency modulation (FM) or whether the negative feedback control modules have evolved to fulfill other functional requirements. Here, we use experimentally calibrated computational models to interrogate the negative feedback loops that regulate the dynamic activity of the transcription factor NF-κB. Linear stability analysis of the model shows that oscillatory frequency is a hard-wired feature of the primary negative feedback loop and not a function of the stimulus, thus arguing against an FM signaling code. Instead, our modeling studies suggest that the two feedback loops may be tuned to provide for rapid activation and inactivation capabilities for transient input signals of a wide range of durations; by minimizing late phase oscillations response durations may be fine-tuned in a graded rather than quantized manner. Further, in the presence of molecular noise the dual delayed negative feedback system minimizes stochastic excursions of the output to produce a robust NF-κB response.

Gene expression in self-repressing system with multiple gene copies

We analyze a simple model of a self-repressing system with multiple gene copies. Protein molecules may bound to DNA promoters and block their own transcription. We derive analytical expressions for the variance of the number of protein molecules in the stationary state in the self-consistent mean-field approximation. We show that the Fano factor (the variance divided by the mean value) is bigger for the one-gene case than for two gene copies and the difference decreases to zero as frequencies of binding and unbinding increase to infinity.

Stochastic analysis of the SOS response in Escherichia coli

BACKGROUND

DNA damage in Escherichia coli evokes a response mechanism called the SOS response. The genetic circuit of this mechanism includes the genes recA and lexA, which regulate each other via a mixed feedback loop involving transcriptional regulation and protein-protein interaction. Under normal conditions, recA is transcriptionally repressed by LexA, which also functions as an auto-repressor. In presence of DNA damage, RecA proteins recognize stalled replication forks and participate in the DNA repair process. Under these conditions, RecA marks LexA for fast degradation. Generally, such mixed feedback loops are known to exhibit either bi-stability or a single steady state. However, when the dynamics of the SOS system following DNA damage was recently studied in single cells, ordered peaks were observed in the promoter activity of both genes (Friedman et al., 2005, PLoS Biol. 3(7):e238). This surprising phenomenon was masked in previous studies of cell populations. Previous attempts to explain these results harnessed additional genes to the system and deployed complex deterministic mathematical models that were only partially successful in explaining the results.

METHODOLOGY/PRINCIPAL FINDINGS

Here we apply stochastic methods, which are better suited for dynamic simulations of single cells. We show that a simple model, involving only the basic components of the circuit, is sufficient to explain the peaks in the promoter activities of recA and lexA. Notably, deterministic simulations of the same model do not produce peaks in the promoter activities.

CONCLUSION/SIGNIFICANCE

We conclude that the double negative mixed feedback loop with auto-repression accounts for the experimentally observed peaks in the promoter activities. In addition to explaining the experimental results, this result shows that including additional regulations in a mixed feedback loop may dramatically change the dynamic functionality of this regulatory module. Furthermore, our results suggests that stochastic fluctuations strongly affect the qualitative behavior of important regulatory modules even under biologically relevant conditions, thus emphasizing the importance of stochastic analysis of regulatory circuits.

Functions of bifans in context of multiple regulatory motifs in signaling networks

Representation of intracellular signaling networks as directed graphs allows for the identification of regulatory motifs. Regulatory motifs are groups of nodes with the same connectivity structure, capable of processing information. The bifan motif, made of two source nodes directly crossregulating two target nodes, is an overrepresented motif in a mammalian cell signaling network and in transcriptional networks. One example of a bifan is the two MAP-kinases, p38, and JNK that phosphorylate and activate the two transcription factors ATF2 and Elk-1. We have used a system of coupled ordinary differential equations to analyze the regulatory capability of this bifan motif by itself, and when it interacts with other motifs such as positive and negative feedback loops. Our results indicate that bifans provide temporal regulation of signal propagation and act as signal sorters, filters, and synchronizers. Bifans that have OR gate configurations show rapid responses whereas AND gate bifans can introduce delays and allow prolongation of signal outputs. Bifans that have AND gates can filter noisy signal inputs. The p38/JNK-ATF2/Elk-1bifan synchronizes the output of activated transcription factors. Synchronization is a robust property of bifans and is exhibited even when the bifan is adjacent to a positive feedback loop. The presence of the bifan promotes the transcription and translation of the dual specificity protein phosphatase MKP-1 that inhibits p38 and JNK thus enabling a negative feedback loop. These results indicate that bifan motifs in cell signaling networks can contribute to signal processing capability both intrinsically and by enabling the functions of other regulatory motifs.

Nonlinear regulation enhances the phenotypic expression of trans-acting genetic polymorphisms

BACKGROUND

Genetic variation explains a considerable part of observed phenotypic variation in gene expression networks. This variation has been shown to be located both locally (cis) and distally (trans) to the genes being measured. Here we explore to which degree the phenotypic manifestation of local and distant polymorphisms is a dynamic feature of regulatory design.

RESULTS

By combining mathematical models of gene expression networks with genetic maps and linkage analysis we find that very different network structures and regulatory motifs give similar cis/trans linkage patterns. However, when the shape of the cis-regulatory input functions is more nonlinear or threshold-like, we observe for all networks a dramatic increase in the phenotypic expression of distant compared to local polymorphisms under otherwise equal conditions.

CONCLUSION

Our findings indicate that genetic variation affecting the form of cis-regulatory input functions may reshape the genotype-phenotype map by changing the relative importance of cis and trans variation. Our approach combining nonlinear dynamic models with statistical genetics opens up for a systematic investigation of how functional genetic variation is translated into phenotypic variation under various systemic conditions.

Stochastic simulations of genetic switch systems

Genetic switch systems with mutual repression of two transcription factors are studied using deterministic methods (rate equations) and stochastic methods (the master equation and Monte Carlo simulations). These systems exhibit bistability, namely two stable states such that spontaneous transitions between them are rare. Induced transitions may take place as a result of an external stimulus. We study several variants of the genetic switch and examine the effects of cooperative binding, exclusive binding, protein-protein interactions, and degradation of bound repressors. We identify the range of parameters in which bistability takes place, enabling the system to function as a switch. Numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems. We show that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding. The average time between spontaneous transitions is evaluated as a function of the biological parameters.

Genetic toggle switch without cooperative binding

Genetic switch systems with mutual repression of two transcription factors are studied using deterministic and stochastic methods. Numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems. Here we show that, for a range of biologically relevant conditions, a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding.