Curl Flux Research Articles

Quantifying Landscape-Flux via Single-Cell Transcriptomics Uncovers the Underlying Mechanism of Cell Cycle.

Recent developments in single-cell sequencing technology enable the acquisition of entire transcriptome data. Understanding the underlying mechanism and identifying the driving force of transcriptional regulation governing cell function directly from these data remains challenging. This study reconstructs a continuous vector field of the cell cycle based on discrete single-cell RNA velocity to quantify the single-cell global nonequilibrium dynamic landscape-flux. It reveals that large fluctuations disrupt the global landscape and genetic perturbations alter landscape-flux, thus identifying key genes in maintaining cell cycle dynamics and predicting associated functional effects. Additionally, it quantifies the fundamental energy cost of the cell cycle initiation and unveils that sustaining the cell cycle requires curl flux and dissipation to maintain the oscillatory phase coherence. This study enables the inference of the cell cycle gene regulatory networks directly from the single-cell transcriptomic data, including the feedback mechanisms and interaction intensity. This provides a golden opportunity to experimentally verify the landscape-flux theory and also obtain its associated quantifications. It also offers a unique framework for combining the landscape-flux theory and single-cell high-through sequencing experiments for understanding the underlying mechanisms of the cell cycle and can be extended to other nonequilibrium biological processes, such as differentiation development and disease pathogenesis.

A robust mode of the winter oceanic mixed layer depth in the North Pacific

Based on a modified mixed layer depth (MLD) calculation method and ocean reanalysis datasets, the second leading empirical orthogonal function (EOF) mode (EOF-2) of the North Pacific winter MLD anomalies is investigated, which shows a robust spatial pattern in three reanalysis datasets with opposite phases in the Kuroshio and the Kuroshio Extension region and the central and east of the North Pacific region. The results presented in this study are from SODA 2.2.4. The explained variance of the EOF-2 mode is comparable to the variance of the leading EOF (EOF-1). We reveal that on the interannual time scale, the corresponding Principal Components of EOF-2 (PC-2) are significantly correlated with Niño 3.4 Index and the MLD EOF-2 mode is dynamically driven by the entrainment processes, which are linked to the anomalies of surface heat fluxes and wind velocity induced by ENSO through atmospheric bridge processes. On decadal time scales, EOF-2 is primarily modulated by the variability of wind stress curl and net heat flux, but the influence of net heat flux is only limited in the Kuroshio Extension region.

Quantifying the Landscape of Decision Making From Spiking Neural Networks.

The decision making function is governed by the complex coupled neural circuit in the brain. The underlying energy landscape provides a global picture for the dynamics of the neural decision making system and has been described extensively in the literature, but often as illustrations. In this work, we explicitly quantified the landscape for perceptual decision making based on biophysically-realistic cortical network with spiking neurons to mimic a two-alternative visual motion discrimination task. Under certain parameter regions, the underlying landscape displays bistable or tristable attractor states, which quantify the transition dynamics between different decision states. We identified two intermediate states: the spontaneous state which increases the plasticity and robustness of changes of minds and the “double-up” state which facilitates the state transitions. The irreversibility of the bistable and tristable switches due to the probabilistic curl flux demonstrates the inherent non-equilibrium characteristics of the neural decision system. The results of global stability of decision-making quantified by barrier height inferred from landscape topography and mean first passage time are in line with experimental observations. These results advance our understanding of the stochastic and dynamical transition mechanism of decision-making function, and the landscape and kinetic path approach can be applied to other cognitive function related problems (such as working memory) in brain networks.

Quantum fluctuation-dissipation theorem far from equilibrium

Fluctuations associated with relaxations in far-from-equilibrium regime is of fundamental interest for a large variety of systems within broad scales. Recent advances in techniques such as spectroscopy have generated the possibility for measuring the fluctuations of the mesoscopic systems in connection to the relaxation processes when driving the underlying quantum systems far from equilibrium. We present a general nonequilibrium Fluctuation-Dissipation Theorem (FDT) for quantum Markovian processes where the detailed-balance condition is violated. Apart from the fluctuations, the relaxation involves extra correlation that is governed by the quantum curl flux emerged in the far-from-equilibrium regime. Such a contribution vanishes for the thermal equilibrium, so that the conventional FDT is recovered. We finally apply the nonequilibrium FDT to the molecular junctions, elaborating the detailed-balance-breaking effects on the optical transmission spectrum. Our results have the advantage of and exceed the scope of the fluctuation-dissipation relation in the perturbative and near equilibrium regimes, and is of broad interest for the study of quantum thermodynamics.ation in the perturbative and near equilibrium regimes, and is of broad interest for the study of quantum thermodynamics.

On the Generation of Daily Gridded Ocean Surface Vector Wind Products From Scatsat-1

An optimal interpolation (OI)-based technique is utilized to generate global daily gridded wind products from Scatsat-1 data. The first guess (i.e., background) for the OI is used from the analyses of National Centre for Medium Range Weather Forecasting (NCMRWF) global operational model. The generated gridded products are subsequently validated using observations from global moored buoys and surface winds analyses from European Centre for Medium-Range Weather Forecasts (ECMWF), and that shows a root mean square error of ~1.3 m/s in wind speed and ~19° in wind direction. Along with the gridded winds, several wind-dependent products (e.g., wind stress, wind divergence, wind stress curl, sensible and latent heat flux, etc.) are also generated. The gridded products from Scatsat-1 are available freely from Meteorological and Oceanographic Satellite Data Archival Center ( <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">http://www.mosdac.gov.in</uri> ) since October 03, 2016.

Curl Flux as a Dynamical Origin of the Bifurcations/Phase Transitions of Nonequilibrium Systems: Cell Fate Decision Making.

The underlying interactions in physical and biological systems often lead to a variety of behaviors and emergent states or phases. Under certain conditions, these phases can be transformed from one to another. The phase transition behaviors can be described by the bifurcation or catastrophe where different stable/unstable states can branch out or meet together with the birth of the new and death of the old states. Despite significant efforts, how the bifurcation and catastrophe actually occur dynamically and the associated mechanisms for nonequilibrium systems are still not very clear. As an example, we study the underlying mechanism of cell differentiation through bifurcations. Cell differentiation is one of the key fate decision-making processes that a cell faces. It is crucial for the development of multicellular organisms. Under induction, gene regulation changes, or stochastic fluctuations, the cell fate decision-making processes can exhibit different types of bifurcations or phase transitions. In order to understand the underlying mechanism, it is crucial to find out where and how the bifurcation occurs. However, this is still largely unknown. In this study, we found that the average of the curl flux as a major component of the driving force for the dynamics in addition to the landscape gradient and the associated entropy production rate both reach maximum near the bifurcation. This indicates that the curl flux and entropy production rate may provide the dynamical and thermodynamic origins of the bifurcation/catastrophe or phase transitions for cell differentiation and this possibly applies to many other nonequilibrium active systems.

Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis–Menten enzyme kinetics

The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.

Exploring the underlying mechanisms of the coupling between cell differentiation and cell cycle.

Differentiation and replication are the two major fates of the cells. They are the fundamental processes for completing the cellular functions. Although the underlying biological processes have been considerably explored for each of these processes and significant progresses have been made, global quantification and physical understanding are still challenging, especially for the relationship among them. In this study, we developed a theoretical framework for both the cell cycle and cell differentiation by exploring the associated global dynamics and their underlying relationship. We found that the dynamics of the cell cycle and cell differentiation is governed by both the landscape gradient and rotational curl flux. While landscape attracts the system down to the stable state basins, the curl flux drives the stable oscillation flow. We uncovered the irregular sombrero-shaped landscapes of the cell cycle at different developmental stages. We studied how the cells develop from undifferentiated cells to differentiated cells and how the cell cycle proceeds at different developmental stages. We investigated how the cell differentiation can influence the cell cycle where more progressive differentiation can lead to the changes of the cell cycle oscillations. In contrast, we can also quantitatively illustrate how the cell cycle can influence the cell differentiation where cell cycle regulation can lead to the changes of the differentiation processes. Through the landscape and flux analysis, we uncovered the key regulatory elements controlling the progression of the cell differentiation and cell cycle. This can help to design an effective strategy for drug discovery against associated diseases.

Non-normality, reactivity, and intrinsic stochasticity in neural dynamics: a non-equilibrium potential approach

Intrinsic stochasticity can induce highly non-trivial effects on dynamical systems, such as stochastic resonance, noise induced bistability, and noise-induced oscillations, to name but a few. Here we revisit a mechanism—first investigated in the context of neuroscience—by which relatively small intrinsic (demographic) fluctuations can lead to the emergence of avalanching behavior in systems that are deterministically characterized by a single stable fixed point (up state). The anomalously large response of such systems to stochasticity stems from (or is strongly associated with) the existence of a ‘non-normal’ stability matrix at the deterministic fixed point, which may induce the system to be ‘reactive’. By employing a number of analytical and computational approaches, we further investigate this mechanism and explore the interplay between non-normality and intrinsic stochasticity. In particular, we conclude that the resulting dynamics of this type of systems cannot be simply derived from a scalar potential but, additionally, one needs to consider a curl flux which describes the essential non-equilibrium nature of this type of noisy non-normal systems. Moreover, we shed further light on the origin of the phenomenon, introduce the novel concept of ‘non-linear reactivity’, and rationalize the observed values of avalanche exponents.

Coherence enhanced quantum metrology in a nonequilibrium optical molecule

We explore the quantum metrology in an optical molecular system coupled to two environments with different temperatures, using a quantum master equation beyond secular approximation. We discover that the steady-state coherence originating from and sustained by the nonequilibrium condition can enhance quantum metrology. We also study the quantitative measures of the nonequilibrium condition in terms of the curl flux, heat current and entropy production at the steady state. They are found to grow with temperature difference. However, an apparent paradox arises considering the contrary behaviors of the steady-state coherence and the nonequilibrium measures in relation to the inter-cavity coupling strength. This paradox is resolved by decomposing the heat current into a population part and a coherence part. Only the latter, the coherence part of the heat current, is tightly connected to the steady-state coherence and behaves similarly with respect to the inter-cavity coupling strength. Interestingly, the coherence part of the heat current flows from the low-temperature reservoir to the high-temperature reservoir, opposite to the direction of the population heat current. Our work offers a viable way to enhance quantum metrology for open quantum systems through steady-state coherence sustained by the nonequilibrium condition, which can be controlled and manipulated to maximize its utility. The potential applications go beyond quantum metrology and extend to areas such as device designing, quantum computation and quantum technology in general.

Exploring the Underlying Mechanisms of the Xenopus laevis Embryonic Cell Cycle.

The cell cycle is an indispensable process in proliferation and development. Despite significant efforts, global quantification and physical understanding are still challenging. In this study, we explored the mechanisms of the Xenopus laevis embryonic cell cycle by quantifying the underlying landscape and flux. We uncovered the Mexican hat landscape of the Xenopus laevis embryonic cell cycle with several local basins and barriers on the oscillation path. The local basins characterize the different phases of the Xenopus laevis embryonic cell cycle, and the local barriers represent the checkpoints. The checkpoint mechanism of the cell cycle is revealed by the landscape basins and barriers. While landscape shape determines the stabilities of the states on the oscillation path, the curl flux force determines the stability of the cell cycle flow. Replication is fundamental for biology of living cells. We quantify the input energy (through the entropy production) as the thermodynamic requirement for initiation and sustainability of single cell life (cell cycle). Furthermore, we also quantify curl flux originated from the input energy as the dynamical requirement for the emergence of a new stable phase (cell cycle). This can provide a new quantitative insight for the origin of single cell life. In fact, the curl flux originated from the energy input or nutrition supply determines the speed and guarantees the progression of the cell cycle. The speed of the cell cycle is a hallmark of cancer. We characterized the quality of the cell cycle by the coherence time and found it is supported by the flux and energy cost. We are also able to quantify the degree of time irreversibility by the cross correlation function forward and backward in time from the stochastic traces in the simulation or experiments, providing a way for the quantification of the time irreversibility and the flux. Through global sensitivity analysis upon landscape and flux, we can identify the key elements for controlling the cell cycle speed. This can help to design an effective strategy for drug discovery against cancer.

Quantifying the potential and flux landscapes of multi-locus evolution

Exploration of multi-locus evolution is critically important for understanding evolutionary dynamics. Recombination and epistasis lead to complex evolutionary dynamics. Quantifying the stability and function of such multi-locus evolutionary systems globally is a long-standing challenge for evolutionary biologists. The conventional Wright, Fisher and quasi-linkage equilibrium (QLE) theories can only be applied to highly restricted, simplified and special evolutionary scenarios. In this study, we developed a non-equilibrium potential and flux landscape theory to explore the multi-locus evolution beyond Wright, Fisher and the QLE. We found that the intrinsic potential landscape as a Lyapunov function under the zero noise limit can be used to describe the global stability of a deterministic multi-locus system. We identified and quantified two driving forces responsible for multi-locus evolution: the underlying landscape and the curl flux. We studied the evolution of different cases under recombination, epistasis and mutation. Recombination, frequency-dependent selection and mutation can give rise to non-zero curl flux. In particular, we investigated the dynamics of a simple example of a two-locus system. We explored the underlying potential landscape which shows stable basins attracting the system down to valleys and rotating curl flux, which aids the global communication under recombination, epistasis and mutation. Discontinuities in the first derivative of the non-equilibrium free energy functional shows the non-equilibrium phase transition as the recombination rate increases. We also explored the effect of epistasis on mono-stability and bi-stability evolution. Mutation may drive the system far from equilibrium and be another source of non-zero probability flux. Entropy production rate can quantify energy consumption or dissipation. We explained the origin of the Red Queen hypothesis for endless evolution using the curl flux. Our landscape and flux framework can be applied more generally to multi-locus evolutionary systems experiencing recombination and epistasis.

Fluctuation-dissipation theorem for nonequilibrium quantum systems

We present a fluctuation-dissipation theorem (FDT) for nonequilibrium quantum systems with detailed-balance breaking, which deviates from the conventional form of the FDT for equilibrium systems preserving detailed balance. Using the phase space formulation of quantum mechanics and the potential-flux landscape framework, we find that the response function of nonequilibrium quantum systems to external perturbations contains a nontrivial contribution from the quantum curl flux quantifying detailed-balance breaking in the steady state, in addition to the correlation function of observables in the steady state representing the contribution of spontaneous fluctuations which is also present in the equilibrium FDT. We illustrate our general formalism with a harmonic oscillator coupled to two heat baths, and show that the nonequilibrium FDT reduces to the conventional expression at the equilibrium condition when the flux contribution vanishes.

Curl flux induced drift in stochastic differential equations in the zero-mass limit

Abstract We consider the nonlinear stochastic dynamics of dissipative Hamiltonian systems with state-dependent friction and diffusion connected by the fluctuation–dissipation relation in high dimensions. The system under study has a close connection to Ao’s framework in constructing a dynamical potential for non-equilibrium processes without detailed balance. We study the limiting case where the mass approaches zero and give a new and complete derivation of effective stochastic differential equations. Using the Ito stochastic integral convention, we show that the limiting effective Langevin equations have a new drift term. This extra term happens to be identical to the corresponding anti-Ito (or isothermal) integral (requiring constant temperature) in one dimension. We, however, cannot obtain this additional drift term using conventional stochastic integrals in high dimension. It is interesting to note that in a high-dimensional system, a curl flux induced drift may appear even if the diffusion matrix is constant. Our findings are supported by numerical simulations. We further analyze and discuss the role of this new drift term in calculating the classic escape time. For the first time, to our knowledge, the relation between the escape rate and the anti-Ito integral is presented. We also demonstrate that the derived diffusion equations give a new sampling algorithm which can increase convergence speed in a simple two-dimensional example.

Physical mechanism of mind changes and tradeoffs among speed, accuracy, and energy cost in brain decision making: Landscape, flux, and path perspectives**Project supported by the National Natural Science Foundation of China (Grant Nos. 21190040, 91430217, and 11305176).

Cognitive behaviors are determined by underlying neural networks. Many brain functions, such as learning and memory, have been successfully described by attractor dynamics. For decision making in the brain, a quantitative description of global attractor landscapes has not yet been completely given. Here, we developed a theoretical framework to quantify the landscape associated with the steady state probability distributions and associated steady state curl flux, measuring the degree of non-equilibrium through the degree of detailed balance breaking for decision making. We quantified the decision-making processes with optimal paths from the undecided attractor states to the decided attractor states, which are identified as basins of attractions, on the landscape. Both landscape and flux determine the kinetic paths and speed. The kinetics and global stability of decision making are explored by quantifying the landscape topography through the barrier heights and the mean first passage time. Our theoretical predictions are in agreement with experimental observations: more errors occur under time pressure. We quantitatively explored two mechanisms of the speed-accuracy tradeoff with speed emphasis and further uncovered the tradeoffs among speed, accuracy, and energy cost. Our results imply that there is an optimal balance among speed, accuracy, and the energy cost in decision making. We uncovered the possible mechanisms of changes of mind and how mind changes improve performance in decision processes. Our landscape approach can help facilitate an understanding of the underlying physical mechanisms of cognitive processes and identify the key factors in the corresponding neural networks.

Bifurcation analysis and potential landscapes of the p53-Mdm2 module regulated by the co-activator programmed cell death 5.

The dynamics of p53 play important roles in the regulation of cell fate decisions in response to various stresses, and programmed cell death 5 (PDCD5) functions as a co-activator of p53 that modulates p53 dynamics. In the present paper, we investigated how p53 dynamics are modulated by PDCD5 during the deoxyribose nucleic acid damage response using methods of bifurcation analysis and potential landscape. Our results revealed that p53 activities display rich dynamics under different PDCD5 levels, including monostability, bistability with two stable steady states, oscillations, and the coexistence of a stable steady state (or two states) and an oscillatory state. The physical properties of the p53 oscillations were further demonstrated by the potential landscape in which the potential force attracts the system state to the limit cycle attractor, and the curl flux force drives coherent oscillation along the cyclic trajectory. We also investigated the efficiency with which PDCD5 induced p53 oscillations. We show that Hopf bifurcation can be induced by increasing the PDCD5 efficiency and that the system dynamics exhibited clear transition features in both barrier height and energy dissipation when the efficiency was close to the bifurcation point.

Shape, orientation and magnitude of the curl quantum flux, the coherence and the statistical correlations in energy transport at nonequilibrium steady state

We provide a quantitative description of the nonequilibriumness based on the model of coupled oscillators interacting with multiple energy sources. This can be applied to the study of vibrational energy transport in molecules. The curl quantum flux quantifying the nonequilibriumness and time-irreversibility is quantified in the coherent representation and we find the geometric description of the shape and polarization of the flux which provides the connection between the microscopic description of quantum nonequilibriumness and the macroscopic observables, i.e., correlation function. We use the Wilson loop integral to quantify the magnitude of curl flux, which is shown to be correlated to the correlation function as well. Coherence contribution is explicitly demonstrated to be non-trivial and to considerably promote the heat transport quantified by heat current and efficiency. This comes from the fact that coherence effect is microscopically reflected by the geometric description of the flux. To uncover the effect of vibron–phonon coupling between the vibrational modes of molecular stretching (vibron) and the molecular chain (phonon), we further explore the influences of localization, which leads to the coherent and incoherent regimes that are characterized by the current–current correlations.

Quantifying the Landscape for Development and Cancer from a Core Cancer Stem Cell Circuit.

Cancer presents a serious threat to human health. The understanding of the cell fate determination during development and tumor-genesis remains challenging in current cancer biology. It was suggested that cancer stem cell (CSC) may arise from normal stem cells or be transformed from normal differentiated cells. This gives hints on the connection between cancer and development. However, the molecular mechanisms of these cell-type transitions and the CSC formation remain elusive. We quantified landscape, dominant paths, and switching rates between cell types from a core gene regulatory network for cancer and development. Stem cell, CSC, cancer, and normal cell types emerge as basins of attraction on associated landscape. The dominant paths quantify the transition processes among CSC, stem cell, normal cell, and cancer cell attractors. Transition actions of the dominant paths are shown to be closely related to switching rates between cell types, but not always to the barriers in between, because of the presence of the curl flux. During the process of P53 gene activation, landscape topography changes gradually from a CSC attractor to a normal cell attractor. This confirms the roles of P53 of preventing the formation of CSC through suppressing self-renewal and inducing differentiation. By global sensitivity analysis according to landscape topography and action, we identified key regulations determining cell-type switchings and suggested testable predictions. From landscape view, the emergence of the CSCs and the associated switching to other cell types are the results of underlying interactions among cancer and developmental marker genes. This indicates that the cancer and development are intimately connected. This landscape and flux theoretical framework provides a quantitative way to understand the underlying mechanisms of CSC formation and interplay between cancer and development.

Landscape, kinetics, paths and statistics of curl flux, coherence, entanglement and energy transfer in non-equilibrium quantum systems

We develop a population and flux landscape theory for general non-equilibrium quantum systems. We illustrate our theory by modelling the quantum transport of donor-acceptor energy transfer. We find two driving forces for the non-equilibrium quantum dynamics. The symmetric part of the driving force corresponds to the population landscape contribution which mainly governs the equilibrium part of dynamics while the anti-symmetric part of the driving force generates the non-equilibrium curl quantum flux which leads to the detailed-balance-breaking and time-irreversibility. The multi-loop structure of the flux emerges forms the flux-landscape. We study the trend of changes in population and flux-landscape with respect to the voltage (temperature difference induced by environments) and electronic coupling. Improving the voltage and electronic coupling in general facilitates the quantum transport by reducing the population landscape barriers between major states and increasing the mean value of the flux. A limit-cycle mode emerges when the underlying flux-landscape becomes funnelled with a significant gap between the largest flux loop and the rest of them. On the kinetic level, we find that multiple kinetic paths between quantum states emerge and illustrate the interference effects. The degree of interference is determined by the landscape and flux. Furthermore, we quantify kinetic rate which strongly correlates with the population landscape and flux. For quantum transport, we demonstrate that as the coherence or the quantum entanglement is enhanced, the flux and energy transfer efficiency are increased. Finally it is surprising that the non-equilibriumness quantified by voltage has a non-trivial contribution on strengthening the entanglement, which is attributed to the non-local feature of the quantum curl flux.

Landscape and flux theory of non-equilibrium dynamical systems with application to biology

We present a review of the recently developed landscape and flux theory for non-equilibrium dynamical systems. We point out that the global natures of the associated dynamics for non-equilibrium system are determined by two key factors: the underlying landscape and, importantly, a curl probability flux. The landscape (U) reflects the probability of states (P) (U = -ln P) and provides a global characterization and a stability measure of the system. The curl flux term measures how much detailed balance is broken and is one of the two main driving forces for the non-equilibrium dynamics in addition to the landscape gradient. Equilibrium dynamics resembles electron motion in an electric field, while non-equilibrium dynamics resembles electron motion in both electric and magnetic fields. The landscape and flux theory has many interesting consequences including (1) the fact that irreversible kinetic paths do not necessarily pass through the landscape saddles; (2) non-equilibrium transition state theory at the new saddle on the optimal paths for small but finite fluctuations; (3) a generalized fluctuation-dissipation relationship for non-equilibrium dynamical systems where the response function is not just equal to the fluctuations at the steady state alone as in the equilibrium case but there is an additional contribution from the curl flux in maintaining the steady state; (4) non-equilibrium thermodynamics where the free energy change is not just equal to the entropy production alone, as in the equilibrium case, but also there is an additional housekeeping contribution from the non-zero curl flux in maintaining the steady state; (5) gauge theory and a geometrical connection where the flux is found to be the origin of the gauge field curvature and the topological phase in analogy to the Berry phase in quantum mechanics; (6) coupled landscapes where non-adiabaticity of multiple landscapes in non-equilibrium dynamics can be analyzed using the landscape and flux theory and an eddy current emerges from the non-zero curl flux; (7) stochastic spatial dynamics where landscape and flux theory can be generalized for non-equilibrium field theory. We provide concrete examples of biological systems to demonstrate the new insights from the landscape and flux theory.