Metabolic networks evolve towards states of maximum entropy production

A metabolic network can be described by a set of elementary modes or pathways representing discrete metabolic states that support cell function. We have recently shown that in the most likely metabolic state the usage probability of individual elementary modes is distributed according to the Boltzmann distribution law while complying with the principle of maximum entropy production. To demonstrate that a metabolic network evolves towards such state we have carried out adaptive evolution experiments with Thermoanaerobacterium saccharolyticum operating with a reduced metabolic functionality based on a reduced set of elementary modes. In such reduced metabolic network metabolic fluxes can be conveniently computed from the measured metabolite secretion pattern. Over a time span of 300 generations the specific growth rate of the strain continuously increased together with a continuous increase in the rate of entropy production. We show that the rate of entropy production asymptotically approaches the maximum entropy production rate predicted from the state when the usage probability of individual elementary modes is distributed according to the Boltzmann distribution. Therefore, the outcome of evolution of a complex biological system can be predicted in highly quantitative terms using basic statistical mechanical principles.     

An optimizing start-up strategy for a bio-methanator

This paper presents an optimizing start-up strategy for a bio-methanator. The goal of the control strategy is to maximize the outflow rate of methane in anaerobic digestion processes, which can be described by a two-population model. The methodology relies on a thorough analysis of the system dynamics and involves the solution of two optimization problems: steady-state optimization for determining the optimal operating point and transient optimization. The latter is a classical optimal control problem, which can be solved using the maximum principle of Pontryagin. The proposed control law is of the bang–bang type. The process is driven from an initial state to a small neighborhood of the optimal steady state by switching the manipulated variable (dilution rate) from the minimum to the maximum value at a certain time instant. Then the dilution rate is set to the optimal value and the system settles down in the optimal steady state. This control law ensures the convergence of the system to the optimal steady state and substantially increases its stability region. The region of attraction of the steady state corresponding to maximum production of methane is considerably enlarged. In some cases, which are related to the possibility of selecting the minimum dilution rate below a certain level, the stability region of the optimal steady state equals the interior of the state space. Aside its efficiency, which is evaluated not only in terms of biogas production but also from the perspective of treatment of the organic load, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any anaerobic digestion process, for which the acidogenesis and methanogenesis are, respectively, characterized by Monod and Haldane kinetics.     

Replicator Equations and the Principle of Minimal Production of Information

Many complex systems in mathematical biology and other areas can be described by the replicator equation. We show that solutions of a wide class of replicator equations minimize the KL-divergence of the initial and current distributions under time-dependent constraints, which in their turn, can be computed explicitly at every instant due to the system dynamics. Therefore, the Kullback principle of minimum discrimination information, as well as the maximum entropy principle, for systems governed by the replicator equations can be derived from the system dynamics rather than postulated. Applications to the Malthusian inhomogeneous models, global demography, and the Eigen quasispecies equation are given.     

Mathematical models of HIV pathogenesis and treatment

We review mathematical models of HIV dynamics, disease progression, and therapy. We start by introducing a basic model of virus infection and demonstrate how it was used to study HIV dynamics and to measure crucial parameters that lead to a new understanding of the disease process. We discuss the diversity threshold model as an example of the general principle that virus evolution can drive disease progression and the destruction of the immune system. Finally, we show how mathematical models can be used to understand correlates of long-term immunological control of HIV, and to design therapy regimes that convert a progressing patient into a state of long-term non-progression.     

Enzyme kinetics and the maximum entropy production principle

A general proof is derived that entropy production can be maximized with respect to rate constants in any enzymatic transition. This result is used to test the assumption that biological evolution of enzyme is accompanied with an increase of entropy production in its internal transitions and that such increase can serve to quantify the progress of enzyme evolution. The state of maximum entropy production would correspond to fully evolved enzyme. As an example the internal transition ES?EP in a generalized reversible Michaelis-Menten three state scheme is analyzed. A good agreement is found among experimentally determined values of the forward rate constant in internal transitions ES→EP for three types of β-Lactamase enzymes and their optimal values predicted by the maximum entropy production principle, which agrees with earlier observations that β-Lactamase enzymes are nearly fully evolved. The optimization of rate constants as the consequence of basic physical principle, which is the subject of this paper, is a completely different concept from a) net metabolic flux maximization or b) entropy production minimization (in the static head state), both also proposed to be tightly connected to biological evolution.     

Joint Modeling of Multiple Social Networks to Elucidate Primate Social Dynamics: I. Maximum Entropy Principle and Network-Based Interactions

In a complex behavioral system, such as an animal society, the dynamics of the system as a whole represent the synergistic interaction among multiple aspects of the society. We constructed multiple single-behavior social networks for the purpose of approximating from multiple aspects a single complex behavioral system of interest: rhesus macaque society. Instead of analyzing these networks individually, we describe a new method for jointly analyzing them in order to gain comprehensive understanding about the system dynamics as a whole. This method of jointly modeling multiple networks becomes valuable analytical tool for studying the complex nature of the interaction among multiple aspects of any system. Here we develop a bottom-up, iterative modeling approach based upon the maximum entropy principle. This principle is applied to a multi-dimensional link-based distributional framework, which is derived by jointly transforming the multiple directed behavioral social network data, for extracting patterns of synergistic inter-behavioral relationships. Using a rhesus macaque group as a model system, we jointly modeled and analyzed four different social behavioral networks at two different time points (one stable and one unstable) from a rhesus macaque group housed at the California National Primate Research Center (CNPRC). We report and discuss the inter-behavioral dynamics uncovered by our joint modeling approach with respect to social stability.     

Agent-Based Analysis of Biomass Feedstock Production Dynamics

The success of the bioenergy sector based on lignocellulosic feedstock will require a sustainable and resilient transition from the current agricultural system focused on food crops to one also producing energy crops. The dynamics of this transition are not well understood. It will be driven significantly by the collective participation, behavior, and interaction of various stakeholders such as farmers within the production system. The objective of this work is to study the system dynamics through the development and application of an agent-based model using the theory of complex adaptive systems. Farmers and biorefinery, two key stakeholders in the system, are modeled as independent agents. The decision making of each agent as well as its interaction with other agents is modeled using a set of rules reflecting the economic, social, and personal attributes of the agent. These rules and model parameters are adapted from literature. Regulatory mechanisms such as Biomass Crop Assistance Program are embedded in the decision-making process. The model is then used to simulate the production of Miscanthus as an energy crop in Illinois. Particular focus has been given on understanding the dynamics of Miscanthus adaptation as an agricultural crop and its impact on biorefinery capacity and contractual agreements. Results showed that only 60% of the maximum regional production capacity could be reached, and it took up to 15 years to establish that capacity. A 25% reduction in the land opportunity cost led to a 63% increase in the steady- state productivity. Sensitivity analysis showed that higher initial conversion of land by farmers to grow energy crop led to faster growth in regional productivity.     

Patch dynamics in a landscape modified by ecosystem engineers

Ecosystem engineers, organisms that modify the environment, have the potential to dramatically alter ecosystem structure and function at large spatial scales. The degree to which ecosystem engineering produces large-scale effects is, in part, dependent on the dynamics of the patches that engineers create. Here we develop a set of models that links the population dynamics of ecosystem engineers to the dynamics of the patches that they create. We show that the relative abundance of different patch types in an engineered landscape is dependent upon the production of successful colonists from engineered patches and the rate at which critical resources are depleted by engineers and then renewed. We also consider the effects of immigration from either outside the system or from engineers that are present in non-engineered patches, and the effects of engineers that can recolonize patches before they are fully recovered on the steady state distribution of different patch types. We use data collected on the population dynamics of a model engineer, the beaver, to estimate the per-patch production rate of new colonists, the decay rate of engineered patches, and the recovery rate of abandoned patches. We use these estimated parameters as a baseline to determine the effects of varying parameters on the distribution of different patch types. We suggest a number of hypotheses that derive from model predictions and that could serve as tests of the model.     

The maximum empower principle: An invisible hand controlling the self-organizing development of forest plantations in south China

Derived from the maximum power principle (MPP), the maximum empower principle (MePP) is considered as the foundation of emergy theory and evaluation methods. However, it has often encountered some doubts since proposed, because of lacking sufficient empirical evidence. To test the validity of the MePP in the self-organization of forest ecosystems, this paper applied a process-based ecosystem model (Biome-BGC) to simulate the dynamics of biomass, litter and soil organic matter (SOM) of three forest plantations in south China during 1985–2007, and attempted to replicate their self-organizing processes. The simulated results and input flows were transformed to emergy as a common basis and, from the viewpoint of emergy synthesis, the dynamics of the production efficiency and empower of the three forest ecosystems were revealed along with their self-organizing developments over time. The results showed that three forest plantations had similar dynamic change patterns of emergy efficiency and empower, but the production efficiencies of them were not always consistent with their empower performances. The production efficiency firstly increased rapidly to maximums, and then decreased to optimal moderate values. However, the empower came to the maximums after the efficiency peaked and then fluctuated up and down, dependent on weather. These results implied that, a forest ecosystem in its self-organizing process tends toward maximum empower at optimal efficiency. Behind the maximum empower of the forest ecosystem is the desynchrony development of different components, e.g., biomass, litter and SOM; leaf, stem and root; biomass and biodiversity. As a whole, the MePP functions like an invisible hand controlling the general self-organizing development of forest ecosystems and pointing out the direction of their development.     

Rare Dissipative Transitions Punctuate the Initiation of Chemical Denaturation in Proteins

Protein unfolding dynamics are bound by their degree of entropy production, a quantity that relates the amount of heat dissipated by a nonequilibrium process to a system’s forward and time-reversed trajectories. We here explore the statistics of heat dissipation that emerge in protein molecules subjected to a chemical denaturant. Coupling large molecular dynamics datasets and Markov state models with the theory of entropy production, we demonstrate that dissipative processes can be rigorously characterized over the course of the urea-induced unfolding of the protein chymotrypsin inhibitor 2. By enumerating full entropy production probability distributions as a function of time, we first illustrate that distinct passive and dissipative regimes are present in the denaturation dynamics. Within the dissipative dynamical region, we next find that chymotrypsin inhibitor 2 is strongly driven into unfolded states in which the protein’s hydrophobic core has been penetrated by urea molecules and disintegrated. Detailed analyses reveal that urea’s interruption of key hydrophobic contacts between core residues causes many of the protein’s native structural features to dissolve.     

An artificial maximum neural network: a winner-take-all neuron model forcing the state of the system in a solution domain

A maximum neuron model is proposed in order to force the state of the system to converge to the solution in neural dynamics. The state of the system is always forced in a solution domain. The artificial maximum neural network is used for the module orientation problem and the bipartite subgraph problem. The usefulness of the maximum neural network is empirically demonstrated by simulating randomly generated massive nstances (examples) in both problems. In randomly generated more than one thousand instances our system always converges to the solution within one hundred iteration steps regardless of the problem size. Our simulation results show the effectiveness of our algorithms and support our claim that one class of NP-complete problems may be solvable in a polynomial time.     

Information-Theoretic Analysis of the Dynamics of an Executable Biological Model

To facilitate analysis and understanding of biological systems, large-scale data are often integrated into models using a variety of mathematical and computational approaches. Such models describe the dynamics of the biological system and can be used to study the changes in the state of the system over time. For many model classes, such as discrete or continuous dynamical systems, there exist appropriate frameworks and tools for analyzing system dynamics. However, the heterogeneous information that encodes and bridges molecular and cellular dynamics, inherent to fine-grained molecular simulation models, presents significant challenges to the study of system dynamics. In this paper, we present an algorithmic information theory based approach for the analysis and interpretation of the dynamics of such executable models of biological systems. We apply a normalized compression distance (NCD) analysis to the state representations of a model that simulates the immune decision making and immune cell behavior. We show that this analysis successfully captures the essential information in the dynamics of the system, which results from a variety of events including proliferation, differentiation, or perturbations such as gene knock-outs. We demonstrate that this approach can be used for the analysis of executable models, regardless of the modeling framework, and for making experimentally quantifiable predictions.     

Protein-Folding Analysis Using Features Obtained by Persistent Homology

Understanding the protein-folding process is an outstanding issue in biophysics; recent developments in molecular dynamics simulation have provided insights into this phenomenon. However, the large freedom of atomic motion hinders the understanding of this process. In this study, we applied persistent homology, an emerging method to analyze topological features in a data set, to reveal protein-folding dynamics. We developed a new, to our knowledge, method to characterize the protein structure based on persistent homology and applied this method to molecular dynamics simulations of chignolin. Using principle component analysis or nonnegative matrix factorization, our analysis method revealed two stable states and one saddle state, corresponding to the native, misfolded, and transition states, respectively. We also identified an unfolded state with slow dynamics in the reduced space. Our method serves as a promising tool to understand the protein-folding process.     

Transient dynamics and early diagnostics in infectious disease

To date, mathematical models of the dynamics of infectious disease have consistently focused on understanding the long-term behavior of the interacting components, where the steady state solutions are paramount. However for most acute infections, the long-term behavior of the pathogen population is of little importance to the host and population health. We introduce the notion of transient pathology, where the short-term dynamics of interaction between the immune system and pathogens is the principal focus. We identify the amplifying effect of the absence of a fully operative immune system on the pathogenesis of the initial inoculum, and its implication for the acute severity of the infection. We then formalize the underlying dynamics, and derive two measures of transient pathogenicity: the peak of infection (maximum pathogenic load) and the time to peak of infection, both crucial to understanding the early dynamics of infection and its consequences for early intervention. Received: 25 January 2000 / Revised version: 30 November 2000 / Published online: 12 October 2001     

Some remarks on the dynamics of molecular self-organization

The dynamics of a self-organizing molecular system is described in terms of its normal modes. Each normal mode is associated with a certain eigenvalue, the average of which reflects most directly the overall process of self-organization. For the temporal change of this quantity a maximum principle holds.     

A general minimum principle for competing populations: Some ecological and evolutionary consequences

The principle introduced by MacArthur (1969. Proc. Natl. Acad. Sci. U.S.A., 64, 1369–1375) that the stable positive equilibrium of an exploitative competition system minimizes the mean squared deviation between available and utilized resources is generalized. It is shown that under less stringent assumptions, the global attractor of a competition system consists of species assortments that minimize a function of consumers' abundances, called inefficient energy utilization. If the minimum is unique (whether or not lying on the boundary of the state space) the global attractor reduces to a globally stable equilibrium. The inefficient energy utilization is the sum of two terms: one, called unutilized productivity, is the squared difference between maximum resource production and consumers' consumption, while the second, called basal energy consumption, is the caloric income to a community necessary for maintaining it at a steady state. The minimum principle is then used to solve some ecological and evolutionary problems. It is demonstrated that if an inverse correlation is assumed between consumers' efficiency and width of the trophic niche, evolution in undisturbed environments leads to assortments of tightly packed specialists.     

Maximum Entropy Principle Based Estimation of Performance Distribution in Queueing Theory

In related research on queuing systems, in order to determine the system state, there is a widespread practice to assume that the system is stable and that distributions of the customer arrival ratio and service ratio are known information. In this study, the queuing system is looked at as a black box without any assumptions on the distribution of the arrival and service ratios and only keeping the assumption on the stability of the queuing system. By applying the principle of maximum entropy, the performance distribution of queuing systems is derived from some easily accessible indexes, such as the capacity of the system, the mean number of customers in the system, and the mean utilization of the servers. Some special cases are modeled and their performance distributions are derived. Using the chi-square goodness of fit test, the accuracy and generality for practical purposes of the principle of maximum entropy approach is demonstrated.     

Selection for intermediate mortality and reproduction rates in a spatially structured population

How local interactions influence both population and evolutionary dynamics is currently a key topic in theoretical ecology. We use a 'well-mixed' analytical model and spatially explicit individual-based models to investigate a system where a population is subject to rare disturbance events. The disturbance can only propagate through regions of the population where the density of individuals is sufficiently high and individuals affected by the disturbance die shortly after. We find that populations where individuals are sessile often exhibit very different dynamic behaviour when compared to populations where individuals are mobile and spatially well mixed. When mutations are allowed which affect either offspring birth rates or mortality rates, the well-mixed populations always evolve to a state where a single disturbance event leads to extinction. Populations often persist substantially longer if individuals are sessile and they disperse their offspring locally. We also find that for sessile populations selection may favour short-lived individuals with limited offspring production. Population dynamics are found to be strongly influenced by the host characters that are evolving and the rate at which host variation is introduced into the system.     

Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics

A formal sensitivity analysis is performed on a delay differential equation model for the viral dynamics of an in vivo HIV infection during protease inhibitor therapy. We present results of both a differential analysis as well as a principle component based analysis and provide evidence that suggests the exact times at which specific parameters have the most influence over the solution. We offer insight into the pairwise mathematical relationships between the productively infected T-cell death rate δ, the viral plasma clearance rate c, and the time delay τ between infection and viral production as they relate to the viral dynamics. The results support the claim that the presence of a nonzero delay has a major impact on the model dynamics. Lastly, we comment upon the inadequacies of an alternative principle component based analysis.