Haidong Feng

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.

Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: Curvature, gauge field, and generalized fluctuation-dissipation theorem

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.

A new mechanism of stem cell differentiation through slow binding/unbinding of regulators to genes.

Understanding differentiation, a biological process from a multipotent stem or progenitor state to a mature cell is critically important. We developed a theoretical framework to quantify the underlying potential landscape and pathways for cell development and differentiation. We proposed a new mechanism of differentiation and found the differentiated states can emerge from the slow binding/unbinding of regulatory proteins to gene promoters. With slow promoter binding/unbinding, we found multiple meta-stable differentiated states, which can explain the origin of multiple states observed in recent experiments. The kinetic time for the differentiation and reprogramming strongly depends on the time scale of the promoter binding/unbinding processes. We discovered an optimal speed for differentiation for certain promoter binding/unbinding rates. Future experiments might be able to tell if cells differentiate at that optimal speed. We also quantified irreversible kinetic pathways for the differentiation and reprogramming, which captures the non-equilibrium dynamics in multipotent stem or progenitor cells.

Landscape and Global Stability of Nonadiabatic and Adiabatic Oscillations in a Gene Network

We quantify the potential landscape to determine the global stability and coherence of biological oscillations. We explore a gene network motif in our experimental synthetic biology studies of two genes that mutually repress and activate each other with self-activation and self-repression. We find that in addition to intrinsic molecular number fluctuations, there is another type of fluctuation crucial for biological function: the fluctuation due to the slow binding/unbinding of protein regulators to gene promoters. We find that coherent limit cycle oscillations emerge in two regimes: an adiabatic regime with fast binding/unbinding and a nonadiabatic regime with slow binding/unbinding relative to protein synthesis/degradation. This leads to two mechanisms of producing the stable oscillations: the effective interactions from averaging the gene states in the adiabatic regime; and the time delays due to slow binding/unbinding to promoters in the nonadiabatic regime, which can be tested by forthcoming experiments. In both regimes, the landscape has a topological shape of the Mexican hat in protein concentrations that quantitatively determines the global stability of limit cycle dynamics. The oscillation coherence is shown to be correlated with the shape of the Mexican hat characterized by the height from the oscillation ring to the central top. The oscillation period can be tuned in a wide range by changing the binding/unbinding rate without changing the amplitude much, which is important for the functionality of a biological clock. A negative feedback loop with time delays due to slow binding/unbinding can also generate oscillations. Although positive feedback is not necessary for generating oscillations, it can make the oscillations more robust.

The energy pump and the origin of the non-equilibrium flux of the dynamical systems and the networks

The global stability of dynamical systems and networks is still challenging to study. We developed a landscape and flux framework to explore the global stability. The potential landscape is directly linked to the steady state probability distribution of the non-equilibrium dynamical systems which can be used to study the global stability. The steady state probability flux together with the landscape gradient determines the dynamics of the system. The non-zero probability flux implies the breaking down of the detailed balance which is a quantitative signature of the systems being in non-equilibrium states. We investigated the dynamics of several systems from monostability to limit cycle and explored the microscopic origin of the probability flux. We discovered that the origin of the probability flux is due to the non-equilibrium conditions on the concentrations resulting energy input acting like non-equilibrium pump or battery to the system. Another interesting behavior we uncovered is that the probabilistic flux is closely related to the steady state deterministic chemical flux. For the monostable model of the kinetic cycle, the analytical expression of the probabilistic flux is directly related to the deterministic flux, and the later is directly generated by the chemical potential difference from the adenosine triphosphate (ATP) hydrolysis. For the limit cycle of the reversible Schnakenberg model, we also show that the probabilistic flux is correlated to the chemical driving force, as well as the deterministic effective flux. Furthermore, we study the phase coherence of the stochastic oscillation against the energy pump, and argue that larger non-equilibrium pump results faster flux and higher coherence. This leads to higher robustness of the biological oscillations. We also uncovered how fluctuations influence the coherence of the oscillations in two steps: (1) The mild fluctuations influence the coherence of the system mainly through the probability flux while maintaining the regular landscape topography. (2) The larger fluctuations lead to flat landscape and the complete loss of the stability of the whole system.

Oscillation, cooperativity, and intermediates in the self-repressing gene

Abstract Biological oscillators are vital to living organisms, which use them as clocks for time-sensitive processes. However, much is unknown about mechanisms which can give rise to coherent oscillatory behavior, with few exceptions (e.g., explicitly delayed self-repressors and simple models of specific organisms’ circadian clocks). We present what may be the simplest possible reliable gene network oscillator, a self-repressing gene. We show that binding cooperativity, which has not been considered in detail in this context, can combine with small numbers of intermediate steps to create coherent oscillation. We also note that noise blurs the line between oscillatory and non-oscillatory behavior.

Multiple coupled landscapes and non-adiabatic dynamics with applications to self-activating genes

Many physical, chemical and biochemical systems (e.g. electronic dynamics and gene regulatory networks) are governed by continuous stochastic processes (e.g. electron dynamics on a particular electronic energy surface and protein (gene product) synthesis) coupled with discrete processes (e.g. hopping among different electronic energy surfaces and on and off switching of genes). One can also think of the underlying dynamics as the continuous motion on a particular landscape and discrete hoppings among different landscapes. The main difference of such systems from the intra-landscape dynamics alone is the emergence of the timescale involved in transitions among different landscapes in addition to the timescale involved in a particular landscape. The adiabatic limit when inter-landscape hoppings are fast compared to continuous intra-landscape dynamics has been studied both analytically and numerically, but the analytical treatment of the non-adiabatic regime where the inter-landscape hoppings are slow or comparable to continuous intra-landscape dynamics remains challenging. In this study, we show that there exists mathematical mapping of the dynamics on 2(N) discretely coupled N continuous dimensional landscapes onto one single landscape in 2N dimensional extended continuous space. On this 2N dimensional landscape, eddy current emerges as a sign of non-equilibrium non-adiabatic dynamics and plays an important role in system evolution. Many interesting physical effects such as the enhancement of fluctuations, irreversibility, dissipation and optimal kinetics emerge due to non-adiabaticity manifested by the eddy current illustrated for an N = 1 self-activator. We further generalize our theory to the N-gene network with multiple binding sites and multiple synthesis rates for discretely coupled non-equilibrium stochastic physical and biological systems.