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A synthetic predator-prey ecosystem diagram. Outer boxes represent cell walls. Arrows represent activation or production, blunt arrows represent inhibition or killing.
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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 so...
Citations
... Energy landscapes not only help visualize the systems' phase space and its structural changes as parameters are varied, but allow to predict the rates of activated processes [1][2][3]. Some fields that benefit from the energy landscape approach are optimization problems [4], neural networks [5], protein folding [6], cell nets [7], gene regulatory networks [8,9], ecology [10], and evolution [11]. The fact that deterministic dissipative systems relax toward their attractors has an expression (known as the ''Lyapunov property'') in terms of the potential ( ):̇< 0 outside the attractors; ( ) is said to be a ''Lyapunov function''. ...
... [62] As found in [65], the entropy production rate in a nonequilibrium steady state (NESS) turns out to be bounded from below by the ratio between the squared mean and the variance of an arbitrary current. 10 This result was followed by several others [100][101][102][103][104], known generically as ''thermodynamic uncertainty relations'' (TUR). Recently, a TUR has been discussed for KPZ [105][106][107]. ...
... We shall not repeat here all the steps done in the indicated reference, but it is worth indicating the use of a vector notation (in particular in 1d, but that could be extended to any dimension). 10 Specifically, after a time in the NESS, a fluctuating integrated current ( ) has a mean ⟨ ( )⟩ = , and a diffusivity = (2 ) −1 lim →∞ ⟨( ( ) − ) 2 ⟩. Then ≥ 2 ∕2 . ...
A brief review is made of the birth and evolution of the “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a “thermodynamic uncertainty relation” (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.
... Energy landscapes not only help visualize the systems' phase space and its structural changes as parameters are varied, but allow to predict the rates of activated processes [1][2][3]. Some fields that benefit from the energy landscape approach are optimization problems [4], neural networks [5], protein folding [6], cell nets [7], gene regulatory networks [8,9], ecology [10], and evolution [11]. The fact that deterministic dissipative systems relax toward their attractors has an expression (known as the ''Lyapunov property'') in terms of the potential ( ):̇< 0 outside the attractors; ( ) is said to be a ''Lyapunov function''. ...
... [62] As found in [65], the entropy production rate in a nonequilibrium steady state (NESS) turns out to be bounded from below by the ratio between the squared mean and the variance of an arbitrary current. 10 This result was followed by several others [100][101][102][103][104], known generically as ''thermodynamic uncertainty relations'' (TUR). Recently, a TUR has been discussed for KPZ [105][106][107]. ...
... We shall not repeat here all the steps done in the indicated reference, but it is worth indicating the use of a vector notation (in particular in 1d, but that could be extended to any dimension). 10 Specifically, after a time in the NESS, a fluctuating integrated current ( ) has a mean ⟨ ( )⟩ = , and a diffusivity = (2 ) −1 lim →∞ ⟨( ( ) − ) 2 ⟩. Then ≥ 2 ∕2 . ...
A brief review is made of the birth and evolution of the ‘‘nonequilibrium potential’’ (NEP) concept. As if
providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension
to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic
dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the
Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a
‘‘thermodynamic uncertainty relation’’ (TUR) in the KPZ equation (the stochastic field theory of kinetic
interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model
and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.
... For evolution, the landscape and flux theory provides a physical foundation for general scenarios [5,6,76,82]. For ecology, the landscape and flux theory can quantify the underlying global stability and associated bifurcations of the ecological states [80,169,170]. For game theory, the landscape and flux theory reveals the global quantification and physical mechanisms of the strategy state switching dynamics [77]. ...
We give a review on the landscape theory of the equilibrium biological systems and landscape-flux theory of the nonequilibrium biological systems as the global driving force. The emergences of the behaviors, the associated thermodynamics in terms of the entropy and free energy and dynamics in terms of the rate and paths have been quantitatively demonstrated. The hierarchical organization structures have been discussed. The biological applications ranging from protein folding, biomolecular recognition, specificity, biomolecular evolution and design for equilibrium systems as well as cell cycle, differentiation and development, cancer, neural networks and brain function, and evolution for nonequilibrium systems, cross-scale studies of genome structural dynamics and experimental quantifications/verifications of the landscape and flux are illustrated. Together, this gives an overall global physical and quantitative picture in terms of the landscape and flux for the behaviors, dynamics and functions of biological systems.
... Feedback mechanisms allow a network to correct or repair nodes and links that are perturbed or become dysfunctional under certain conditions. Common examples include negative feedbacks in predator-prey systems that result in population oscillations (Li et al. 2011), gene regulation systems that lead to constant gene expression outputs (Gjusvland et al. 2007;Hensel et al. 2012), or DNA proofreading and repair systems (Ashour and Mosammaparast 2021), and positive feedbacks in excitable organism behaviors (O'Boyle et al. 2020) or memories in gene regulatory networks (Qiao et al. 2020). Diversity: Diversity within a network can be regarded as the number, variations, and complexity of nodes of differential identities or functions. ...
Why do some biological systems and communities persist while others fail? Robustness, a system's stability, and resilience, the ability to return to a stable state, are key concepts that span multiple disciplines within and outside the biological sciences. Discovering and applying common rules that govern the robustness and resilience of biological systems is a critical step toward creating solutions for species survival in the face of climate change, as well as the for the ever-increasing need for food, health, and energy for human populations. We propose that network theory provides a framework for universal scalable mathematical models to describe robustness and resilience and the relationship between them, and hypothesize that resilience at lower organization levels contribute to robust systems. Insightful models of biological systems can be generated by quantifying the mechanisms of redundancy, diversity, and connectivity of networks, from biochemical processes to ecosystems. These models provide pathways towards understanding how evolvability can both contribute to and result from robustness and resilience under dynamic conditions. We now have an abundance of data from model and non-model systems and the technological and computational advances for studying complex systems. Several conceptual and policy advances will allow the research community to elucidate the rules of robustness and resilience. Conceptually, a common language and data structure that can be applied across levels of biological organization needs to be developed. Policy advances such as cross-disciplinary funding mechanisms, access to affordable computational capacity, and the integration of network theory and computer science within the standard biological science curriculum will provide the needed research environments. This new understanding of biological systems will allow us to derive ever more useful forecasts of biological behaviors and revolutionize the engineering of biological systems that can survive changing environments or disease, navigate the deepest oceans, or sustain life throughout the solar system.
... In practise, the steady-state distribution is obtained either by solving (analytically) the associated Fokker-Planck equation (FPE) [20,21] or, more commonly, by way of simulations of the corresponding SDE [15,16,22,19,23]. ...
The metaphor of the Waddington epigenetic landscape has become an iconic representation of the cellular differentiation process. Recent accessibility of single-cell transcriptomic data has provided new opportunities for quantifying this originally conceptual tool that could offer insight into the gene regulatory networks underlying cellular development. While a number of methods for constructing the landscape have been proposed, by far the most commonly employed approach is based on computing the landscape as the negative logarithm of the steady-state probability distribution. Here, we use a simple model to highlight the complexities and limitations that arise when reconstructing the potential landscape in the presence of stochastic fluctuations. We consider how the landscape changes in accordance with different stochastic systems, and show that it is the subtle interplay between the deterministic and stochastic components of the system that ultimately shapes the landscape. We further discuss how the presence of noise has important implications for the identifiability of the regulatory dynamics from experimental data.
... Experiments in microchemostats confirmed distinct types of population dynamics, namely, coexistence, extinction, and oscillation. The underlying landscape for a global description of the dynamics was also quantified (Li et al., 2011b;Xu et al., 2014a). Beyond well-mixed cultures, microbial populations have also been used to study spatial ecology. ...
Life is characterized by a myriad of complex dynamic processes allowing organisms to grow, reproduce, and evolve. Physical approaches for describing systems out of thermodynamic equilibrium have been increasingly applied to living systems, which often exhibit phenomena unknown from those traditionally studied in physics. Spectacular advances in experimentation during the last decade or two, for example, in microscopy, single cell dynamics, in the reconstruction of sub- and multicellular systems outside of living organisms, or in high throughput data acquisition have yielded an unprecedented wealth of data about cell dynamics, genetic regulation, and organismal development. These data have motivated the development and refinement of concepts and tools to dissect the physical mechanisms underlying biological processes. Notably, the landscape and flux theory as well as active hydrodynamic gel theory have proven very useful in this endeavour. Together with concepts and tools developed in other areas of nonequilibrium physics, significant progresses have been made in unraveling the principles underlying efficient energy transport in photosynthesis, cellular regulatory networks, cellular movements and organization, embryonic development and cancer, neural network dynamics, population dynamics and ecology, as well as ageing, immune responses and evolution. Here, we review recent advances in nonequilibrium physics and survey their application to biological systems. We expect many of these results to be important cornerstones as the field continues to build our understanding of life.
... Energy landscapes not only help visualize the systems' phase space and its structural changes as parameters are varied, but allow predicting the rates of activated processes [2][3][4]. Some fields that benefit from the energy landscape approach are optimization problems [5], neural networks [6], protein folding [7], cell nets [8], gene regulatory networks [9,10], ecology [11], and evolution [12]. ...
... Early successful examples are the complex Ginzburg-Landau equation (CGLE) [21,22] and the FitzHugh-Nagumo (FHN) model [23,24] 5 . This scheme has been later reformulated [29], extended [30], and exploited in many interesting cases [6][7][8][9][10][11][12]. The goal of this work is to show that a NEP exists for a broad class of rate models of neural networks, of the type proposed by Wilson and Cowan [31]. ...
... All the parameters are real and moreover, the j kl are positive (j 11 and j 22 are recurrent interactions, j 12 and j 21 are cross-population interactions). The above definitions are such that for M = 0, x = 0 is a stable fixed point. ...
Energy landscapes are a highly useful aid for the understanding of dynamical systems, and a particularly valuable tool for their analysis. For a broad class of rate neural-network models of relevance in neuroscience, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained “nonequilibrium potential” (NEP)—the first one obtained for a model of neural circuits—predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability—with implications for working memory and slow neural oscillations—and population bursts, associated with signal detection in neuroscience. Instead, limit cycles are not found for the conditions in which the NEP is defined. Their nonexistence can be proven by resorting to the Bendixson–Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. This NEP constitutes a powerful tool to understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.
... A number of methods exist for determining the decomposition in which the potential is defined; however, by far the most common is via the steady state probability distribution over the state space, where U / −ln(P s (X)). Numerous studies have obtained landscapes based upon this approach, either using methods that directly obtain the steady state distribution [25,28,29], or use extensive simulations to estimate the distribution empirically [11,[30][31][32]. Alternative approaches include those based upon variational principles and the action in moving between any two points [12,23,28], empirical Lyapunov functions [33], approximations to the Lyapunov functions [34], or an orthogonal decomposition of the vector field [35]. ...
Models describing the process of stem-cell differentiation are plentiful, and may offer insights into the underlying mechanisms and experimentally observed behaviour. Waddington’s epigenetic landscape has been providing a conceptual framework for differentiation processes since its inception. It also allows, however, for detailed mathematical and quantitative analyses, as the landscape can, at least in principle, be related to mathematical models of dynamical systems. Here we focus on a set of dynamical systems features that are intimately linked to cell differentiation, by considering exemplar dynamical models that capture important aspects of stem cell differentiation dynamics. These models allow us to map the paths that cells take through gene expression space as they move from one fate to another, e.g. from a stem-cell to a more specialized cell type. Our analysis highlights the role of the transition state (TS) that separates distinct cell fates, and how the nature of the TS changes as the underlying landscape changes—change that can be induced by e.g. cellular signaling. We demonstrate that models for stem cell differentiation may be interpreted in terms of either a static or transitory landscape. For the static case the TS represents a particular transcriptional profile that all cells approach during differentiation. Alternatively, the TS may refer to the commonly observed period of heterogeneity as cells undergo stochastic transitions.
... This approach is based on the Fokker-Planck equation, which justifies computing a potential as U (x) ∝ − ln[P S (x)]. In practice the steady-state distribution is either obtained by solving the Fokker-Planck equation directly [28][29][30] or found from extensive simulations of the corresponding SDE [9,[31][32][33]. While this method is easy to implement, it may be impractical for higher-dimensional systems and there is generally no guarantee that the derived landscape relates directly to the fixed points of the deterministic system or that a steady state distribution can even be obtained [34]. ...
The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasipotential, a scalar function that quantifies the likelihood of reaching each point in the state space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the sum-of-squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based on a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining "curl" component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasipotential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasipotential for high-dimensional linear systems and also for a number of nonlinear examples.
... Energy landscapes not only help visualize the systems' phase space and its structural changes as parameters are varied, but allow predicting the rates of activated processes (Wio HS "Nonequilibrium potential in reaction-diffusion systems" In Garrido P, Marro J, 1997; Wio and Deza, 2007;Wio et al., 2012). Some fields that benefit from the energy landscape approach are optimization problems (Kirkpatrick et al., 1983), neural networks (Yan et al., 2013), protein folding , cell nets , gene regulatory networks (Kim and Wang, 2007;Wang et al., 2010), ecology (Li et al., 2011), and evolution (Zhang et al., 2012). ...
... Early successful examples are the complex Ginzburg-Landau equation (CGLE) (Descalzi and Graham, 1992;Montagne et al., 1996) and the FitzHugh-Nagumo (FHN) model (Izús et al., 1998;Izús et al., 1999) 4 . This scheme has been later reformulated (Ao, 2004), extended (Wu and Wang, 2013), and exploited in many interesting cases (Yan et al., 2013;Wang et al., 2012;Wang et al., 2006;Kim and Wang, 2007;Wang et al., 2010;Li et al., 2011;Zhang et al., 2012). ...
Energy landscapes are a useful aid for the understanding of dynamical systems, and a valuable tool for their analysis. For a broad class of rate models of neural networks, we derive a global Lyapunov function which provides an energy landscape without any symmetry constraint. This newly obtained `nonequilibrium potential' (NEP) predicts with high accuracy the outcomes of the dynamics in the globally stable cases studied here. Common features of the models in this class are bistability --with implications for working memory and slow neural oscillations --and `population burst', also relevant in neuroscience. Instead, limit cycles are not found. Their nonexistence can be proven by resort to the Bendixson--Dulac theorem, at least when the NEP remains positive and in the (also generic) singular limit of these models. Hopefully, this NEP will help understand average neural network dynamics from a more formal standpoint, and will also be of help in the description of large heterogeneous neural networks.
