Effects of coupling strength and space on the dynamics of coupled toggle switches in stochastic gene networks with multiple-delayed reactions

Single-cell kinetics of a repressilator when implemented in a single-copy plasmid

Synthetic genetic clocks, such as the Elowitz-Leibler repressilator, will be key regulatory components of future synthetic circuits. We constructed a single-copy repressilator (SCR) by implementing the original repressilator circuit on a single-copy F-plasmid. After verifying its functionality, we studied its behaviour as a function of temperature and compared it with that of the original low-copy-number repressilator (LCR). Namely, we compared the period of oscillations, functionality (the fraction of cells exhibiting oscillations) and robustness to internal fluctuations (the fraction of expected oscillations that would occur). We found that, under optimal temperature conditions, the dynamics of the two systems differs significantly, although qualitatively they respond similarly to temperature changes. Exception to this is in the functionality, in which the SCR is higher at lower temperatures but lower at higher temperatures. Next, by adding IPTG to the medium at low and high concentrations during microscopy sessions, we showed that the functionality of the SCR is more robust to external perturbations, which indicates that the oscillatory behaviour of the LCR can be disrupted by affecting only a few of the copies in a cell. We conclude that the SCR, the first functional, synthetic, single-copy, ring-type genetic clock, is more robust to lower temperatures and to external perturbations than the original LCR. The SCR will be of use in future synthetic circuits, since it complements the array of tasks that the LCR can perform.

Effects of multimerization on the temporal variability of protein complex abundance

We explore whether the process of multimerization can be used as a means to regulate noise in the abundance of functional protein complexes. Additionally, we analyze how this process affects the mean level of these functional units, response time of a gene, and temporal correlation between the numbers of expressed proteins and of the functional multimers. We show that, although multimerization increases noise by reducing the mean number of functional complexes it can reduce noise in comparison with a monomer, when abundance of the functional proteins are comparable. Alternatively, reduction in noise occurs if both monomeric and multimeric forms of the protein are functional. Moreover, we find that multimerization either increases the response time to external signals or decreases the correlation between number of functional complexes and protein production kinetics. Finally, we show that the results are in agreement with recent genome-wide assessments of cell-to-cell variability in protein numbers and of multimerization in essential and non-essential genes in Escherichia coli, and that the effects of multimerization are tangible at the level of genetic circuits.

Kinetics of gene expression in bacteria — From models to measurements, and back again

Dennis Salahub has contributed to several scientific areas. Likely, one of the lesser known contributions is the one on the study of the kinetics of gene expression in prokaryotes, which resulted in a delayed stochastic model of transcription and translation dynamics in bacteria. The model has become the basis for modeling stochastic genetic circuits and has provided a framework for interpreting recent measurements of transcripts production dynamics in live cells, one event at a time. Here, we review the contributions by Salahub and colleagues to the modeling strategies of gene expression as a multi-delayed stochastic process. Next, we describe recent findings, which build upon this work and provide experimental validation of the models. Finally, we discuss potential future developments in this field of research.

Effects of gene length on the dynamics of gene expression

In Escherichia coli, the nucleotide length of a gene is bound to affect its expression dynamics. From simulations of a stochastic model of gene expression at the nucleotide and codon levels we show that, within realistic parameter values, the nucleotide length affects RNA and protein mean levels, as well as the expected transient time for RNA and protein numbers to change, following a signal. Fluctuations in RNA and protein numbers are found to be minimized for a small range of lengths, which matches the means of the distributions of lengths found in E. coli of both essential and non-essential genes. The variance of the length distribution for essential genes is found to be smaller than for non-essential genes, implying that these distributions are far from random. Finally, gene lengths are shown to affect the kinetics of a genetic switch, namely, the correlation between temporal proteins numbers, the stability of the two noisy attractors of the switch, and how biased is the choice of noisy attractor. The stability increases with gene length due to increased 'memory' about the previous states of the switch. We argue that, by affecting the dynamics of gene expression and of genetic circuits, gene lengths are subject to selection.

Effects of codon sequence on the dynamics of genetic networks

In prokaryotes, the rate at which codons are translated varies from one codon to the next. Using a stochastic model of transcription and translation at the nucleotide and codon levels, we investigate the effects of the codon sequence on the dynamics of protein numbers. For sequences generated according to the codon frequencies in Escherichia coli, we find that mean protein numbers at near equilibrium differ with the codon sequence, due to the mean codon translation efficiencies, in particular of the codons at the ribosome binding site region. We find close agreement between these predictions and measurements of protein expression levels as a function of the codon sequence. Next, we investigate the effects of short codon sequences at the start/end of the RNA sequence with linearly increasing/decreasing translation efficiencies, known as slow ramps. The ramps affect the mean, but not the fluctuations, in proteins numbers by affecting the rate of translation initiation. Finally, we show that slow ramps affect the dynamics of small genetic circuits, namely, switches and clocks. In switches, ramps affect the frequency of switching and bias the robustness of the noisy attractors. In repressilators, ramps alter the robustness of periodicity. We conclude that codon sequences affect the dynamics of gene expression and genetic circuits and, thus, are likely to be under selection regarding both mean codon frequency as well as spatial arrangement along the sequence.

Cell-to-cell diversity in protein levels of a gene driven by a tetracycline inducible promoter

BACKGROUND

Gene expression in Escherichia coli is regulated by several mechanisms. We measured in single cells the expression level of a single copy gene coding for green fluorescent protein (GFP), integrated into the genome and driven by a tetracycline inducible promoter, for varying induction strengths. Also, we measured the transcriptional activity of a tetracycline inducible promoter controlling the transcription of a RNA with 96 binding sites for MS2-GFP.

RESULTS

The distribution of GFP levels in single cells is found to change significantly as induction reaches high levels, causing the Fano factor of the cells' protein levels to increase with mean level, beyond what would be expected from a Poisson-like process of RNA transcription. In agreement, the Fano factor of the cells' number of RNA molecules target for MS2-GFP follows a similar trend. The results provide evidence that the dynamics of the promoter complex formation, namely, the variability in its duration from one transcription event to the next, explains the change in the distribution of expression levels in the cell population with induction strength.

CONCLUSIONS

The results suggest that the open complex formation of the tetracycline inducible promoter, in the regime of strong induction, affects significantly the dynamics of RNA production due to the variability of its duration from one event to the next.

Effects of the promoter open complex formation on gene expression dynamics

Little is known about the biological mechanisms that shape the distribution of intervals between the completion of RNA molecules (T(p)RNA) , and thus transcriptional noise. We characterize numerically and analytically how the promoter open complex delay (tau(P)) and the transcription initiation rate (k(t)) shape T(p)RNA. From this, we assess the noise and mean of transcript levels and show that these can be tuned both independently and simultaneously by tau(P) and k(t). Finally, we characterize how tau(P) affects bursting in RNA production and show that the tau(P) measured for a lac promoter best fits independent measurements of the burst distribution of the same promoter. Since tau(P) affects noise in gene expression, and given that it is sequence dependent, it is likely to be evolvable.

Analysis of genetic toggle switch systems encoded on plasmids

Genetic switch systems with mutual repression of two transcription factors, encoded on plasmids, are studied using stochastic methods. The plasmid copy number is found to strongly affect the behavior of these systems. More specifically, the average time between spontaneous switching events quickly increases with the number of plasmids. It was shown before that for a single copy encoded on the chromosome, the exclusive switch is more stable than the general switch. Here we show that when the switch is encoded on a sufficiently large number of plasmids, the situation is reversed and the general switch is more stable than the exclusive switch. These predictions can be tested experimentally using methods of synthetic biology.

Delayed stochastic model of transcription at the single nucleotide level

We present a delayed stochastic model of transcription at the single nucleotide level. The model accounts for the promoter open complex formation and includes alternative pathways to elongation, namely pausing, arrest, misincorporation and editing, pyrophosphorolysis, and premature termination. We confront the dynamics of this detailed model with a single-step multi-delayed stochastic model and with measurements of expression of a repressed gene at the single molecule level. At low expression rates both models match the experiments but, at higher rates the two models differ significantly, with consequences to cell-to-cell phenotypic variability. The alternative pathway reactions, due to, for example, causing polymerases to collide more often on the template, are the cause for the difference in dynamical behaviors. Next, we confront the model with measurements of the transcriptional dynamics at the single RNA level of an induced gene and show that RNA production, besides its bursting dynamics, also exhibits pulses (2 or more RNAs produced in intervals smaller than the smallest interval between initiations). The distribution of occurrences and amplitudes of pulses match the experimental measurements. This pulsing and the noise at the elongation stage are shown to play a role in the dynamics of a genetic switch.

Statistical physics of a model binary genetic switch with linear feedback

We study the statistical properties of a simple genetic regulatory network that provides heterogeneity within a population of cells. This network consists of a binary genetic switch in which stochastic flipping between the two switch states is mediated by a "flipping" enzyme. Feedback between the switch state and the flipping rate is provided by a linear feedback mechanism: the flipping enzyme is only produced in the on switch state and the switching rate depends linearly on the copy number of the enzyme. This work generalizes the model of Visco [Phys. Rev. Lett. 101, 118104 (2008)] to a broader class of linear feedback systems. We present a complete analytical solution for the steady-state statistics of the number of enzyme molecules in the on and off states, for the general case where the enzyme can mediate flipping in either direction. For this general case we also solve for the flip time distribution, making a connection to first passage and persistence problems in statistical physics. We show that the statistics are non-Poissonian, leading to a peak in the flip time distribution. The occurrence of such a peak is analyzed as a function of the parameter space. We present a relation between the flip time distributions measured for two relevant choices of initial condition. We also introduce a correlation measure and use this to show that this model can exhibit long-lived temporal correlations, thus providing a primitive form of cellular memory. Motivated by DNA replication as well as by evolutionary mechanisms involving gene duplication, we study the case of two switches in the same cell. This results in correlations between the two switches; these can be either positive or negative depending on the parameter regime.

Converting genetic network oscillations into somite spatial patterns

The segmentation of vertebrate embryos, a process known as somitogenesis, depends on a complex genetic network that generates highly dynamic gene expression in an oscillatory manner. A recent proposal for the mechanism underlying these oscillations involves negative-feedback regulation at transcriptional translational levels, also known as the "delay model" [J. Lewis Curr. Biol. 13, 1398 (2003)]. In addition, in the zebrafish a longitudinal positional information signal in the form of an Fgf8 gradient constitutes a determination front that could be used to transform these coupled intracellular temporal oscillations into the observed spatial periodicity of somites. Here we consider an extension of the delay model by taking into account the interaction of the oscillation clock with the determination front. Comparison is made with the known properties of somite formation in the zebrafish embryo. We also show that the model can mimic the anomalies formed when progression of the determination wave front is perturbed and make an experimental prediction that can be used to test the model.

Dynamics of a two-dimensional model of cell tissues with coupled stochastic gene networks

Gene expression and differentiation were shown to be stochastic processes. However, cells in a tissue can coordinate their behavior, including gene expression and differentiation pathways choice. A tissue of coupled cells is modeled as a two-dimensional regular square lattice of identical cells, each a three-dimensional compartment with a gene regulatory network (GRN) and a toggle switch (TS). The dynamics is driven by a delayed stochastic simulation algorithm, nearest neighbor cells are coupled by normally distributed time delayed reactions allowing interchange of proteins, and gene expression is a multiple time delayed reaction. It is defined the coupling strength (C), to characterize the lattice structure as a function of the rate constants of the reactions coupling nearest neighbor cells. Conditions are investigated for the emergence of synchronization and stable differentiation of cells within a tissue. From the time series of the cells GRNs, the tissue dynamical stability (S) is computed from the average toggling period of each GRN. The synchronization of cells' proteins expression levels is measured by their time series entropy (H). It is shown that the tissue goes through various dynamical regimes as C is increased, by measuring H and S . For null C, the cells GRNs toggle asynchronously. For weak C, cells synchronize by regions of space. Increasing C, the tissue becomes homogeneously synchronous. As C is further increased, S goes through a phase transition, from synchronized toggling to stable, where all cells produce one and the same protein. Finally, increasing C even further, a new stable state emerges where both genes of all cells are expressed and bistability is lost. This state, resembling an infinitely long transient, is an emergent behavior not observable in a single TS. The results provide an explanation of how cells with bistable GRNs, inherently stochastic, can synchronize or uniformly differentiate into stable states, by interacting with nearest neighbors.