Mapping the Dynamics Landscape of Conformational Transitions in Enzyme: The Adenylate Kinase Case

Conformational transition describes the essential dynamics and mechanism of enzymes in pursuing their various functions. The fundamental and practical challenge to researchers is to quantitatively describe the roles of large-scale dynamic transitions for regulating the catalytic processes. In this study, we tackled this challenge by exploring the pathways and free energy landscape of conformational changes in adenylate kinase (AdK), a key ubiquitous enzyme for cellular energy homeostasis. Using explicit long-timescale (up to microseconds) molecular dynamics and bias-exchange metadynamics simulations, we determined at the atomistic level the intermediate conformational states and mapped the transition pathways of AdK in the presence and absence of ligands. There is clearly chronological operation of the functional domains of AdK. Specifically in the ligand-free AdK, there is no significant energy barrier in the free energy landscape separating the open and closed states. Instead there are multiple intermediate conformational states, which facilitate the rapid transitions of AdK. In the ligand-bound AdK, the closed conformation is energetically most favored with a large energy barrier to open it up, and the conformational population prefers to shift to the closed form coupled with transitions. The results suggest a perspective for a hybrid of conformational selection and induced fit operations of ligand binding to AdK. These observations, depicted in the most comprehensive and quantitative way to date, to our knowledge, emphasize the underlying intrinsic dynamics of AdK and reveal the sophisticated conformational transitions of AdK in fulfilling its enzymatic functions. The developed methodology can also apply to other proteins and biomolecular systems.     

陆地生态系统临界转换理论及其生态学机制研究进展

随着气候变化和人类活动对陆地生态系统双重扰动的不断加剧,越来越多的研究已经意识到生态系统结构和功能会发生难以预知的突变,并且恢复起来需要很长时间.开发判别典型生态系统临界转换的早期预警模型及理解其生态学机制成为生态学研究的热点.目前,基于跨越多个时空尺度的理论和实验研究,提出了多种预警陆地生态系统临界转换的理论框架和指...     

Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data

Many dynamical systems, including lakes, organisms, ocean circulation patterns, or financial markets, are now thought to have tipping points where critical transitions to a contrasting state can happen. Because critical transitions can occur unexpectedly and are difficult to manage, there is a need for methods that can be used to identify when a critical transition is approaching. Recent theory shows that we can identify the proximity of a system to a critical transition using a variety of so-called 'early warning signals', and successful empirical examples suggest a potential for practical applicability. However, while the range of proposed methods for predicting critical transitions is rapidly expanding, opinions on their practical use differ widely, and there is no comparative study that tests the limitations of the different methods to identify approaching critical transitions using time-series data. Here, we summarize a range of currently available early warning methods and apply them to two simulated time series that are typical of systems undergoing a critical transition. In addition to a methodological guide, our work offers a practical toolbox that may be used in a wide range of fields to help detect early warning signals of critical transitions in time series data.     

生态系统状态跃变的早期预警信号及检测

生态系统在气候变化和土地利用及人类活动等的影响下其状态会由某一稳态转变到另一稳态。由于环境压力的复杂性、非线性、随机性等特征，往往导致状态转变表现为非线性、突变、跃变等特点。准确界定系统状态跃变的拐点或阈值点存在很大的挑战，而捕捉接近临界拐点前的生态系统结构和属性上的变化特征作为早期预警信号是切实可行的。早期预警信号理论经历理论框架构建、方法确立、机理认知等近半个多世纪的探索，已经由最初的通过仅依赖检测临界点恢复力的速率减慢、方差增加、系统自相关增强等统计学信号过度到更加多样化的检测方法，如检测系统组分属性的变化特征，诊断系统组分各属性之间的关系变化，系统组分的性状变化、系统组分网络结构变化等等，并且试图整合多信号提高预警的精确性。利用来自自然生态系统的长时间高密度数据集和空间代替时间的数据集，基于多度及性状信号的早期预警，结合稳定性、临界恢复力的减速、以及统计参数的指示作用对系统跃变进行早期诊断和预警是预测生态学的主旨。早期预警信号的深入研究不仅能够完善已有理论的不足，同时还能够为生态系统的保护和管理提供切实有效的理论指导。     

Quantifying Cell Fate Decisions for Differentiation and Reprogramming of a Human Stem Cell Network: Landscape and Biological Paths

Cellular reprogramming has been recently intensively studied experimentally. We developed a global potential landscape and kinetic path framework to explore a human stem cell developmental network composed of 52 genes. We uncovered the underlying landscape for the stem cell network with two basins of attractions representing stem and differentiated cell states, quantified and exhibited the high dimensional biological paths for the differentiation and reprogramming process, connecting the stem cell state and differentiated cell state. Both the landscape and non-equilibrium curl flux determine the dynamics of cell differentiation jointly. Flux leads the kinetic paths to be deviated from the steepest descent gradient path, and the corresponding differentiation and reprogramming paths are irreversible. Quantification of paths allows us to find out how the differentiation and reprogramming occur and which important states they go through. We show the developmental process proceeds as moving from the stem cell basin of attraction to the differentiation basin of attraction. The landscape topography characterized by the barrier heights and transition rates quantitatively determine the global stability and kinetic speed of cell fate decision process for development. Through the global sensitivity analysis, we provided some specific predictions for the effects of key genes and regulation connections on the cellular differentiation or reprogramming process. Key links from sensitivity analysis and biological paths can be used to guide the differentiation designs or reprogramming tactics.     

生态系统多稳态研究进展

在多稳态的生态系统中,外力可能导致生态系统状态突然之间发生不可逆转的转变,从而达到一个新的平衡状态。但目前对多稳态理论的系统研究很少,如何使用预警信号来预测生态系统的状态转变依旧是个难题。通过多稳态理论的梳理提出了一个更加综合的多稳态定义,并以放牧模型为例,系统总结了多稳态理论的相关概念,将多稳态理论应用在生态系统演替和扰沌理论的解释中;通过对生态系统稳态转换预警信号的原理、优缺点和应用条件的分析,对不同尺度下多稳态的研究方法进行了归纳;最后提出了目前多稳态领域的研究问题和未来的研究重点。结果表明:(1)将时间和空间预警信号结合在一起,并量化正确预警信号的概率,对错误预警信号的比例进行加权,可能会提供更准确的稳态转换的预报。(2)定量观测试验适用于小尺度的研究,而较大尺度的研究则采用简化的模型来模拟研究,选择正确的尺度极有可能改变预警信号的可靠性。(3)结合多稳态理论研究生态系统临界转换和反馈控制机制,并将基于性状的特征指标和进化动力学纳入其中,是生态系统修复实践的重要研究方向。(4)将多稳态相关理论和生态保护管理政策的实践相结合,是多稳态理论未来应用的前景。本研究为多稳态理论和实践的...     

Interspecific facilitation and critical transitions in arid ecosystems

Climate change and intensified land‐use impose severe stress on arid ecosystems, resulting in relatively rapid degradation which is difficult to reverse. To prevent such critical transitions it is crucial to detect early warning signals. Increased ‘patchiness’– smaller and fewer vegetated patches – is thought to be such a signal, but the underlying mechanisms are still poorly understood. Facilitation between plants is known to be an important mechanism driving the patchiness of the vegetation, but we lack understanding of how interactions between plants change in response to combined effects of drought and consumer pressure – the main stressors in many arid ecosystems. Over the last decade numerous experimental studies have tested how intensity of facilitation between plants changes with increasing stress. The most recent synthesis predicts a decline in facilitation intensity at the severe end of a drought stress gradient. Adding consumer pressure may result in even earlier and faster declines in facilitation intensity. So far, studies on critical transitions and plant–plant interactions have developed separately. The relationship between stress and facilitation intensity has been overlooked in critical transition theory, while facilitation intensity may determine the position of a critical transition threshold. In this study, we incorporate experimental studies on the relation between stress and facilitation intensity into the critical transition framework, to improve our ability to predict critical transitions. Moreover, we propose that a decline in facilitation intensity at the severe end of a stress gradient may occur prior to a critical transition. Inclusion of consumer pressure will speed up this process, leading to earlier and faster degradation. In‐field monitoring of seedling–facilitator associations and declines in facilitator recruitment can indicate declines in facilitation intensity and may thus provide additional early warning signals for imminent critical transitions, besides increased patchiness.     

The Potential Landscape of Genetic Circuits Imposes the Arrow of Time in Stem Cell Differentiation

Differentiation from a multipotent stem or progenitor state to a mature cell is an essentially irreversible process. The associated changes in gene expression patterns exhibit time-directionality. This “arrow of time” in the collective change of gene expression across multiple stable gene expression patterns (attractors) is not explained by the regulated activation, the suppression of individual genes which are bidirectional molecular processes, or by the standard dynamical models of the underlying gene circuit which only account for local stability of attractors. To capture the global dynamics of this nonequilibrium system and gain insight in the time-asymmetry of state transitions, we computed the quasipotential landscape of the stochastic dynamics of a canonical gene circuit that governs branching cell fate commitment. The potential landscape reveals the global dynamics and permits the calculation of potential barriers between cell phenotypes imposed by the circuit architecture. The generic asymmetry of barrier heights indicates that the transition from the uncommitted multipotent state to differentiated states is inherently unidirectional. The model agrees with observations and predicts the extreme conditions for reprogramming cells back to the undifferentiated state.     

Early warning signals for critical transitions: a generalized modeling approach

Critical transitions are sudden, often irreversible, changes that can occur in a large variety of complex systems; signals that warn of critical transitions are therefore highly desirable. We propose a new method for early warning signals that integrates multiple sources of information and data about the system through the framework of a generalized model. We demonstrate our proposed approach through several examples, including a previously published fisheries model. We regard our method as complementary to existing early warning signals, taking an approach of intermediate complexity between model-free approaches and fully parameterized simulations. One potential advantage of our approach is that, under appropriate conditions, it may reduce the amount of time series data required for a robust early warning signal.     

Ensemble perspective for catalytic promiscuity: calorimetric analysis of the active site conformational landscape of a detoxification enzyme

Enzymological paradigms have shifted recently to acknowledge the biological importance of catalytic promiscuity. However, catalytic promiscuity is a poorly understood property, and no thermodynamic treatment has described the conformational landscape of promiscuous versus substrate-specific enzymes. Here, two structurally similar glutathione transferase (GST, glutathione S-transferase) isoforms with high specificity or high promiscuity are compared. Differential scanning calorimetry (DSC) indicates a reversible low temperature transition for the promiscuous GSTA1-1 that is not observed with substrate-specific GSTA4-4. This transition is assigned to rearrangement of the C terminus at the active site of GSTA1-1 based on the effects of ligands and mutations. Near-UV and far-UV circular dichroism indicate that this transition is due to repacking of tertiary contacts with the remainder of the subunit, rather than "unfolding" of the C terminus per se. Analysis of the DSC data using a modified Landau theory indicates that the local conformational landscape of the active site of GSTA1-1 is smooth, with barrierless transitions between states. The partition function of the C-terminal states is a broad unimodal distribution at all temperatures within this DSC transition. In contrast, the remainder of the GSTA1-1 subunit and the GSTA4-4 protein exhibit folded and unfolded macrostates with a significant energy barrier separating them. Their partition function includes a sharp unimodal distribution of states only at temperatures that yield either folded or unfolded macrostates. At intermediate temperatures the partition function includes a bimodal distribution. The barrierless rearrangement of the GSTA1-1 active site within a local smooth energy landscape suggests a thermodynamic basis for catalytic promiscuity.     

Exploring the Mechanisms of Differentiation,Dedifferentiation, Reprogramming and Transdifferentiation

We explored the underlying mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation (cell type switchings) from landscape and flux perspectives. Lineage reprogramming is a new regenerative method to convert a matured cell into another cell including direct transdifferentiation without undergoing a pluripotent cell state and indirect transdifferentiation with an initial dedifferentiation-reversion (reprogramming) to a pluripotent cell state. Each cell type is quantified by a distinct valley on the potential landscape with higher probability. We investigated three driving forces for cell fate decision making: stochastic fluctuations, gene regulation and induction, which can lead to cell type switchings. We showed that under the driving forces the direct transdifferentiation process proceeds from a differentiated cell valley to another differentiated cell valley through either a distinct stable intermediate state or a certain series of unstable indeterminate states. The dedifferentiation process proceeds through a pluripotent cell state. Barrier height and the corresponding escape time from the valley on the landscape can be used to quantify the stability and efficiency of cell type switchings. We also uncovered the mechanisms of the underlying processes by quantifying the dominant biological paths of cell type switchings on the potential landscape. The dynamics of cell type switchings are determined by both landscape gradient and flux. The flux can lead to the deviations of the dominant biological paths for cell type switchings from the naively expected landscape gradient path. As a result, the corresponding dominant paths of cell type switchings are irreversible. We also classified the mechanisms of cell fate development from our landscape theory: super-critical pitchfork bifurcation, sub-critical pitchfork bifurcation, sub-critical pitchfork with two saddle-node bifurcation, and saddle-node bifurcation. Our model showed good agreements with the experiments. It provides a general framework to explore the mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation.     

All-atom replica exchange molecular simulation of protein BBL

Downhill folding is one of the most important predictions of energy landscape theory. Recently, the Escherichia coli 2-oxoglutarate dehydrogenase PSBD was described as a first example of global downhill folding (Garcia-Mira et al., Science 2002;298:2191), classification that has been later subject of significant controversy. To help resolve this problem, by using intensive all-atom simulation with explicit water model and the replica exchange method, we sample the phase space of protein BBL and depict the free energy landscape. We give an estimate of the free energy barrier height of 1-2 k(B)T, dependent on the way the energy landscape is projected. We also study the conformational distribution of the transition region and find that the three helices generally take the similar positions as that in the native states whereas their spatial orientations show large variability. We further detect the inconsistency between different signals by individually fitting the thermal denaturation curves of five structural features using two-state model, which gives a wide spread melting temperature of 19 K. All of these features are consistent with a picture of folding with very low cooperativities. Compared with the experimental data (Sadqi et al., Nature 2006; 442:317), our results indicate that the Naf-BBL (pH5.3) may have an even lower barrier height and cooperativity.     

Rate of forcing and the forecastability of critical transitions

Critical transitions are qualitative changes of state that occur when a stochastic dynamical system is forced through a critical point. Many critical transitions are preceded by characteristic fluctuations that may serve as model‐independent early warning signals, implying that these events may be predictable in applications ranging from physics to biology. In nonbiological systems, the strength of such early warning signals has been shown partly to be determined by the speed at which the transition occurs. It is currently unknown whether biological systems, which are inherently high dimensional and typically display low signal‐to‐noise ratios, also exhibit this property, which would have important implications for how ecosystems are managed, particularly where the forces exerted on a system are anthropogenic. We examine whether the rate of forcing can alter the strength of early warning signals in (1) a model exhibiting a fold bifurcation where a state shift is driven by the harvesting of individuals, and (2) a model exhibiting a transcritical bifurcation where a state shift is driven by increased grazing pressure. These models predict that the rate of forcing can alter the detectability of early warning signals regardless of the underlying bifurcation the system exhibits, but that this result may be more pronounced in fold bifurcations. These findings have important implications for the management of biological populations, particularly harvested systems such as fisheries, and suggest that knowing the class of bifurcations a system will manifest may help discriminate between true‐positive and false‐positive signals.     

Potential landscape and probabilistic flux of a predator prey network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design.     

Flickering as an early warning signal

Most work on generic early warning signals for critical transitions focuses on indicators of the phenomenon of critical slowing down that precedes a range of catastrophic bifurcation points. However, in highly stochastic environments, systems will tend to shift to alternative basins of attraction already far from such bifurcation points. In fact, strong perturbations (noise) may cause the system to “flicker” between the basins of attraction of the system’s alternative states. As a result, under such noisy conditions, critical slowing down is not relevant, and one would expect its related generic leading indicators to fail, signaling an impending transition. Here, we systematically explore how flickering may be detected and interpreted as a signal of an emerging alternative attractor. We show that—although the two mechanisms differ—flickering may often be reflected in rising variance, lag-1 autocorrelation and skewness in ways that resemble the effects of critical slowing down. In particular, we demonstrate how the probability distribution of a flickering system can be used to map potential alternative attractors and their resilience. Thus, while flickering systems differ in many ways from the classical image of critical transitions, changes in their dynamics may carry valuable information about upcoming major changes.     

Dynamics and stability of polymorphic human telomeric G-quadruplex under tension

As critical DNA structures capping the human chromosome ends, the stability and structural polymorphism of human telomeric G-quadruplex (G4) have drawn increasing attention in recent years. This work characterizes the equilibrium transitions of single-molecule telomeric G4 at physiological K⁺ concentration. We report three folded states of telomeric G4 with markedly different lifetime and mechanical stability. Our results show that the kinetically favored folding pathway is through a short-lived intermediate state to a longer-lived state. By examining the force dependence of transition rates, the force-dependent transition free energy landscape for this pathway is determined. In addition, an ultra-long-lived form of telomeric G4 structure with a much stronger mechanical stability is identified.     

Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation

Circadian rhythms with a period of ∼24 h, are natural timing machines. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. However, the global and physical natures of the oscillation under fluctuations are still not very clear. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. The potential landscape of the network is uncovered and has a global Mexican-hat shape. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.     

Indicators of transitions in biological systems

In the face of global biodiversity declines, predicting the fate of biological systems is a key goal in ecology. One popular approach is the search for early warning signals (EWSs) based on alternative stable states theory. In this review, we cover the theory behind nonlinearity in dynamic systems and techniques to detect the loss of resilience that can indicate state transitions. We describe the research done on generic abundance‐based signals of instability that are derived from the phenomenon of critical slowing down, which represent the genesis of EWSs research. We highlight some of the issues facing the detection of such signals in biological systems – which are inherently complex and show low signal‐to‐noise ratios. We then document research on alternative signals of instability, including measuring shifts in spatial autocorrelation and trait dynamics, and discuss potential future directions for EWSs research based on detailed demographic and phenotypic data. We set EWSs research in the greater field of predictive ecology and weigh up the costs and benefits of simplicity vs. complexity in predictive models, and how the available data should steer the development of future methods. Finally, we identify some key unanswered questions that, if solved, could improve the applicability of these methods.