Single Peak Fitting Research Articles

Viral replication modes in single-peak fitness landscapes: A dynamical systems analysis

Positive-sense, single-stranded RNA viruses are important pathogens infecting almost all types of organisms. Experimental evidence from distributions of mutations and from viral RNA amplification suggest that these pathogens may follow different RNA replication modes, ranging from the stamping machine replication (SMR) to the geometric replication (GR) mode. Although previous theoretical work has focused on the evolutionary dynamics of RNA viruses amplifying their genomes with different strategies, little is known in terms of the bifurcations and transitions involving the so-called error threshold (mutation-induced dominance of mutants) and lethal mutagenesis (extinction of all sequences due to mutation accumulation and demographic stochasticity). Here we analyze a dynamical system describing the intracellular amplification of viral RNA genomes evolving on a single-peak fitness landscape focusing on three cases considering neutral, deleterious, and lethal mutants. We analytically derive the critical mutation rates causing lethal mutagenesis and error threshold, governed by transcritical bifurcations that depend on parameters α (parameter introducing the mode of replication), replicative fitness of mutants (k1), and on the spontaneous degradation rates of the sequences (ϵ). Our results relate the error catastrophe with lethal mutagenesis in a model with continuous populations of viral genomes. The former case involves dominance of the mutant sequences, while the latter, a deterministic extinction of the viral RNAs during replication due to increased mutation. For the lethal case the critical mutation rate involving lethal mutagenesis is μc=1−ɛ/α. Here, the SMR involves lower critical mutation rates, being the system more robust to lethal mutagenesis replicating closer to the GR mode. This result is also found for the neutral and deleterious cases, but for these later cases lethal mutagenesis can shift to the error threshold once the replication mode surpasses a threshold given by α=ϵ/k1.

Lattice parameter evolution during heating of Ti-45Al-7.5Nb-0.25/0.5C alloys under atmospheric and high pressures

Lattice strain evolution during deformation processing of Ti-Al alloys at high temperature is important in terms of its microstructural evolution and microstructural stability. It is shown here that careful evaluation of lattice parameters is critical for the understanding of thermal expansion, crystallographic order, chemical composition and response to pressure, allowing to identify phase transitions and segregation, in addition to the measurement of the more conventional quantities phase composition and order parameter. The lattice parameters of Ti-45Al-7.5Nb-0.25/0.5C (at. %) alloys were calculated using both Rietveld and single peak fitting methods from data obtained by in-situ synchrotron diffraction experiments at high temperature under atmospheric and high pressure respectively. The lattice strain evolution as a function of temperature in a Ti-45Al-7.5Nb-0.25C (at. %) alloy under high pressure was compared with that of a Ti-45Al-7.5Nb-0.5C (at. %) alloy heated under atmospheric pressure. The contribution of each of the four lattice strain factors is semi-quantitatively assessed in the temperature range investigated.

Solution of the Crow-Kimura model with changing population size and Allee effect.

The Crow-Kimura model is commonly used in the modeling of genetic evolution in the presence of mutations and associated selection pressures. We consider a modified version of the Crow-Kimura model, in which population sizes are not fixed and Allee saturation effects are present. We demonstrate the evolutionary dynamics in this system through an analytical approach, examining both symmetric and single-peak fitness landscape cases. Especially interesting are the dynamics of the populations near extinction. A special version of the model with saturation and degradation on the single-peak fitness landscape is investigated as a candidate of the Allee effect in evolution, revealing reduction tendencies of excessively large populations, and extinction tendencies for small populations. The analytical solutions for these dynamics are presented with accuracy O(1/N), where N is the number of nucleotides in the genome.

Decoupling of superposed textures in an electrically biased piezoceramic with a 100 preferred orientation

A method leveraging Rietveld full-pattern texture analysis to decouple induced domain texture from a preferred grain orientation is presented. The proposed method is demonstrated by determining the induced domain texture in a polar polymorph of 100 oriented 0.91Bi1/2Na1/2TiO3-0.07BaTiO3-0.02K0.5Na0.5NbO3. Domain textures determined using the present method are compared with results obtained via single peak fitting. Texture determined using single peak fitting estimated more domain alignment than that determined using the Rietveld based method. These results suggest that the combination of grain texture and phase transitions can lead to single peak fitting under or over estimating domain texture. While demonstrated for a bulk piezoelectric, the proposed method can be applied to quantify domain textures in multi-component systems and thin films.

In Situ X-ray Diffraction Study of the Tensile Deformation of U-5.8Nb Alloy

Abstract The in situ X-ray diffraction (XRD) was used during uniaxial tensile loading to study the lattice response and deformation behavior of α ″ U-5.8 wt% Nb (U5.8Nb) alloy. Diffraction patterns were recorded at incremental macroscopic strains until fracture of the sample, and the XRD data were analyzed by Rietveld method and single-peak fitting. The changes regulation of lattice parameter, diffraction peak intensity and interplanar spacing with applied stress during deformation were analyzed. It is found that the elastic deformation and the twinning deformation (reorientation) coexist in the initial stage of the first yield platform. The lattice response suggests that the remarkable anisotropy of elastic limit in various directions should be the genesis of double yield curve.

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics.

We formulate the Crow-Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton-Jacobi equation (HJE). We apply the method to the Crow-Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.

Adaptive evolution of molecular phenotypes

Molecular phenotypes link genomic information with organismic functions, fitness, and evolution. Quantitative traits are complex phenotypes that depend on multiple genomic loci. In this paper, we study the adaptive evolution of a quantitative trait under time-dependent selection, which arises from environmental changes or through fitness interactions with other co-evolving phenotypes. We analyze a model of trait evolution under mutations and genetic drift in a single-peak fitness seascape. The fitness peak performs a constrained random walk in the trait amplitude, which determines the time-dependent trait optimum in a given population. We derive analytical expressions for the distribution of the time-dependent trait divergence between populations and of the trait diversity within populations. Based on this solution, we develop a method to infer adaptive evolution of quantitative traits. Specifically, we show that the ratio of the average trait divergence and the diversity is a universal function of evolutionary time, which predicts the stabilizing strength and the driving rate of the fitness seascape. From an information-theoretic point of view, this function measures the macro-evolutionary entropy in a population ensemble, which determines the predictability of the evolutionary process. Our solution also quantifies two key characteristics of adapting populations: the cumulative fitness flux, which measures the total amount of adaptation, and the adaptive load, which is the fitness cost due to a population's lag behind the fitness peak.

Resolving structural contributions to the electric-field-induced strain in lead-free (1 − x)Ba(Zr0.2Ti0.8)O3 − x(Ba0.7Ca0.3)TiO3 piezoceramics

The large signal macroscopic strain response during an applied bipolar electric field is calculated from field-dependent in situ XRD data using expressions for, first, a lattice strain contribution and, second, a ferroelastic strain contribution. The lattice strain contribution is estimated using a weighted average of the lattice strains for the observed reflections along the field direction. The ferroelastic strain contribution is calculated by integrating the lattice parameter changes weighted with the ferroelastic domain distribution over all orientations relative to the direction of the applied field. Structural parameters are determined by means of both single peak fitting and Rietveld refinements. A large ferroelastic contribution is found for tetragonal (1−x)Ba(Zr0.2Ti0.8)O3−x(Ba0.7Ca0.3)TiO3 materials that appears to be the dominant origin for the large signal macroscopic strain. The strong changes in lattice parameters and the decrease in tetragonality as a function of orientation and electric field also indicate a large influence of microstructure constraints on the macroscopic strain response. The total strain calculated from X-ray diffraction using both methods is in good agreement with macroscopic strain measurements.

Analysis of Preferred Orientations in Linear Friction Welded (LFW) Aluminium Alloy Specimens using “One-shot” Multi-element Energy Dispersive Synchrotron X-ray Diffraction

Linear Friction Welding (LFW) has tremendous potential for joining components from similar and dissimilar materials, avoiding material melting and introducing minimal distortion and only moderate levels of residual stress. However, the significant amount of attendant shear introduces preferred crystal orientations that have not yet been well studied. The “one-shot” approach to the interpretation of multi-element energy-dispersive X-ray powder diffraction data allows preferred orientation analysis without any sample preparation (cutting or polishing) or sample rotation. The key step for texture analysis by X-ray powder diffraction is the derivation of the orientation distribution function (ODF) from experimental data. Matlab toolbox “MTEX” provides a powerful function “calcODF” based on the harmonics method for this purpose. In the study reported in this paper, energy dispersive X-ray diffraction patterns were collected using the “horseshoe” multi-element energy-dispersive Ge detector installed on the JEEP beamline at Diamond Light Source. A single exposure was used for each gauge volume of interest, and a line was scanned across an Aluminum 2024 alloy LFW sample. The patterns were converted into raw pole figures through single peak fitting and equal area projection. The ODF calculation was performed based on these pole figures using Matlab toolbox “MTEX”. As a result, full pole figures obtained after ODF calculation were obtained. These are presented and discussed. The results show that the thermal-mechanical processes that occur during the LFW process lead to significant modification of the orientation distribution, but cause only moderate changes in the texture index.

Finding the boundary between evolutionary basins of attraction, and implications forWright’s fitness landscape analogy

In 1932 Wright introduced the notion of the fitness landscape. By analogy with a physicallandscape, whose gradient predicts a rolling marble’s spatial trajectory, the contours of thefitness landscape are meant to predict an evolving population’s genetic trajectory. Wright’schief interest was in the possibility that mutational interactions might frustrate naturalselection, giving rise to multiple maxima on the fitness landscape. Here we study adynamical system over the state space defined by allele frequencies and linkagedisequilibria between alleles. We first analytically locate the saddle between basins ofattraction in infinite-sized populations evolving under the influence of selectionand recombination for the simplest two-locus case. We further show numericallythat the boundary between basins is approximately linear with respect to linkagedisequilibrium, though not allele frequency. We also employ this framework to developnovel perspectives on two venerable results for single-peaked fitness landscapes.Finally we sought the potential function whose contours would predict evolutionarytrajectories through this state space. Importantly not every dynamical system canbe described by a potential function, and the present problem is provably onesuch case. Thus in the parlance of Wright’s analogy, in locating the floor of thefitness valley we have lost the landscape, and this conclusion is not limited to ourchoice of parameterization, nor of problem. This result motivates us to carefullyreview the formal implications and requirements of this widely used analogy.

ANTIBIOTIC RESISTANCE AND STRESS IN THE LIGHT OF FISHER'S MODEL

The role of mutations in evolution depends upon the distribution of their effects on fitness. This distribution is likely to depend on the environment. Indeed genotype-by-environment interactions are key for the process of local adaptation and ecological specialization. An important trait in bacterial evolution is antibiotic resistance, which presents a clear case of change in the direction of selection between environments with and without antibiotics. Here, we study the distribution of fitness effects of mutations, conferring antibiotic resistance to Escherichia coli, in benign and stressful environments without drugs. We interpret the distributions in the light of a fitness landscape model that assumes a single fitness peak. We find that mutation effects (s) are well described by a shifted gamma distribution, with a shift parameter that reflects the distance to the fitness peak and varies across environments. Consistent with the theoretical predictions of Fisher's geometrical model, with a Gaussian relationship between phenotype and fitness, we find that the main effect of stress is to increase the variance in s. Our findings are in agreement with the results of a recent meta-analysis, which suggest that a simple fitness landscape model may capture the variation of mutation effects across species and environments.

On the Relation Between the Eigen Model and the Asexual Wright–Fisher Model

We show that the Eigen model and the asexual Wright-Fisher model can be obtained as two different limit cases of a single stochastic model. This result is used to enlighten the mathematical similarities and differences among these two models.The two key concepts introduced with the Eigen model, the error threshold and the quasispecies, are not invalidated by these differences, so that they are naturally present also in population genetics models. According to this fact, we use the classical diploid mutation-selection equation and the single peak fitness approximation to obtain the error threshold for sexual diploids. Finally, we perform the same analysis in the asexual case and compare the results.

In situ room temperature tensile deformation of a 1% CrMoV bainitic steel using synchrotron and neutron diffraction

Neutron and synchrotron X-ray diffraction spectra have been acquired during room temperature tensile deformation of a creep-resistant bainitic 1% CrMoV steel, in order to study the evolution of internal microstresses and load-sharing mechanisms between the ferrite matrix and the various carbides. Cementite takes load from the plastifying matrix at the onset of macroscopic plasticity resulting in residual interphase stresses. Single peak fitting indicates an elastic anisotropic behaviour of cementite.

Reciprocal sign epistasis is a necessary condition for multi-peaked fitness landscapes

Having multiple peaks within fitness landscapes critically affects the course of evolution, but whether their presence imposes specific requirements at the level of genetic interactions remains unestablished. Here we show that to exhibit multiple fitness peaks, a biological system must contain reciprocal sign epistatic interactions, which are defined as genetic changes that are separately unfavorable but jointly advantageous. Using Morse theory, we argue that it is impossible to formulate a sufficient condition for multiple peaks in terms of local genetic interactions. These findings indicate that systems incapable of reciprocal sign epistasis will always possess a single fitness peak. However, reciprocal sign epistasis should be pervasive in nature as it is a logical consequence of specificity in molecular interactions. The results thus predict that specific molecular interactions may yield multiple fitness peaks, which can be tested experimentally.

Effect of the SOS Response on the Mean Fitness of Unicellular Populations: A Quasispecies Approach

The goal of this paper is to develop a mathematical model that analyzes the selective advantage of the SOS response in unicellular organisms. To this end, this paper develops a quasispecies model that incorporates the SOS response. We consider a unicellular, asexually replicating population of organisms, whose genomes consist of a single, double-stranded DNA molecule, i.e. one chromosome. We assume that repair of post-replication mismatched base-pairs occurs with probability , and that the SOS response is triggered when the total number of mismatched base-pairs is at least . We further assume that the per-mismatch SOS elimination rate is characterized by a first-order rate constant . For a single fitness peak landscape where the master genome can sustain up to mismatches and remain viable, this model is analytically solvable in the limit of infinite sequence length. The results, which are confirmed by stochastic simulations, indicate that the SOS response does indeed confer a fitness advantage to a population, provided that it is only activated when DNA damage is so extensive that a cell will die if it does not attempt to repair its DNA.

Recombination in one- and two-dimensional fitness landscapes

We consider many-site mutation-recombination models of molecular evolution, where fitness is a function of a Hamming distance from one (one-dimensional case) or two (two-dimensional case) sequences. For the one-dimensional case, we calculate the population distribution dynamics for a model with zero fitness and an arbitrary symmetric initial distribution and find an error threshold transition point in the single-peak fitness model for a given initial symmetric distribution. We calculate the recombination period in the case of a single-peak fitness function, when the original population is located at one sequence, at some Hamming distance from the peak configuration. Steady-state fitness is calculated with finite genome length corrections. We derive analytical equations for the two-dimensional mutation-recombination model.

Relaxation Property and Stability Analysis of the Quasispecies Models

The relaxation property of both Eigen model and Crow–Kimura model with a single peak fitness landscape is studied from phase transition point of view. We first analyze the eigenvalue spectra of the replication mutation matrices. For sufficiently long sequences, the almost crossing point between the largest and second-largest eigenvalues locates the error threshold at which critical slowing down behavior appears. We calculate the critical exponent in the limit of infinite sequence lengths and compare it with the result from numerical curve fittings at sufficiently long sequences. We find that for both models the relaxation time diverges with exponent 1 at the error (mutation) threshold point. Results obtained from both methods agree quite well. From the unlimited correlation length feature, the first order phase transition is further confirmed. Finally with linear stability theory, we show that the two model systems are stable for all ranges of mutation rate. The Eigen model is asymptotically stable in terms of mutant classes, and the Crow–Kimura model is completely stable.

Quasispecies theory for horizontal gene transfer and recombination

We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasispecies theory to modern biology are analytically solvable. For two-parent recombination, we find two selected phases, one of which is spectrally rigid. We present exact analytical formulas for the equilibrium mean fitness of the population, in terms of a maximum principle, which are generally applicable to any permutation invariant replication rate function. For smooth fitness landscapes, we show that when positive epistatic interactions are present, recombination or horizontal gene transfer introduces a mild load against selection. Conversely, if the fitness landscape exhibits negative epistasis, horizontal gene transfer or recombination introduces an advantage by enhancing selection towards the fittest genotypes. These results prove that the mutational deterministic hypothesis holds for quasispecies models. For the discontinuous single sharp peak fitness landscape, we show that horizontal gene transfer has no effect on the fitness, while recombination decreases the fitness, for both the parallel and the Eigen models. We present numerical and analytical results as well as phase diagrams for the different cases.

S28 Elastic-Plastic Deformation of a Polycrystalline Titanium Alloy Studied In Situ by Energy-Dispersive Synchrotron X-ray Diffraction

Elastic lattice strain in polycrystalline assemblies evolves as a function of applied load due to elastic and plastic deformation mechanisms. The detailed nature of this deformation is complex and depends on the local arrangement of grains, their morphology and orientation, etc. To obtain mesoscopic (intermediate) scale description of polycrystal deformation, one approach that can be adopted is to measure the average lattice strain response of grain groups sharing common orientation by diffraction as the sample is subjected to increasing uniaxial load [1]. This reveals the anisotropy of the response of individual reflections (orientation grain groups), and enables the determination of the inhomogeneous fields of elastic and eventually plastic strain, if the measurements are combined with suitable interpretation and modeling approaches. Regions of localized macroscopic plasticity may be identified, and the prediction of fatigue crack initiation may also be addressed using energy dissipation at the micro-scale [2]. In the present study in situ loading and synchrotron X-ray diffraction experiments on Ti-6Al-4V alloy were performed. Multiple diffraction peaks were obtained from the experiments and both single peak fitting and Pawley pattern refinement were used for interpretation. The calculation of stress-strain response of hkl lattice planes based on the a and c lattice parameters obtained from Pawley pattern refinement was in good agreement with the results obtained by single peak fitting. The orientation dependence of polycrystal lattice stiffness and strain is discussed and a modified profile function for pattern refinement is proposed to provide an improved description of anisotropic behavior. Diffraction peak broadening was considered and the correlation between peak width and plastic strain was found (Fig 1). The intergranular strain responses for differently oriented crystallites obtained from the diffraction experiments were compared to the crystal plasticity finite element model predictions and used for validation of the model.

Dynamic Recrystallization and Dynamic Precipitation Behaviors of a 17Ni–0.2C Martensite Steel Studied by In Situ Neutron Diffraction

The austenite precipitation and dynamic recrystallization behaviors of 17Ni‐0.2C martensite steel during isothermal tempering at 500°C followed by isothermal warm compression at 8.3� 10 � 4 /s were investigated in situ using neutron diffraction. The Rietveld analysis of Time-Of-Flight neutron diffraction profiles revealed that the austenite amount increased by about 1.5% during 10 min isothermal holding while the warm compression at 500°C accelerated the austenite precipitation. Splitting of the austenite (111) peak occurred and then disappeared during warm compression. Based on integrated intensities obtained by single peak fitting of neutron spectra, it was found from the axial neutron spectra that the ferrite (110) peak intensity rapidly decreased and the ferrite (200) and (211) peak intensities slowly increased during warm compression; the occurrence of dynamic recrystallization led to an evident deviation from this trend. The comparison between lattice strains and texture indexes of austenite and ferrite suggested that the austenite was harder than the ferrite at 500°C. Thus, heterogeneous deformation occurred in the ferrite, leading to accelerated dynamic recrystallization.