Dynamics and Intrinsic Statistical Fluctuations of a Gene Switch

Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation

Cellular differentiation, reprogramming and transdifferentiation are determined by underlying gene regulatory networks. Non-adiabatic regulation via slow binding/unbinding to the gene can be important in these cell fate decision-making processes. Based on a stem cell core gene network, we uncovered the stem cell developmental landscape. As the binding/unbinding speed decreases, the landscape topography changes from bistable attractors of stem and differentiated states to more attractors of stem and other different cell states as well as substates. Non-adiabaticity leads to more differentiated cell types and provides a natural explanation for the heterogeneity observed in the experiments. We quantified Waddington landscapes with two possible cell fate decision mechanisms by changing the regulation strength or regulation timescale (non-adiabaticity). Transition rates correlate with landscape topography through barrier heights between different states and quantitatively determine global stability. We found the optimal speeds of these cell fate decision-making processes. We quantified biological paths and predict that differentiation and reprogramming go through an intermediate state (IM1), whereas transdifferentiation goes through another intermediate state (IM2). Some predictions are confirmed by recent experimental studies.

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra- and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steady-state probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.

Stochastic expression dynamics of a transcription factor revealed by single-molecule noise analysis

Gene expression is inherently stochastic; precise gene regulation by transcription factors is important for cell-fate determination. Many transcription factors regulate their own expression, suggesting that autoregulation counters intrinsic stochasticity in gene expression. Using a new strategy, cotranslational activation by cleavage (CoTrAC), we probed the stochastic expression dynamics of cI, which encodes the bacteriophage λ repressor CI, a fate-determining transcription factor. CI concentration fluctuations influence both lysogenic stability and induction of bacteriophage λ. We found that the intrinsic stochasticity in cI expression was largely determined by CI expression level irrespective of autoregulation. Furthermore, extrinsic, cell-to-cell variation was primarily responsible for CI concentration fluctuations, and negative autoregulation minimized CI concentration heterogeneity by counteracting extrinsic noise and introducing memory. This quantitative study of transcription factor expression dynamics sheds light on the mechanisms cells use to control noise in gene regulatory networks.

Adiabatic and non-adiabatic non-equilibrium stochastic dynamics of single regulating genes

We explore the stochastic dynamics of self-regulative genes from fluctuations of molecular numbers and of on and off switching of gene states due to regulatory protein binding/unbinding to the genes. We found when the binding/unbinding is relatively fast (slow) compared with the synthesis/degradation of proteins in adiabatic (nonadiabatic) case the self-regulators can exhibit one or two peak (two peak) distributions in protein concentrations. This phenomena can also be quantified through Fano factors. This shows that even with the same architecture (topology of wiring) networks can have quite different functions (phenotypes), consistent with recent single molecule single gene experiments. We further found the inhibition and activation curves to be consistent with previous results (monomer binding) in adiabatic regime, but, in nonadiabatic regimes, show significantly different behaviors with previous predictions (monomer binding). Such difference is due to the slow (nonadiabatic) dimer binding/unbinding effect, and it has never been reported before. We derived the nonequilibrium phase diagrams of monostability and bistability in adiabatic and nonadiabatic regimes. We studied the dynamical trajectories of the self-regulating genes on the underlying landscapes from nonadiabatic to adiabatic limit, and we provide a global picture of understanding and show an analogy to the electron transfer problem. We studied the stability and robustness of the systems through mean first passage time (MFPT) from one peak (basin of attraction) to another and found both monotonic and nonmonotonic turnover behavior from adiabatic to nonadiabatic regimes. For the first time, we explore global dissipation by entropy production and the relation with binding/unbinding processes. Our theoretical predictions for steady state peaks, fano factos, inhibition/activation curves, and MFPT can be probed and tested from experiments.

Amplification and detection of single-molecule conformational fluctuation through a protein interaction network with bimodal distributions

A protein undergoes conformational dynamics with multiple time scales, which results in fluctuating enzyme activities. Recent studies in single-molecule enzymology have observe this "age-old" dynamic disorder phenomenon directly. However, the single-molecule technique has its limitation. To be able to observe this molecular effect with real biochemical functions in situ, we propose to couple the fluctuations in enzymatic activity to noise propagations in small protein interaction networks such as a zeroth-order ultrasensitive phosphorylation-dephosphorylation cycle. We show that enzyme fluctuations can indeed be amplified by orders of magnitude into fluctuations in the level of substrate phosphorylation, a quantity of wide interest in cellular biology. Enzyme conformational fluctuations sufficiently slower than the catalytic reaction turnover rate result in a bimodal concentration distribution of the phosphorylated substrate. In return, this network-amplified single-enzyme fluctuation can be used as a novel biochemical "reporter" for measuring single-enzyme conformational fluctuation rates.

Cell fate decision mediated by p53 pulses

The tumor suppressor p53 plays a crucial role in cellular response to various stresses. Recent experiments have shown that p53 level exhibits a series of pulses after DNA damage caused by ionizing radiation (IR). However, how the p53 pulses govern cell survival and death remains unclear. Here, we develop an integrated model with four modules for the p53 network and explore the mechanism for cell fate decision based on the dynamics of the network. By numerical simulations, the following processes are characterized. First, DNA repair proteins bind to IR-induced double-strand breaks, forming complexes, which are then detected by ataxia telangiectasia mutated (ATM). Activated ATM initiates the p53 oscillator to produce pulses. Consequently, the target genes of p53 are selectively induced to control cell fate. We propose that p53 promotes the repair of minor DNA damage but suppresses the repair of severe damage. We demonstrate that cell fate is determined by the number of p53 pulses relying on the extent of DNA damage. At low damage levels, few p53 pulses evoke cell cycle arrest by inducing p21 and promote cell survival, whereas at high damage levels, sustained p53 pulses trigger apoptosis by inducing p53AIP1. We find that p53 can effectively maintain genomic integrity by regulating the efficiency and fidelity of DNA repair. We also show that stochasticity in the generation and repair of DNA damage leads to variability in cell fate. These findings are consistent with experimental observations and advance our understanding of the dynamics and functions of the p53 network.

A molecular noise generator

Recent studies have demonstrated that intracellular variations in the rate of gene expression are of fundamental importance to cellular function and development. While such 'noise' is often considered detrimental in the context of perturbing genetic systems, it can be beneficial in processes such as species diversification and facilitation of evolution. A major difficulty in exploring such effects is that the magnitude and spectral properties of the induced variations arise from some intrinsic cellular process that is difficult to manipulate. Here, we present two designs of a molecular noise generator that allow for the flexible modulation of the noise profile of a target gene. The first design uses a dual-signal mechanism that enables independent tuning of the mean and variability of an output protein. This is achieved through the combinatorial control of two signals that regulate transcription and translation separately. We then extend the design to allow for DNA copy-number regulation, which leads to a wider tuning spectrum for the output molecule. To gain a deeper understanding of the circuit's functionality in a realistic environment, we introduce variability in the input signals in order to ascertain the degree of noise induced by the control process itself. We conclude by illustrating potential applications of the noise generator, demonstrating how it could be used to ascertain the robust or fragile properties of a genetic circuit.

Intrinsic noise, dissipation cost, and robustness of cellular networks: the underlying energy landscape of MAPK signal transduction

We develop a probabilistic method for analyzing global features of a cellular network under intrinsic statistical fluctuations, which is important when there are finite numbers of molecules. By making a self-consistent mean field approximation of splitting the variables in order to reduce the large number of degrees of freedom, which is reasonable for a not very strongly interacting network, we discovered that the underlying energy landscape of the mitogen-activated protein kinases (MAPKs) signal transduction network (with experimentally measured or inferred parameters such as chemical reaction rate coefficients in the network) is funneled toward a global minimum characterized by the nonequilibrium steady-state fixed point of the system at the end of the signal transduction process. For this system, we also show that the energy landscape is robust against intrinsic fluctuations and random perturbation to the inherent chemical reaction rates. The ratio of the slope versus the roughness of the energy landscape becomes a quantitative measure of robustness and stability of the network. Furthermore, we quantify the dissipation cost of this nonequilibrium system through entropy production, caused by the nonequilibrium flux in the system. We found that a lower dissipation cost corresponds to a more robust network. This least dissipation property might provide a design principle for robust and functional networks. Finally, we find the possibility of bistable and oscillatory-like solutions, which are important for cell fate decisions, upon perturbations. The method described here can be used in a variety of biological networks.

Transcriptional control of noise in gene expression

Cis-regulatory control of transcription is the dominant form of regulation of gene expression. Recent experimental results suggest that, in addition to the mean expression level, cell-to-cell variability might also be transcriptionally regulated. Here, we develop a stochastic model of transcriptional regulation that allows us to calculate closed-form analytical expressions for the mean and variance of the protein and mRNA distributions for an arbitrarily complex cis-regulatory motif. Our model allows us to investigate how noise may be transcriptionally regulated independently from the mean expression. We show that our approach is in excellent agreement with stochastic simulations and experiment, and leads to an experimentally testable formula for the noise in gene expression as a function of inducer-molecule concentrations.