Entropy and barrier-controlled fluctuations determine conformational viscoelasticity of single biomolecules

Biological macromolecules have complex and nontrivial energy landscapes, endowing them with a unique conformational adaptability and diversity in function. Hence, understanding the processes of elasticity and dissipation at the nanoscale is important to molecular biology and emerging fields such as nanotechnology. Here we analyze single molecule fluctuations in an atomic force microscope, using a generic model of biopolymer viscoelasticity that includes local "internal" conformational dissipation. Comparing two biopolymers, dextran and cellulose (polysaccharides with and without local bistable transitions), demonstrates that signatures of simple conformational change are minima in both the elastic and internal friction constants around a characteristic force. A novel analysis of dynamics on a bistable energy landscape provides a simple explanation: an elasticity driven by the entropy, and friction by a barrier-controlled hopping time of populations between states, which is surprisingly distinct to the well-known relaxation time. This nonequilibrium microscopic analysis thus provides a means of quantifying new dynamical features of the energy landscape of the glucopyranose ring, revealing an unexpected underlying roughness and information on the shape of the barrier of the chair-boat transition in dextran. The results presented herein provide a basis toward probing the viscoelasticity of macromolecular conformational transitions on more complex energy landscapes, such as during protein folding.     

Timescale analysis of rule-based biochemical reaction networks

The flow of information within a cell is governed by a series of protein–protein interactions that can be described as a reaction network. Mathematical models of biochemical reaction networks can be constructed by repetitively applying specific rules that define how reactants interact and what new species are formed on reaction. To aid in understanding the underlying biochemistry, timescale analysis is one method developed to prune the size of the reaction network. In this work, we extend the methods associated with timescale analysis to reaction rules instead of the species contained within the network. To illustrate this approach, we applied timescale analysis to a simple receptor–ligand binding model and a rule‐based model of interleukin‐12 (IL‐12) signaling in naïve CD4+ T cells. The IL‐12 signaling pathway includes multiple protein–protein interactions that collectively transmit information; however, the level of mechanistic detail sufficient to capture the observed dynamics has not been justified based on the available data. The analysis correctly predicted that reactions associated with Janus Kinase 2 and Tyrosine Kinase 2 binding to their corresponding receptor exist at a pseudo‐equilibrium. By contrast, reactions associated with ligand binding and receptor turnover regulate cellular response to IL‐12. An empirical Bayesian approach was used to estimate the uncertainty in the timescales. This approach complements existing rank‐ and flux‐based methods that can be used to interrogate complex reaction networks. Ultimately, timescale analysis of rule‐based models is a computational tool that can be used to reveal the biochemical steps that regulate signaling dynamics. © 2011 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

A recurring problem in population biology - as well as other stochastic dynamical systems in biology, the physical and social sciences - is the distinction between the ‘true’ dynamics of a system and observational noise: i.e. can we from present data reliably infer e.g. biological mechanisms, or are signals swamped by noise.Here, we approach this problem using the canonical model for simple systems that exhibit complex behaviour, the logistic map. At each time-point noise is added, which allows us to study the long-term behaviour of a system which exhibits both non-linear dynamics and intrinsic noise.We show that the interplay between deterministic non-linear dynamics and simple Gaussian noise results in a perplexingly simple system when viewed statistically.In particular we show that for the case of Gaussian noise it is possible to derive at very reliable approximations for the time until the system has reached an absorbing state. This generic model allows us, for example, to study the life-time of molecular species involved in noisy feedback loops.     

Data-driven flow-map models for data-efficient discovery of dynamics and fast uncertainty quantification of biological and biochemical systems

Computational models are increasingly used to investigate and predict the complex dynamics of biological and biochemical systems. Nevertheless, governing equations of a biochemical system may not be (fully) known, which would necessitate learning the system dynamics directly from, often limited and noisy, observed data. On the other hand, when expensive models are available, systematic and efficient quantification of the effects of model uncertainties on quantities of interest can be an arduous task. This paper leverages the notion of flow-map (de)compositions to present a framework that can address both of these challenges via learning data-driven models useful for capturing the dynamical behavior of biochemical systems. Data-driven flow-map models seek to directly learn the integration operators of the governing differential equations in a black-box manner, irrespective of structure of the underlying equations. As such, they can serve as a flexible approach for deriving fast-to-evaluate surrogates for expensive computational models of system dynamics, or, alternatively, for reconstructing the long-term system dynamics via experimental observations. We present a data-efficient approach to data-driven flow-map modeling based on polynomial chaos Kriging. The approach is demonstrated for discovery of the dynamics of various benchmark systems and a coculture bioreactor subject to external forcing, as well as for uncertainty quantification of a microbial electrosynthesis reactor. Such data-driven models and analyses of dynamical systems can be paramount in the design and optimization of bioprocesses and integrated biomanufacturing systems.     

Simulation modelling of ecological hierarchies in constructive dynamical systems

Organized complexity is a characteristic feature of ecological systems with heterogeneous components interacting at several spatio-temporal scales. The hierarchy theory is a powerful epistemological framework to describe such systems by decomposing them vertically into levels and horizontally into holons. It was at first developed in a temporal and functional perspective and then, in the context of landscape ecology, extended to a spatial and structural approach. So far, most ecological applications of this theory were restricted to observational purposes, using multi-scale analysis to describe hierarchies. In spite of an increasing attention to dynamics of hierarchically structured ecological systems, current simulation models are still very limited in their representation of self-organization in complex adaptive systems. An ontological conceptualization of the hierarchy theory is outlined, focusing on key concepts, such as levels of organization and the compound and component faces of the holons. Various existing formalisms are currently used in simulation modelling, such as system dynamics, discrete event and agent based paradigms. Their ability to express the hierarchical organization of dynamical ecological systems is discussed. It turns out that a multi-modelling approach linking all these formalisms and oriented toward the specification of a constructive dynamical system would be able to express the dynamical structure of the hierarchy (creation, destruction and change of holons) and the functional and structural links between levels of organization.     

Dynamical Analysis and Visualization of Tornadoes Time Series

In this paper we analyze the behavior of tornado time-series in the U.S. from the perspective of dynamical systems. A tornado is a violently rotating column of air extending from a cumulonimbus cloud down to the ground. Such phenomena reveal features that are well described by power law functions and unveil characteristics found in systems with long range memory effects. Tornado time series are viewed as the output of a complex system and are interpreted as a manifestation of its dynamics. Tornadoes are modeled as sequences of Dirac impulses with amplitude proportional to the events size. First, a collection of time series involving 64 years is analyzed in the frequency domain by means of the Fourier transform. The amplitude spectra are approximated by power law functions and their parameters are read as an underlying signature of the system dynamics. Second, it is adopted the concept of circular time and the collective behavior of tornadoes analyzed. Clustering techniques are then adopted to identify and visualize the emerging patterns.     

Graphical approach to model reduction for nonlinear biochemical networks

Model reduction is a central challenge to the development and analysis of multiscale physiology models. Advances in model reduction are needed not only for computational feasibility but also for obtaining conceptual insights from complex systems. Here, we introduce an intuitive graphical approach to model reduction based on phase plane analysis. Timescale separation is identified by the degree of hysteresis observed in phase-loops, which guides a "concentration-clamp" procedure for estimating explicit algebraic relationships between species equilibrating on fast timescales. The primary advantages of this approach over Jacobian-based timescale decomposition are that: 1) it incorporates nonlinear system dynamics, and 2) it can be easily visualized, even directly from experimental data. We tested this graphical model reduction approach using a 25-variable model of cardiac β(1)-adrenergic signaling, obtaining 6- and 4-variable reduced models that retain good predictive capabilities even in response to new perturbations. These 6 signaling species appear to be optimal "kinetic biomarkers" of the overall β(1)-adrenergic pathway. The 6-variable reduced model is well suited for integration into multiscale models of heart function, and more generally, this graphical model reduction approach is readily applicable to a variety of other complex biological systems.     

Creative dynamics approach to neural intelligence

The thrust of this paper is to introduce and discuss a substantially new type of dynamical system for modelling biological behavior. The approach was motivated by an attempt to remove one of the most fundamental limitations of artificial neural networks — their rigid behavior compared with even simplest biological systems. This approach exploits a novel paradigm in nonlinear dynamics based upon the concept of terminal attractors and repellers. It was demonstrated that non-Lipschitzian dynamics based upon the failure of Lipschitz condition exhibits a new qualitative effect — a multi-choice response to periodic external excitations. Based upon this property, a substantially new class of dynamical systems — the unpredictable systems — was introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics. Firstly, such systems have zero Jacobian. As a result of that, they have an infinite number of equilibrium points which occupy curves, surfaces or hypersurfaces. Secondly, at all these equilibrium points, the Lipschitz conditions fails, so the equilibrium points become terminal attractors or repellers depending upon the sign of the periodic excitation. Both of these pathological characteristics result in multi-choice response of unpredictable dynamical systems. It has been shown that the unpredictable systems can be controlled by sign strings which uniquely define the system behaviors by specifying the direction of the motions in the critical points. By changing the combinations of signs in the code strings the system can reproduce any prescribed behavior to a prescribed accuracy. That is why the unpredictable systems driven by sign strings are extremely flexible and are highly adaptable to environmental changes. It was also shown that such systems can serve as a powerful tool for temporal pattern memories and complex pattern recognition. It has been demonstrated that new architecture of neural networks based upon non-Lipschitzian dynamics can be utilized for modelling more complex patterns of behavior which can be associated with phenomenological models of creativity and neural intelligence.     

Predictive landscapes hidden beneath biological cellular automata

To celebrate Hans Frauenfelder’s achievements, we examine energy(-like) “landscapes” for complex living systems. Energy landscapes summarize all possible dynamics of some physical systems. Energy(-like) landscapes can explain some biomolecular processes, including gene expression and, as Frauenfelder showed, protein folding. But energy-like landscapes and existing frameworks like statistical mechanics seem impractical for describing many living systems. Difficulties stem from living systems being high dimensional, nonlinear, and governed by many, tightly coupled constituents that are noisy. The predominant modeling approach is devising differential equations that are tailored to each living system. This ad hoc approach faces the notorious “parameter problem”: models have numerous nonlinear, mathematical functions with unknown parameter values, even for describing just a few intracellular processes. One cannot measure many intracellular parameters or can only measure them as snapshots in time. Another modeling approach uses cellular automata to represent living systems as discrete dynamical systems with binary variables. Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g., Cellular Potts Model). But numerous biological features, in current practice, are qualitatively described rather than quantitatively (e.g., gene is (highly) expressed or not (highly) expressed). Cellular automata governed by verbal rules are useful representations for living systems and can mitigate the parameter problem. However, they can yield complex dynamics that are difficult to understand because the automata-governing rules are not quantitative and much of the existing mathematical tools and theorems apply to continuous but not discrete dynamical systems. Recent studies found ways to overcome this challenge. These studies either discovered or suggest an existence of predictive “landscapes” whose shapes are described by Lyapunov functions and yield “equations of motion” for a “pseudo-particle.” The pseudo-particle represents the entire cellular lattice and moves on the landscape, thereby giving a low-dimensional representation of the cellular automata dynamics. We outline this promising modeling strategy.

     

An integrative approach to understanding host–parasitoid population dynamics in real landscapes

Many of our advances regarding the spatial ecology of predators and prey have been attributed to research with insect parasitoids and their hosts. Host–parasitoid systems are ideal for spatial-ecological studies because of the small size of the organisms, the often discrete distribution of their resources, and the relative ease with which host mortality from parasitoids can be determined. We outline an integrated approach to studying host–parasitoid interactions in heterogeneous natural landscapes. This approach involves conducting experiments to obtain critically important information on dispersal and boundary behavior of the host and parasitoid, large-scale manipulations of landscape structure to reveal the impacts of landscape change on host–parasitoid interactions and temporal population dynamics, and the development of spatially realistic, behavior-based landscape models. The dividends from such an integrative approach are far reaching, as is illustrated in our research on the prairie planthopper Prokelisia crocea and its egg parasitoid Anagrus columbi that occurs in the tall-grass prairies of North America. Here, we describe the population structure of this system which is based on a long-term survey of planthoppers and parasitoids among host–plant patches. We also outline novel approaches to experimentally quantify and model the movement and boundary behavior of animals in general. The value of this information is revealed in a landscape-level field experiment that was designed to test predictions about how landscape change affects the spatial and temporal population dynamics of the host and parasitoid. Finally, with these empirical data as the foundation, we describe novel simulation models that are spatially realistic and behavior based. Drawing from this integrated approach and case study, we identify key research questions for the future.     

Periodic orbits: a new language for neuronal dynamics.

A new nonlinear dynamical analysis is applied to complex behavior from neuronal systems. The conceptual foundation of this analysis is the abstraction of observed neuronal activities into a dynamical landscape characterized by a hierarchy of "unstable periodic orbits" (UPOs). UPOs are rigorously identified in data sets representative of three different levels of organization in mammalian brain. An analysis based on UPOs affords a novel alternative method of decoding, predicting, and controlling these neuronal systems.     

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.     

Analysis of Sub-τc and Supra-τc Motions in Protein Gβ1 Using Molecular Dynamics Simulations

The functions of proteins depend on the dynamical behavior of their native states on a wide range of timescales. To investigate these dynamics in the case of the small protein Gβ1, we analyzed molecular dynamics simulations with the model-free approach of nuclear magnetic relaxation. We found amplitudes of fast timescale motions (sub-τ_c, where τ_c is the rotational correlation time) consistent with S² obtained from spin relaxation measurements as well as amplitudes of slow timescale motions (supra-τ_c) in quantitative agreement with S² order parameters derived from residual dipolar coupling measurements. The slow timescale motions are associated with the large variations of the ³J couplings that follow transitions between different conformational substates. These results provide further characterization of the large structural fluctuations in the native states of proteins that occur on timescales longer than the rotational correlation time.     

A method for representing and developing process models

Scientists investigate the dynamics of complex systems with quantitative models, employing them to synthesize knowledge, to explain observations, and to forecast future system behavior. Complete specification of systems is impossible, so models must be simplified abstractions. Thus, the art of modeling involves deciding which system elements to include and determining how they should be represented. We view modeling as search through a space of candidate models that is guided by model objectives, theoretical knowledge, and empirical data. In this contribution, we introduce a method for representing process-based models that facilitates the discovery of structures that explain observed behavior. This representation casts dynamic systems as interacting sets of processes that act on entities. Using this approach, a modeler first encodes relevant ecological knowledge into a library of generic entities and processes, then instantiates these theoretical components, and finally assembles candidate models from these elements. We illustrate this methodology with a model of the Ross Sea ecosystem.     

Some remarks on the compatibility between determinism and unpredictability

Determinism and unpredictability are compatible since deterministic flows can produce, if sensitive to initial conditions, unpredictable behaviors. Within this perspective, the notion of scenario to chaos transition offers a new form of predictability for the behavior of sensitive to initial condition systems under the variation of a control parameter. In this paper I first shed light on the genesis of this notion, based on a dynamical systems approach and on considerations of structural stability. I then suggest a link to the figure of epigenetic landscape, partially inspired by a dynamical systems perspective, and offering a theoretical framework to apprehend developmental noise.     

Mesoscopic biochemical basis of isogenetic inheritance and canalization: stochasticity, nonlinearity, and emergent landscape

Biochemical reaction systems in mesoscopic volume, under sustained environmental chemical gradient(s), can have multiple stochastic attractors. Two distinct mechanisms are known for their origins: (a) Stochastic single-molecule events, such as gene expression, with slow gene on-off dynamics; and (b) nonlinear networks with feedbacks. These two mechanisms yield different volume dependence for the sojourn time of an attractor. As in the classic Arrhenius theory for temperature dependent transition rates, a landscape perspective provides a natural framework for the system's behavior. However, due to the nonequilibrium nature of the open chemical systems, the landscape, and the attractors it represents, are all themselves emergent properties of complex, mesoscopic dynamics. In terms of the landscape, we show a generalization of Kramers' approach is possible to provide a rate theory. The emergence of attractors is a form of self-organization in the mesoscopic system; stochastic attractors in biochemical systems such as gene regulation and cellular signaling are naturally inheritable via cell division. Delbrück-Gillespie's mesoscopic reaction system theory, therefore, provides a biochemical basis for spontaneous isogenetic switching and canalization.     

Femtochemistry in enzyme catalysis: DNA photolyase

Photolyase uses light energy to split UV-induced cyclobutane pyrimidine dimers in damaged DNA. This photoenzyme encompasses a series of elementary dynamical processes during repair function from early photoinitiation by a photoantenna molecule to enhance repair efficiency, to in vitro photoreduction through aromatic residues to reconvert the cofactor to the active form, and to final photorepair to fix damaged DNA. The corresponding series of dynamics include resonance energy transfer, intraprotein electron transfer, and intermolecular electron transfer, bond breaking-making rearrangements and back electron return, respectively. We review here our recent direct studies of these dynamical processes in real time, which showed that all these elementary reactions in the enzyme occur within subnanosecond timescale. Active-site solvation was observed to play a critical role in the continuous modulation of catalytic reactions. As a model system for enzyme catalysis, we isolated the enzyme–substrate complex in the transition-state region and mapped out the entire evolution of unmasked catalytic reactions of DNA repair. These observed synergistic motions in the active site reveal a perfect correlation of structural integrity and dynamical locality to ensure maximum repair efficiency on the ultrafast time scale.     

Comparison of Three Ionic Liquid-Tolerant Cellulases by Molecular Dynamics

We have employed molecular dynamics to investigate the differences in ionic liquid tolerance among three distinct family 5 cellulases from Trichoderma viride, Thermogata maritima, and Pyrococcus horikoshii. Simulations of the three cellulases were conducted at a range of temperatures in various binary mixtures of the ionic liquid 1-ethyl-3-methyl-imidazolium acetate with water. Our analysis demonstrates that the effects of ionic liquids on the enzymes vary in each individual case from local structural disturbances to loss of much of one of the enzyme’s secondary structure. Enzymes with more negatively charged surfaces tend to resist destabilization by ionic liquids. Specific and unique structural changes in the enzymes are induced by the presence of ionic liquids. Disruption of the secondary structure, changes in dynamical motion, and local changes in the binding pocket are observed in less tolerant enzymes. Ionic-liquid-induced denaturation of one of the enzymes is indicated over the 500 ns timescale. In contrast, the most tolerant cellulase behaves similarly in water and in ionic-liquid-containing mixtures. Unlike the heuristic approaches that attempt to predict enzyme stability using macroscopic properties, molecular dynamics allows us to predict specific atomic-level structural and dynamical changes in an enzyme’s behavior induced by ionic liquids and other mixed solvents. Using these insights, we propose specific experimentally testable hypotheses regarding the origin of activity loss for each of the systems investigated in this study.