Nonlinear Scaling Relationship Research Articles

Allometry of behavior and niche differentiation among congeneric African antelopes

AbstractSize‐structured differences in resource use stabilize species coexistence in animal communities, but what behavioral mechanisms underpin these niche differences? Behavior is constrained by morphological and physiological traits that scale allometrically with body size, yet the degree to which behaviors exhibit allometric scaling remains unclear; empirical datasets often encompass broad variation in environmental context and phylogenetic history, which complicates the detection and interpretation of scaling relationships between size and behavior. We studied the movement and foraging behaviors of three sympatric, congeneric spiral‐horned antelope species (Tragelaphusspp.) that differ in body mass—bushbuck (26–40 kg), nyala (57–83 kg), and kudu (80–142 kg)—in an African savanna ecosystem where (i) food was patchily distributed due to ecosystem engineering by fungus‐farming termites and (ii) predation risk was low due to the extirpation of several large carnivores. Because foraging behavior is directly linked to traits that scale allometrically with size (e.g., metabolic rate, locomotion), we hypothesized that habitat use and diet selection would likewise exhibit nonlinear scaling relationships. All three antelope species selected habitat near termitaria, which are hotspots of abundant, high‐quality forage. Experimental removal of forage from termite mounds sharply reduced use of those mounds by bushbuck, confirming that habitat selection was resource driven. Strength of selection for termite mounds scaled negatively and nonlinearly with body mass, as did recursion (frequency with which individuals revisited locations), whereas home‐range area and mean step length scaled positively and nonlinearly with body mass. All species disproportionately ate mound‐associated plant taxa; nonetheless, forage selectivity and dietary composition, richness, and quality all differed among species, reflecting the partitioning of shared food resources. Dietary protein exhibited the theoretically predicted negative allometric relationship with body mass, whereas digestible‐energy content scaled positively. Our results demonstrate cryptic size‐based separation along spatial and dietary niche axes—despite superficial similarities among species—consistent with the idea that body‐size differentiation is driven by selection for divergent resource‐acquisition strategies, which in turn underpin coexistence. Foraging and space‐use behaviors were nonlinearly related to body mass, supporting the hypothesis that behavior scales allometrically with size. However, explaining the variable functional forms of these relationships is a challenge for future research.

Rediscovering the Scaling Law of Urban Land from a Multi-Scale Perspective—A Case Study of Wuhan

The law of urban scaling implies that there is a universally applicable nonlinear scaling relationship between population size and urban indicators, which is a method of quantitative analysis that can reflect the growth law and internal logic of the urban system. However, most present research is conducted at the municipal scale, and studies of scaling law in the inner-city system are scarce, especially from the perspective of compact urban form development. The goal of this paper is to discover the scaling law within urban systems from a multi-scale perspective. Through the empirical analysis of Wuhan, this paper examines the internal scale law of the urban system from the municipal and district scales. Moreover, we use the landscape expansion index to perform spatial autocorrelation analysis. In this way, we assess the relationship between the compactness of urban morphological development and the urban scaling law. The results indicate that the temporal scaling law on the city scale has a more significant linear law than the single-year scaling law. The analysis also shows the scaling law relationship within the inner-city system. Nevertheless, there is a deviation between the temporal scaling law and the cross-section scaling law. Namely, the time series development of a district does not follow the section scaling law of the urban system. Furthermore, the urban scaling law shows a negative correlation with the compactness of the urban form development. It is crucial to understand the current economic development and resource endowment of an urban system in the urbanization process, as it significantly contributes to urban development and regional coordinated planning.

Mapping Active Site Geometry to Activity in Immobilized Frustrated Lewis Pair Catalysts.

The immobilization of molecular catalysts imposes spatial constraints on their active site. We reveal that in bifunctional catalysis such constraints can also be utilized as an appealing handle to boost intrinsic activity through judicious control of the active site geometry. To demonstrate this, we develop a pragmatic approach, based on nonlinear scaling relationships, to map the spatial arrangements of the acid–base components of frustrated Lewis pairs (FLPs) to their performance in the catalytic hydrogenation of CO2. The resulting activity map shows that fixing the donor–acceptor centers at specific distances and locking them into appropriate orientations leads to an unforeseen many‐fold increase in the catalytic activity of FLPs compared to their unconstrained counterparts.

Mapping Active Site Geometry to Activity in Immobilized Frustrated Lewis Pair Catalysts

AbstractThe immobilization of molecular catalysts imposes spatial constraints on their active site. We reveal that in bifunctional catalysis such constraints can also be utilized as an appealing handle to boost intrinsic activity through judicious control of the active site geometry. To demonstrate this, we develop a pragmatic approach, based on nonlinear scaling relationships, to map the spatial arrangements of the acid–base components of frustrated Lewis pairs (FLPs) to their performance in the catalytic hydrogenation of CO2. The resulting activity map shows that fixing the donor–acceptor centers at specific distances and locking them into appropriate orientations leads to an unforeseen many‐fold increase in the catalytic activity of FLPs compared to their unconstrained counterparts.

Neuroanatomical norms in the UK Biobank: The impact of allometric scaling, sex, and age.

Few neuroimaging studies are sufficiently large to adequately describe population‐wide variations. This study's primary aim was to generate neuroanatomical norms and individual markers that consider age, sex, and brain size, from 629 cerebral measures in the UK Biobank (N = 40,028). The secondary aim was to examine the effects and interactions of sex, age, and brain allometry—the nonlinear scaling relationship between a region and brain size (e.g., total brain volume)—across cerebral measures. Allometry was a common property of brain volumes, thicknesses, and surface areas (83%) and was largely stable across age and sex. Sex differences occurred in 67% of cerebral measures (median |β| = .13): 37% of regions were larger in males and 30% in females. Brain measures (49%) generally decreased with age, although aging effects varied across regions and sexes. While models with an allometric or linear covariate adjustment for brain size yielded similar significant effects, omitting brain allometry influenced reported sex differences in variance. Finally, we contribute to the reproducibility of research on sex differences in the brain by replicating previous studies examining cerebral sex differences. This large‐scale study advances our understanding of age, sex, and brain allometry's impact on brain structure and provides data for future UK Biobank studies to identify the cerebral regions that covary with specific phenotypes, independently of sex, age, and brain size.

Assessing the Attraction of Cities on Venture Capital From a Scaling Law Perspective

Cities are centers for the integration of capital and incubators of inventions. Attracting venture capital (VC) is of great importance for cities to advance in innovative technology and business models towards a sustainable and prosperous future. Yet we still lack a quantitative understanding of the relationship between urban characteristics and VC activities. In this paper, we find a clear nonlinear scaling relationship between VC activities and the urban population of Chinese cities. In such nonlinear systems, the widely applied linear per capita indicators would be either biased to larger cities or smaller cities depends on whether it is superlinear or sublinear, while the residual of cities relative to the prediction of scaling law is a more objective and scale-invariant metric. Such a metric can distinguish the effects of local dynamics and scaled growth induced by the change of population size. The spatiotemporal evolution of such metrics on VC activities reveals three distinct groups of cities, two of which stand out with increasing and decreasing trends, respectively. The taxonomy results together with spatial analysis also signify different development modes between large urban agglomeration regions. Besides, we notice the evolution of scaling exponents on VC activities are of much larger fluctuations than on socioeconomic output of cities, and a conceptual model that focuses on the growth dynamics of different sized cities can well explain it, which we assume would be general to other scenarios.

Functional exploration of heterotrimeric kinesin-II in IFT and ciliary length control in Chlamydomonas.

Heterodimeric motor organization of kinesin-II is essential for its function in anterograde IFT in ciliogenesis. However, the underlying mechanism is not well understood. In addition, the anterograde IFT velocity varies significantly in different organisms, but how this velocity affects ciliary length is not clear. We show that in Chlamydomonas motors are only stable as heterodimers in vivo, which is likely the key factor for the requirement of a heterodimer for IFT. Second, chimeric CrKinesin-II with human kinesin-II motor domains functioned in vitro and in vivo, leading to a ~ 2.8 fold reduced anterograde IFT velocity and a similar fold reduction in IFT injection rate that supposedly correlates with ciliary assembly activity. However, the ciliary length was only mildly reduced (~15%). Modeling analysis suggests a nonlinear scaling relationship between IFT velocity and ciliary length that can be accounted for by limitation of the motors and/or its ciliary cargoes, e.g. tubulin.

Adjusting for allometric scaling in ABIDE I challenges subcortical volume differences in autism spectrum disorder.

Inconsistencies across studies investigating subcortical correlates of autism spectrum disorder (ASD) may stem from small sample size, sample heterogeneity, and omitting or linearly adjusting for total brain volume (TBV). To properly adjust for TBV, brain allometry—the nonlinear scaling relationship between regional volumes and TBV—was considered when examining subcortical volumetric differences between typically developing (TD) and ASD individuals. Autism Brain Imaging Data Exchange I (ABIDE I; N = 654) data was analyzed with two methodological approaches: univariate linear mixed effects models and multivariate multiple group confirmatory factor analyses. Analyses were conducted on the entire sample and in subsamples based on age, sex, and full scale intelligence quotient (FSIQ). A similar ABIDE I study was replicated and the impact of different TBV adjustments on neuroanatomical group differences was investigated. No robust subcortical allometric or volumetric group differences were observed in the entire sample across methods. Exploratory analyses suggested that allometric scaling and volume group differences may exist in certain subgroups defined by age, sex, and/or FSIQ. The type of TBV adjustment influenced some reported volumetric and scaling group differences. This study supports the absence of robust volumetric differences between ASD and TD individuals in the investigated volumes when adjusting for brain allometry, expands the literature by finding no group difference in allometric scaling, and further suggests that differing TBV adjustments contribute to the variability of reported neuroanatomical differences in ASD.

Functional Exploration of Heterotrimeric Kinesin-Ii on Intraflagellar Transport and Ciliary Length in <i>Chlamydomonas reinhardtii</i>

Heterotrimeric kinesin-2 (kinesin-II) serves as the anterograde motor for intraflagellar transport (IFT) that is essential for ciliary assembly. The functional significance of requiring two non-identical motor subunits in kinesin-II for IFT is not clear. In addition, the velocity of anterograde IFT varies significantly in different organisms, e.g. ~2.2 mm s-1 in Chlamydomonas and ~0.5 mm s-1 in mammalian cells. The mechanism underlying this difference and the impact of kinesin-II velocity on IFT and ciliary assembly remain elusive. To address these questions, we have studied human (HsKinesin-II), Chlamydomonas (CrKinesin-II) and chimeric kinesin-IIs in vitro and in vivo. We showed that the motor subunits (FLA10 and FLA8) of CrKinesin-II could self-interact respectively and the resulting homodimeric motors with two identical heads could function in vitro and in vivo. However, these motors could not interact with KAP (for FLA8) or with IFT complex (for FLA10), which necessitates heterodimer formation. Motility analyses of Crkinesin-II, HsKinesin-II and chimeric kinesin-IIs in vitro and in vivo suggest that the difference in the velocity of anterograde IFT is accounted for by the species-specificity of kinesin-II. Expressing the chimeric kinesin-II (KIF3B’/FLA10) reduced the velocity of anterograde IFT in Chlamydomonas by almost three-fold, which led to a significant reduction in IFT injection rate but only a mild decrease in ciliary length (~15%). Finally, theoretical modeling analysis reveals a non-linear scaling relationship between IFT velocity and ciliary length, implying a negative feedback in which the level of ciliary IFT proteins, in return, regulates the IFT injection rate.

Modelling the metabolism: allometric relationships between total daily energy expenditure, body mass, and height.

Accurately predicting energy requirements form a critical component for initializing dynamic mathematical models of metabolism. The majority of such existing estimates rely on linear regression models that predict total daily energy expenditure (TDEE) from age, gender, height, and body mass, however, there is evidence these predictors obey a power function. Baseline, free-living TDEE measured by doubly labeled water (DLW) in 20 studies with no overlapping subjects were obtained from the core lab at the University of Chicago and the University of Wisconsin-Madison (N = 2501 adults, 628 males, 1873 females). Linear regression models of log-transformed equations of the form: [Formula: see text] and [Formula: see text] were developed to determine the values of the exponents of body mass (M (kg)) and height (H (cm)) along with a gender effect (Sex). A nonlinear curve fit was performed to develop a power model that also includes age [Formula: see text]. The power for body mass, β1 = 0.45 and the power for height was β2 = 1.52 in the database with both genders combined. Adding gender reduced these to β1 = 0.43 and β2 = 1.04. All terms were significant (p < 0.01) except for height when including gender. The powers for height in the additive gender-specific models were both closer to 1 and the power for body mass was similar across all models ranging between 0.41 and 0.57. A nonlinear scaling relationship was found to hold for body mass and needs to be considered when adjusting TDEE for body mass or predicting human energy requirements as a function of body mass especially in individuals with obesity.

Matrix Mechanosensing: From Scaling Concepts in ’Omics Data to Mechanisms in the Nucleus, Regeneration, and Cancer

Many of the most important molecules of life are polymers. In animals, the most abundant of the proteinaceous polymers are the collagens, which constitute the fibrous matrix outside cells and which can also self-assemble into gels. The physically measurable stiffness of gels, as well as tissues, increases with the amount of collagen, and cells seem to sense this stiffness. An understanding of this mechanosensing process in complex tissues, including fibrotic disease states with high collagen, is now utilizing 'omics data sets and is revealing polymer physics-type, nonlinear scaling relationships between concentrations of seemingly unrelated biopolymers. The nuclear structure protein lamin A provides one example, with protein and transcript levels increasing with collagen 1 and tissue stiffness, and with mechanisms rooted in protein stabilization induced by cytoskeletal stress. Physics-based models of fibrous matrix, cytoskeletal force dipoles, and the lamin A gene circuit illustrate the wide range of testable predictions emerging for tissues, cell cultures, and even stem cell-based tissue regeneration. Beyond the epigenetics of mechanosensing, the scaling in cancer of chromosome copy number variations and other mutations with tissue stiffness suggests that genomic changes are occurring by mechanogenomic processes that now require elucidation.

(Invited) Electrolysis with Plasma Cathodes: Modeling and Experiments to Understand the Electrochemical Interface

In the conventional picture of electrolysis, two metal electrodes are submerged into an electrolyte solution and an applied voltage drives redox reactions at the cathode and anode, respectively. Plasma electrolysis, also called glow discharge electrolysis, changes this convention by replacing one or both of the solid electrodes with a plasma or gas discharge. Reduction and/or oxidation reactions, therefore, are no longer mediated by a solid/liquid interface but by a gaseous/liquid interface, requiring that free charges traverse from the gas to liquid and vice versa. The plasma cathode configuration replaces the typical metal cathode with a plasma, such that free gaseous electrons in the plasma must be injected into the solution phase to initiate reduction reactions at the “cathode”. Recently, this approach to electrolysis has received significant attention for nanomaterials synthesis and emerging applications include the ‘catalyst-free’ electrochemical processing of carbon dioxide (CO2) into useful chemicals. However, there are significant questions that persist about the nature of the electron reactions at the plasma-liquid interface. Recent studies have confirmed that the plasma electrons solvate in the solution before reacting with other dissolved species, and interfacial measurements of the absorption spectrum show that it is similar to, but different from, the standard solvated electron spectrum measured using radiolysis techniques. This anomaly suggests that the plasma-liquid interface is a wholly different environment for solvated electrons from that they experience in bulk radiolysis. To understand this interface, a model of the interfacial electrostatic Debye layer was constructed by considering both the electrostatic potential variation on the plasma side and the liquid side along with electron injection across the liquid interface. This analytical model predicts a non-linear scaling relationship between the plasma current density and electrolyte conductivity, and experimental measurements confirm the model. Extracted values from the model, including the interfacial electric field (~105 V/m) and the electrolyte cation concentration, offer insight into the electrochemical interface, but do not fully resolve questions regarding the anomalous absorption spectrum of plasma-injected solvated electrons.

Interfacial stiffening of polymer thin films under nanoconfinement

Changes in the mechanical properties of polymer thin films due to substrate and free surfaces determine the performance of soft nanomaterials. Here we systematically investigate how these interfaces influence the elasticity of supported polymer thin films. Our simulation results demonstrate that the substrate–film interfacial interaction has a strong effect on the elastic response of the films, in which the modulus increases with the strength of interfacial interaction with a nonlinear scaling relationship. The investigation of the local stiffness within the film reveals that there exists an interfacial stiffened region near the substrate and a soft region with reduced stiffness near the free surface. Both regions exhibit similar length scales on the order of nanometers. Our analyses explain the molecular origins of recently observed interfacial stiffening and softening phenomena in polymer thin films, and provide a high-resolution description of the local stiffness gradients in polymer thin films.

Dependence of Polymer Thin Film Adhesion Energy on Cohesive Interactions between Chains

Substrate–film interfacial properties play a fundamental role in the adhesion of glassy polymer thin films onto supporting substrates. Utilizing steered molecular dynamics (SMD) simulations, here we uncover that the cohesive noncovalent forces between polymer chains also have a significant effect on the adhesive properties of polymer films in contact with flat substrates. We demonstrate that weaker interchain interactions, all else being the same, can induce higher adhesion energy within the interface. Three different regimes in the adhesion energy profile can be characterized by a nonlinear scaling relationship. In the weak substrate–film interaction regime, the adhesion energy of the film exhibits near independence of cohesive force variations, and the entropic contributions to the surface free energy are consequential. In the intermediate regime, films with weaker cohesive forces exhibit higher adhesion energy due to the ability of polymer chains to pack more effectively in the interfacial region, thereby increasing the adhesive interaction density. In the strong interaction regime, the adhesion energy increases linearly with the adhesive interaction strength because of saturation of local packing in the interfacial region. These findings corroborate recent polymer dewetting observations that have hinted on the importance of local relaxation and packing effects on interfacial properties.

Linking phenological shifts to species interactions through size‐mediated priority effects

Interannual variation in seasonal weather patterns causes shifts in the relative timing of phenological events of species within communities, but we currently lack a mechanistic understanding of how these phenological shifts affect species interactions. Identifying these mechanisms is critical to predicting how interannual variation affects populations and communities. Species' phenologies, particularly the timing of offspring arrival, play an important role in the annual cycles of community assembly. We hypothesize that shifts in relative arrival of offspring can alter interspecific interactions through a mechanism called size-mediated priority effects (SMPE), in which individuals that arrive earlier can grow to achieve a body size advantage over those that arrive later. In this study, we used an experimental approach to isolate and quantify the importance of SMPE for species interactions. Specifically, we simulated shifts in relative arrival of the nymphs of two dragonfly species to determine the consequences for their interactions as intraguild predators. We found that shifts in relative arrival altered not only predation strength but also the nature of predator-prey interactions. When arrival differences were great, SMPE allowed the early arriver to prey intensely upon the late arriver, causing exclusion of the late arriver from nearly all habitats. As arrival differences decreased, the early arriver's size advantage also decreased. When arrival differences were smallest, there was mutual predation, and the two species coexisted in similar abundances across habitats. Importantly, we also found a nonlinear scaling relationship between shifts in relative arrival and predation strength. Specifically, small shifts in relative arrival caused large changes in predation strength while subsequent changes had relatively minor effects. These results demonstrate that SMPE can alter not only the outcome of interactions but also the demographic rates of species and the structure of communities. Elucidating the mechanisms that link phenological shifts to species interactions is crucial for understanding the dynamics of seasonal communities as well as for predicting the effects of climate change on these communities.

Chapter Three - Measuring Changes in Plasmodium falciparum Transmission: Precision, Accuracy and Costs of Metrics

As malaria declines in parts of Africa and elsewhere, and as more countries move towards elimination, it is necessary to robustly evaluate the effect of interventions and control programmes on malaria transmission. To help guide the appropriate design of trials to evaluate transmission-reducing interventions, we review 11 metrics of malaria transmission, discussing their accuracy, precision, collection methods and costs and presenting an overall critique. We also review the nonlinear scaling relationships between five metrics of malaria transmission: the entomological inoculation rate, force of infection, sporozoite rate, parasite rate and the basic reproductive number, R0. Our chapter highlights that while the entomological inoculation rate is widely considered the gold standard metric of malaria transmission and may be necessary for measuring changes in transmission in highly endemic areas, it has limited precision and accuracy and more standardised methods for its collection are required. In areas of low transmission, parasite rate, seroconversion rates and molecular metrics including MOI and mFOI may be most appropriate. When assessing a specific intervention, the most relevant effects will be detected by examining the metrics most directly affected by that intervention. Future work should aim to better quantify the precision and accuracy of malaria metrics and to improve methods for their collection.

Back-and-forth micromotion of aqueous droplets in a dc electric field

Recently, it was reported that an aqueous droplet in an oil phase exhibited rhythmic back-and-forth motion under stationary dc voltage on the order of 100 V. Here, we demonstrate that the threshold voltage for inducing such oscillation is successfully decreased to the order of 10 V through downsizing of the experimental system. Notably, the threshold electric field tends to decrease with a nonlinear scaling relationship accompanied by the downsizing. We derive a simple theoretical model to interpret the system size dependence of the threshold voltage. This model equation suggests the unique effect of additional noise, which is qualitatively characterized as a coherent resonance by an actual experiment as a kind of coherent resonance. Our result would provide insight into the construction of micrometer-sized self-commutating motors and actuators in microfluidic and micromechanical devices.

Functional morphology of the feeding apparatus and evolution of proboscis length in metalmark butterflies (Lepidoptera: Riodinidae)

An assessment of the anatomical costs of extremely long proboscid mouthparts can contribute to the understanding of the evolution of form and function in the context of insect feeding behaviour. An integrative analysis of expenses relating to an exceptionally long proboscis in butterflies includes all organs involved in fluid feeding, such as the proboscis plus its musculature, sensilla, and food canal, as well as organs for proboscis movements and the suction pump for fluid uptake. In the present study, we report a morphometric comparison of derived long-tongued (proboscis approximately twice as long as the body) and short-tongued Riodinidae (proboscis half as long as the body), which reveals the non-linear scaling relationships of an extremely long proboscis. We found no elongation of the tip region, low numbers of proboscis sensilla, short sensilla styloconica, and no increase of galeal musculature in relation to galeal volume, but a larger food canal, as well as larger head musculature in relation to the head capsule. The results indicate the relatively low extra expense on the proboscis musculature and sensilla equipment but significant anatomical costs, such as reinforced haemolymph and suction pump musculature, as well as thick cuticular proboscis walls, which are functionally related to feeding performance in species possessing an extremely long proboscis. © 2013 The Linnean Society of London, Biological Journal of the Linnean Society, 2013, 110, 291–304.

Nonlinear Scaling Relationships of a Railgun

Scaling methods are widely used in railgun technology. But nonlinear problems are seldom studied because of the complexity and uncertainty. To employ the scaling method better, this paper brings the nonlinear drag model to the solid armature railgun circuit. Several drag models are compared including friction drag model (FDM), atmosphere drag model (ADM), and current drag model (CDM). Then, nonlinear scaling relationships of these drag models are discussed. Furthermore, a novel experiment is constructed to compare the drag models at low velocity. At last, the nonlinear scaling method is verified by an example at high velocity. FDM is accurate enough to simulate a railgun system at any condition. Coefficient $\varepsilon$ in FDM should be used considering different test conditions. Scaling relationships of FDM could not be derived at low velocity because of nonlinearity. Although at high velocity, FDM could be simplified to CDM and realize linear scaling method (LSM). Further research indicates that the FDM could use the scaling method directly and realize approximate LSM. CDM is workable which satisfies the conditions of LSM in nature. Due to the work range of the solid armature, usually under 2.5 km/s, nonlinear scaling method could be used in a railgun with FDM. For a solid armature railgun, the $f_{\rm ADM}$ is too small compared with $f_{\rm FDM}$ and it could be ignored in analysis of a railgun, especially at low velocity. What should be noted is that the velocity scaling factor must be controlled above about one-quarter to avoid the nonlinear field.

Foldamer simulations: Novel computational methods and applications to poly-phenylacetylene oligomers

We apply several methods to probe the ensemble kinetic and structural properties of a model system of poly-phenylacetylene (pPA) oligomer folding trajectories. The kinetic methods employed included a brute force accounting of conformations, a Markovian state matrix method, and a nonlinear least squares fit to a minimalist kinetic model used to extract the folding time. Each method gave similar measures for the folding time of the 12-mer chain, calculated to be on the order of 7 ns for the complete folding of the chain from an extended conformation. Utilizing both a linear and a nonlinear scaling relationship between the viscosity and the folding time to correct for a low simulation viscosity, we obtain an upper and a lower bound for the approximate folding time within the range 70 ns<tau<350 ns. This is in agreement with the experimentally measured folding time on the order of 160 ns. The kinetic model used to fit the kinetic behavior of the ensemble of trajectories provides a framework to describe the bulk folding mechanism. We were able to identify two unique clusters of conformations that provide a structural basis to account for the appearance of a kinetic intermediate in the mechanism. We discuss the implications of these findings in the context of helix-coil theory.