Evolutionary stability under the replicator and the gradient dynamics

We unite two general models for evolutionary change under the forces of selection, mutation and reproduction, a genetic model (replicator dynamics) and a cultural model (gradient dynamics). Under the assumption of normality, we find that the mean and variance dynamics are essentially identical under the two models and we relate these to the ESS and convergence stability conditions.     

Conformational sampling and dynamics of membrane proteins from 10-nanosecond computer simulations

In the current report, we provide a quantitative analysis of the convergence of the sampling of conformational space accomplished in molecular dynamics simulations of membrane proteins of duration in the order of 10 nanoseconds. A set of proteins of diverse size and topology is considered, ranging from helical pores such as gramicidin and small beta-barrels such as OmpT, to larger and more complex structures such as rhodopsin and FepA. Principal component analysis of the C(alpha)-atom trajectories was employed to assess the convergence of the conformational sampling in both the transmembrane domains and the whole proteins, while the time-dependence of the average structure was analyzed to obtain single-domain information. The membrane-embedded regions, particularly those of small or structurally simple proteins, were found to achieve reasonable convergence. By contrast, extra-membranous domains lacking secondary structure are often markedly under-sampled, exhibiting a continuous structural drift. This drift results in a significant imprecision in the calculated B-factors, which detracts from any quantitative comparison to experimental data. In view of such limitations, we suggest that similar analyses may be valuable in simulation studies of membrane protein dynamics, in order to attach a level of confidence to any biologically relevant observations.     

Statistical prediction and molecular dynamics simulation

We describe a statistical approach to the validation and improvement of molecular dynamics simulations of macromolecules. We emphasize the use of molecular dynamics simulations to calculate thermodynamic quantities that may be compared to experimental measurements, and the use of a common set of energetic parameters across multiple distinct molecules. We briefly review relevant results from the theory of stochastic processes and discuss the monitoring of convergence to equilibrium, the obtaining of confidence intervals for summary statistics corresponding to measured quantities, and an approach to validation and improvement of simulations based on out-of-sample prediction. We apply these methods to replica exchange molecular dynamics simulations of a set of eight helical peptides under the AMBER potential using implicit solvent. We evaluate the ability of these simulations to quantitatively reproduce experimental helicity measurements obtained by circular dichroism. In addition, we introduce notions of statistical predictive estimation for force-field parameter refinement. We perform a sensitivity analysis to identify key parameters of the potential, and introduce Bayesian updating of these parameters. We demonstrate the effect of parameter updating applied to the internal dielectric constant parameter on the out-of-sample prediction accuracy as measured by cross-validation.     

From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics

We study the large population limit of the Moran process, under the assumption of weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.     

Planar polarity and actin dynamics in the epidermis of Drosophila

Dorsal closure is a morphogenetic process involving the coordinated convergence of two epithelial sheets to enclose the Drosophila melanogaster embryo. Specialized populations of cells at the edges of each epithelial sheet, the dorsal-most epidermal cells, emit actin-based processes that are essential for the proper enclosure of the embryo. Here we show that actin dynamics at the leading edge is preceded by a planar polarization of the dorsal-most epidermal cells associated with a reorganization of the cytoskeleton. An important consequence of this planar polarization is the formation of actin-nucleating centres at the leading edge, which are important in the dynamics of actin. We show that Wingless (Wg) signalling and Jun amino-terminal kinase (JNK) signalling have overlapping but different roles in these events.     

Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics

We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes—asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.     

Entropy and convergence in dynamics and demography

Demographic dynamics is formally equivalent to the dynamics of a Markov chain, as is true of some nonlinear dynamical systems. Convergence to demographic equilibrium can be studied in terms of convergence in the Markov chain. Tuljapurkar (1982) showed that population entropy (Kolmogorov-Sinai entropy) provides information on the rate of this convergence. This paper begins by considering finite state Markov chains, providing elementary proofs of the relationship between convergence rate and entropy, and discusses in detail the uses and limitations of entropy as a convergence measure; these results also apply to Markovian dynamical systems. Next, new qualitative and quantitative arguments are used to discuss the demographic meaning of entropy. An exact relationship is established giving population entropy in terms of the eigenvalues of the Leslie matrix characteristic equation. Finally, the significance of imprimitive and periodic limits is discussed in relation to population entropy.     

DNA methylation dynamics in human induced pluripotent stem cells over time

Epigenetic reprogramming is a critical event in the generation of induced pluripotent stem cells (iPSCs). Here, we determined the DNA methylation profiles of 22 human iPSC lines derived from five different cell types (human endometrium, placental artery endothelium, amnion, fetal lung fibroblast, and menstrual blood cell) and five human embryonic stem cell (ESC) lines, and we followed the aberrant methylation sites in iPSCs for up to 42 weeks. The iPSCs exhibited distinct epigenetic differences from ESCs, which were caused by aberrant methylation at early passages. Multiple appearances and then disappearances of random aberrant methylation were detected throughout iPSC reprogramming. Continuous passaging of the iPSCs diminished the differences between iPSCs and ESCs, implying that iPSCs lose the characteristics inherited from the parent cells and adapt to very closely resemble ESCs over time. Human iPSCs were gradually reprogrammed through the "convergence" of aberrant hyper-methylation events that continuously appeared in a de novo manner. This iPS reprogramming consisted of stochastic de novo methylation and selection/fixation of methylation in an environment suitable for ESCs. Taken together, random methylation and convergence are driving forces for long-term reprogramming of iPSCs to ESCs.     

Strong convergence of integrators for nonequilibrium Langevin dynamics

Several numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the strong rate of convergence for each scheme is analyzed. The schemes considered here employ specialised periodic boundary conditions that deform with the flow, namely Lees-Edwards and Kraynik-Reinelt boundary conditions and their generalisations. We show that care must be taken when implementing standard stochastic integration schemes with these boundary conditions in order to avoid a breakdown in the strong order of convergence.     

Oligomorphic dynamics for analyzing the quantitative genetics of adaptive speciation

Ecological interaction, including competition for resources, often causes frequency-dependent disruptive selection, which, when accompanied by reproductive isolation, may act as driving forces of adaptive speciation. While adaptive dynamics models have added new perspectives to our understanding of the ecological dimensions of speciation processes, it remains an open question how best to incorporate and analyze genetic detail in such models. Conventional approaches, based on quantitative genetics theory, typically assume a unimodal character distribution and examine how its moments change over time. Such approximations inevitably fail when a character distribution becomes multimodal. Here, we propose a new approximation, oligomorphic dynamics, to the quantitative genetics of populations that include several morphs and that therefore exhibit multiple peaks in their character distribution. To this end, we first decompose the character distribution into a sum of unimodal distributions corresponding to individual morphs. Characterizing these morphs by their frequency (fraction of individuals belonging to each morph), position (mean character of each morph), and width (standard deviation of each morph), we then derive the coupled eco-evolutionary dynamics of morphs through a double Taylor expansion. We show that the demographic, convergence, and evolutionary stability of a population’s character distribution correspond, respectively, to the asymptotic stability of frequencies, positions, and widths under the oligomorphic dynamics introduced here. As first applications of oligomorphic dynamics theory, we analytically derive the effects (a) of the strength of disruptive selection on waiting times until speciation, (b) of mutation on conditions for speciation, and (c) of the fourth moments of competition kernels on patterns of speciation.     

Comparison of multiple molecular dynamics trajectories calculated for the drug-resistant HIV-1 integrase T66I/M154I catalytic domain

HIV-1 integrase (IN) is an essential enzyme for the viral replication and an interesting target for the design of new pharmaceuticals for multidrug therapy of AIDS. Single and multiple mutations of IN at residues T66, S153, or M154 confer degrees of resistance to several inhibitors that prevent the enzyme from performing its normal strand transfer activity. Four different conformations of IN were chosen from a prior molecular dynamics (MD) simulation on the modeled IN T66I/M154I catalytic core domain as starting points for additional MD studies. The aim of this article is to understand the dynamic features that may play roles in the catalytic activity of the double mutant enzyme in the absence of any inhibitor. Moreover, we want to verify the influence of using different starting points on the MD trajectories and associated dynamical properties. By comparison of the trajectories obtained from these MD simulations we have demonstrated that the starting point does not affect the conformational space explored by this protein and that the time of the simulation is long enough to achieve convergence for this system.     

Actomyosin contractility spatiotemporally regulates actin network dynamics in migrating cells

Coupling interactions among mechanical and biochemical factors are important for the realization of various cellular processes that determine cell migration. Although F-actin network dynamics has been the focus of many studies, it is not yet clear how mechanical forces generated by actomyosin contractility spatiotemporally regulate this fundamental aspect of cell migration. In this study, using a combination of fluorescent speckle microscopy and particle imaging velocimetry techniques, we perturbed the actomyosin system and examined quantitatively the consequence of actomyosin contractility on F-actin network flow and deformation in the lamellipodia of actively migrating fish keratocytes. F-actin flow fields were characterized by retrograde flow at the front and anterograde flow at the back of the lamellipodia, and the two flows merged to form a convergence zone of reduced flow intensity. Interestingly, activating or inhibiting actomyosin contractility altered network flow intensity and convergence, suggesting that network dynamics is directly regulated by actomyosin contractility. Moreover, quantitative analysis of F-actin network deformation revealed that the deformation was significantly negative and predominant in the direction of cell migration. Furthermore, perturbation experiments revealed that the deformation was a function of actomyosin contractility. Based on these results, we suggest that the actin cytoskeletal structure is a mechanically self-regulating system, and we propose an elaborate pathway for the spatiotemporal self-regulation of the actin cytoskeletal structure during cell migration. In the proposed pathway, mechanical forces generated by actomyosin interactions are considered central to the realization of the various mechanochemical processes that determine cell motility.     

Molecular dynamics simulation of hen egg white lysozyme: a test of the GROMOS96 force field against nuclear magnetic resonance data

Biomolecular force fields for use in molecular dynamics (MD) simulations of proteins, DNA, or membranes are generally parametrized against ab initio quantum-chemical and experimental data for small molecules. The application of a force field in a simulation of a biomolecular system, such as a protein in solution, may then serve as a test of the quality and transferability of the force field. Here, we compare various properties obtained from two MD simulations of the protein hen egg white lysozyme (HEWL) in aqueous solution using the latest version, GROMOS96, of the GROMOS force field and an earlier version, GROMOS87+, with data derived from nuclear magnetic resonance (NMR) experiments: NOE atom-atom distance bounds, (3)J(HNalpha)-coupling constants, and backbone and side-chain order parameters. The convergence of these quantities over a 2-ns period is considered, and converged values are compared to experimental ones. The GROMOS96 simulation shows better agreement with the NMR data and also with the X-ray crystal structure of HEWL than the GROMOS87+ simulation, which was based on an earlier version of the GROMOS force field.     

Developmental dynamics of ecomorphological convergence in a transcontinental lizard radiation

Phenotypic convergence has confounded evolutionary biologists for centuries, explained as adaptations to shared selective pressures, or alternatively, the result of limited developmental pathways. We tested the relative roles of adaptation and constraint in generating convergent cranial morphologies across a large lizard radiation, the Lacertidae, whose members inhabit diverse environments throughout the Old World and display high amounts of homoplasy associated with ecological niche. Using 3D X‐ray computed tomography, we quantified cranial shape variation associated with ontogeny, allometry, and ecology, covering all lacertid genera and one‐third of species diversity. Landmark‐based geometric morphometrics showed that cranial shape varied significantly among biomes, with substantial convergence among arid‐dwelling lineages. Comparisons of species cranial growth trajectories between biomes revealed that allometric postdisplacement, as evidenced by decreased elevation of a constant ontogenetic slope, drives the convergent paedomorphic appearance of independent arid‐dwelling forms. We hypothesize that observed heterochronic changes reflect temporal compression of ancestral life history in response to extreme environments, with associated phenotypes occurring as by‐products of adaptive shifts in reproductive investment. Although allometry has long been considered a developmental constraint, our results demonstrate that allometric flexibility during early ontogeny produces convergent ecomorphologies over vast temporal and spatial scales, thus dramatically obscuring underlying phylogenetic signals.     

Hybrid bounded error observers for uncertain bioreactor models

In this paper, we build bounded error observers for a common class of partially known bioreactor models. The main idea is to construct hybrid bounded observers “between” high gain observer, which has an adjustable convergence rate but requires perfect knowledge of the model, and asymptotic observer which is very robust towards uncertainty but has a fixed convergence rate. An hybrid bounded error observer which reconstructs the two state variables is constructed considering two steps: first step is similar to a high gain observer meaning that fast convergence rate but error depending on the knowledge of the model are obtained; second step is a switch to an observer similar to the asymptotic one meaning that fixed convergence rate towards an error as small as desired is obtained. Thus, a better convergence rate of estimated variables than the classical asymptotic observer is obtained.     

Multiply accelerated ReaxFF molecular dynamics: coupling parallel replica dynamics with collective variable hyper dynamics

ABSTRACT

To tackle the time scales required to study complex chemical reactions, methods performing accelerated molecular dynamics are necessary even with the recent advancement in high-performance computing. A number of different acceleration techniques are available. Here we explore potential synergies between two popular acceleration methods – Parallel Replica Dynamics (PRD) and Collective Variable Hyperdynamics (CVHD), by analysing the speedup obtained for the pyrolysis of n-dodecane. We observe that PRD?+?CVHD provides additional speedup to CVHD by reaching the required time scales for the reaction at an earlier wall-clock time. Although some speedup is obtained with the additional replicas, we found that the effectiveness of the inclusion of PRD is depreciated for systems where there is a dramatic increase in reaction rates induced by CVHD. Similar observations were made in the simulation of ethylene-carbonate/Li system, which is inherently more reactive than pyrolysis, indicate that the speedup obtained via the combination of the two acceleration methods can be generalised to most practical chemical systems.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Adaptive dynamics of competition for nutritionally complementary resources: character convergence, displacement, and parallelism

Consumers acquire essential nutrients by ingesting the tissues of resource species. When these tissues contain essential nutrients in a suboptimal ratio, consumers may benefit from ingesting a mixture of nutritionally complementary resource species. We investigate the joint ecological and evolutionary consequences of competition for complementary resources, using an adaptive dynamics model of two consumers and two resources that differ in their relative content of two essential nutrients. In the absence of competition, a nutritionally balanced diet rarely maximizes fitness because of the dynamic feedbacks between uptake rate and resource density, whereas in sympatry, nutritionally balanced diets maximize fitness because competing consumers with different nutritional requirements tend to equalize the relative abundances of the two resources. Adaptation from allopatric to sympatric fitness optima can generate character convergence, divergence, and parallel shifts, depending not on the degree of diet overlap but on the match between resource nutrient content and consumer nutrient requirements. Contrary to previous verbal arguments that suggest that character convergence leads to neutral stability, coadaptation of competing consumers always leads to stable coexistence. Furthermore, we show that incorporating costs of consuming or excreting excess nonlimiting nutrients selects for nutritionally balanced diets and so promotes character convergence. This article demonstrates that resource-use overlap has little bearing on coexistence when resources are nutritionally complementary, and it highlights the importance of using mathematical models to infer the stability of ecoevolutionary dynamics.     

Investigation of the conformational dynamics of the apo A2A adenosine receptor

The activation/deactivation processes for G-protein coupled receptors (GPCRs) have been computationally studied for several different classes, including rhodopsin, the β2 adrenergic receptor, and the M2 muscarinic receptor. Despite determined cocrystal structures of the adenosine A_2A receptor (A_2AAR) in complex with antagonists, agonists and an antibody, the deactivation process of this GPCR is not completely understood. In this study, we investigate the convergence of two apo simulations, one starting with an agonist-bound conformation (PDB: 3QAK)¹⁴ and the other starting with an antagonist-bound conformation (PDB: 3EML)¹¹. Despite the two simulations not completely converging, we were able to identify distinct intermediate steps of the deactivation process characterized by the movement of Y288^7.53 in the NPxxY motif. We find that Y288^7.53 contributes to the process by forming hydrogen bonds to residues in transmembrane helices 2 and 7 and losing these interactions upon full deactivation. Y197^5.58 also plays a role in the process by forming a hydrogen bond only once the side chain moves from the lipid interface to the middle of the helical bundle.