Molecular transformations and self-association in anthocyanin pigment patterns.

Spatial patterns of red, purple, and blue colors due to plant pigments called anthocyanins appear in a wide variety of flower petals. Activator and inhibitor proteins involved in anthocyanin synthesis in Mimulus (monkeyflowers) have been identified, and an activator-inhibitor system based on the classic Gierer-Meinhardt system has been proposed as a mathematical model. Analysis in this paper provides a prediction for the critical value of a dimensionless parameter, the ratio of the degradation rate constants of the inhibitor and activator, for pattern formation to occur, and numerical simulations demonstrate the potential for this system to form disordered hexagonal or stripe patterns. We provide experimental evidence for spatial variation in total anthocyanin concentration and for concentration-dependent anthocyanin association. Extending the mathematical model to include anthocyanin transport and diffusion, a series of molecular transformations encompassing acid-base and hydration (speciation) reactions, and self association, we predict that spatial color patterns are accompanied by complex spatial variation in the degree of self association. An important consequence of these studies is a proposal that anthocyanin association allows for colored anthocyanin species to be present in large mole fractions in cell vacuoles despite the fact that the typical vacuolar pH range favors the formation of colorless species.

Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation.

Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE). The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results.

Topological measures of order for imperfect two-dimensional Bravais lattices.

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

Theory of the oscillatory instability of a rigid tip scraped over a polymer surface.

The patterns produced by dragging an atomic force microscope (AFM) tip over a polymer surface are studied using a mesoscopic model introduced by Gnecco and co-workers [E. Gnecco et al., New J. Phys. 17, 032001 (2015)1367-263010.1088/1367-2630/17/3/032001]. We show that the problem can be reduced to solving a closed integrodifferential equation for a single degree of freedom, the position of the AFM tip. We find the steady-state solution to this equation and then carry out a linear stability analysis of it. The steady state is only stable if the dimensionless indentation rate α is less than a critical value α_{c} which depends on the dimensionless velocity of the rigid support r. Conversely, for α>α_{c}, periodic stick-slip motion sets in after a transient. Simulations show that the amplitude of these oscillations is proportional to (α-α_{c})^{1/2} for α just above α_{c}. Our analysis also yields a closed equation that can be solved for the critical value α_{c}=α_{c}(r). If the steady-state motion is perturbed, as long as the deviation from the steady state is small, the deviation of the tip's position from the steady state can be written as a linear superposition of terms of the form exp(λ_{k}t), where the complex constants λ_{k} are solutions to an integral equation. Finally, we demonstrate that the results obtained for the two-dimensional model of Gnecco et al. carry over in a straightforward way to the generalization of the model to three dimensions.

Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach.

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of "how well do we know our parameters, along with their uncertainties?." The results reveal that Bayesian inversion statistical analysis on an example, prototype Ir0n nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a "Gold Standard" Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.

Determining the contributions of protein synthesis and breakdown to muscle atrophy requires non-steady-state equations.

BACKGROUND: Ageing and cachexia cause a loss of muscle mass over time, indicating that protein breakdown exceeds protein synthesis. Deuterium oxide (D2 O) is used for studies of protein turnover because of the advantages of long-term labelling, but these methods introduce considerations that have been largely overlooked when studying conditions of protein gain or loss. The purpose of this study was to demonstrate the importance of accounting for a change in protein mass, a non-steady state, during D2 O labelling studies while also exploring the contribution of protein synthesis and breakdown to denervation-induced muscle atrophy. METHODS: Adult (6 months) male C57BL/6 mice (n = 14) were labelled with D2 O for a total of 7 days following unilateral sciatic nerve transection to induce denervation of hindlimb muscles. The contralateral sham limb and nonsurgical mice (n = 5) were used as two different controls to account for potential crossover effects of denervation. We calculated gastrocnemius myofibrillar and collagen protein synthesis and breakdown assuming steady-state or using non-steady-state modelling. We measured RNA synthesis rates to further understand ribosomal turnover during atrophy. RESULTS: Gastrocnemius mass was less in denervated muscle (137 ± 9 mg) compared with sham (174 ± 15 mg; P < 0.0001) or nonsurgical control (162 ± 5 mg; P < 0.0001). With steady-state calculations, fractional synthesis and breakdown rates (FSR and FBR) were lower in the denervated muscle (1.49 ± 0.06%/day) compared with sham (1.81 ± 0.09%/day; P < 0.0001) or nonsurgical control (2.27 ± 0.04%/day; P < 0.0001). When adjusting for change in protein mass, FSR was 4.21 ± 0.19%/day in denervated limb, whereas FBR was 4.09 ± 0.22%/day. When considering change in protein mass (ksyn ), myofibrillar synthesis was lower in denervated limb (2.44 ± 0.14 mg/day) compared with sham (3.43 ± 0.22 mg/day; P < 0.0001) and non-surgical control (3.74 ± 0.12 mg/day; P < 0.0001), whereas rate of protein breakdown (kdeg, 1/t) was greater in denervated limb (0.050 ± 0.003) compared with sham (0.019 ± 0.001; P < 0.0001) and nonsurgical control (0.023 ± 0.000; P < 0.0001). Muscle collagen breakdown was completely inhibited during denervation. There was a strong correlation (r = 0.83, P < 0.001) between RNA and myofibrillar protein synthesis in sham but not denervated muscle. CONCLUSIONS: We show conflicting results between steady- and non-steady-state calculations on myofibrillar protein synthesis and breakdown during periods of muscle loss. We also found that collagen accumulation was largely from a decrease in collagen breakdown. Comparison between sham and non-surgical control demonstrated a crossover effect of denervation on myofibrillar protein synthesis and ribosomal biogenesis, which impacts study design for unilateral atrophy studies. These considerations are important because not accounting for them can mislead therapeutic attempts to maintain muscle mass.

Dust Effects on Ir(0)n Nanoparticle Formation Nucleation and Growth Kinetics and Particle Size-Distributions: Analysis by and Insights from Mechanism-Enabled Population Balance Modeling.

The effects of microfiltration removal of filterable dust on nanoparticle formation kinetics and particle-size distribution, in a polyoxometalate polyanion (P2W15Nb3O629-)-stabilized Ir(0)n nanoparticle formation system, are analyzed by the newly developed method of Mechanism-Enabled Population Balance Modeling (ME-PBM). The [(Bu4N)5Na3(1,5-COD)Ir·P2W15Nb3O62] precatalyst system produces on average Ir(0)∼200 nanoparticles of 1.74 ± 0.33 nm and hence a particle-size distribution (PSD) of ±19% dispersion when the precatalyst is reduced under H2 in unfiltered propylene carbonate solvent. But if the precatalyst is reduced in microfiltered solvent and microfiltered reagent solutions (where the filtered solvent is then also used to rinse dust from the glassware), then larger Ir(0)∼300 1.96 ± 0.16 nm nanoparticles are produced with a remarkable, 2.4-fold lowered ±8% dispersion. The results and effects of the microfiltration reduction of dust are analyzed by the newly developed method of ME-PBM. More specifically, the studies reported herein address eight outstanding questions that are listed in the Introduction. Those questions include: how easy or difficult it is to fit PSD data? What is the ability of the recently discovered alternative termolecular nucleation and two size-dependent growth steps mechanism to account for the effects of dust on the PSD? What types and amount of PSD kinetics data are needed to deconvolute the PSD into the parameters of the ME-PBM? What is the reliability of the resulting rate constants? Additional questions addressed include: if the ME-PBM results offer insights into the remarkable 2.4-fold narrowing of the PSD post simple microfiltration lowering of the dust, and if the results are likely to be more general? The Summary and Conclusions section lists nine specific insights that include comments on needed future studies.

Mechanism-Enabled Population Balance Modeling of Particle Formation en Route to Particle Average Size and Size Distribution Understanding and Control.

The concept of Mechanism-Enabled Population Balance Modeling (ME-PBM) is reported, illustrated by its application to a prototype Ir(0)n nanoparticle formation reaction. ME-PBM is defined herein as the use of now available, experimentally established, disproof-based, deliberately minimalistic mechanisms of particle formation as the required input for more rigorous Population Balance models, critically including an experimentally established nucleation mechanism. ME-PBM achieves the long-sought goal of connecting such now available experimental minimum mechanisms to the understanding and rational control of particles size and size distributions. Twelve pseudoelementary step, particle-formation mechanisms are considered so that the approach to the ME-PBM is also extensively disproof-based. Resurrection of Smoluchowski's 1918 full Ordinary Differential Equation (ODE) approach to the PBM is another, critical aspect of our approach which, in turn, allows unbiased fitting of the information-laden particle-size distribution (PSD) including its shape. The results provide one solution to the "inverse problem" in which the PSD informs one as to the correct particle formation mechanism: A new, deliberately minimalistic 3-step particle-formation mechanism has been uncovered that is a single-additional-step expansion of the now broadly used Finke-Watzky (FW) 2-step mechanism, the new 3-step mechanism being: A → B (rate constant k1), A + B → C (rate constant k2), and A + C → 1.5C (rate constant k3), where A represents the monomeric nanoparticle precursor, B represents "small" nanoparticles, and C represents "larger" nanoparticles. The results strongly support three paradigm shifts for nucleation and growth of particles, the most critical paradigm shift being that the "burst" nucleation assumption in LaMer's 1950s model of particle formation is not required to produce narrow, near-monodisperse PSDs. Instead, narrow PSDs can be and are achieved despite continuous nucleation because smaller particles grow faster than larger ones, k2 > k3, thereby allowing the smaller particles to catch up in size to the more slowly growing larger particles.

The counterdiffusion of HCl and NH3: An experimental and modeling analysis of topochemistry, diffusion, reaction, and phase transitions.

Vapor-phase ammonia, NH3(g), and hydrochloric acid, HCl(g), undergo a series of complex reactions, including nucleation and growth, to form solid ammonium chloride, NH4Cl(s). The counterdiffusional experiment, whereby HCl(g) and NH3(g) diffuse from opposite ends of a tube and react to form spatiotemporally complex patterns, has a rich history of study. In this paper, we combine experimental data, molecular simulations, and analysis and simulations of a partial differential equation model to address the questions of where the first unobserved vapor product NH4Cl(g) and visually observable precipitate NH4Cl(s) form and how these positions depend on experimental parameters. These analyses yield a consistent picture which involves a moving reaction front as well as previously unobserved heterogeneous nucleation, wall nucleation, and homogeneous nucleation. The experiments combined with modeling allow for an estimate of the heterogeneous and homogeneous nucleation thresholds for the vapor-to-solid phase transition. The results, synthesized with the literature on this vapor-to-particle reaction, inform a discussion of the details of the reaction mechanism, including the role of water, which concludes the paper.

Highly ordered square arrays of nanoscale pyramids produced by ion bombardment of a crystalline binary material.

A theory is developed for the nanoscale patterns formed when the (001) surface of a crystalline binary material with fourfold rotational symmetry is subjected to normal-incidence ion bombardment. The deterministic nonlinear continuum equations account for the Ehrlich-Schwoebel barrier, which produces uphill atomic currents on the crystal surface. We demonstrate that highly ordered square arrays of nanopyramids can form in a certain region of parameter space. An Ehrlich-Schwoebel barrier is required for patterns of this kind to develop. For another range of parameters, a disordered square array of nanodots forms and the pattern coarsens over time.

Modeling the contribution of individual proteins to mixed skeletal muscle protein synthetic rates over increasing periods of label incorporation.

Advances in stable isotope approaches, primarily the use of deuterium oxide ((2)H2O), allow for long-term measurements of protein synthesis, as well as the contribution of individual proteins to tissue measured protein synthesis rates. Here, we determined the influence of individual protein synthetic rates, individual protein content, and time of isotopic labeling on the measured synthesis rate of skeletal muscle proteins. To this end, we developed a mathematical model, applied the model to an established data set collected in vivo, and, to experimentally test the impact of different isotopic labeling periods, used (2)H2O to measure protein synthesis in cultured myotubes over periods of 2, 4, and 7 days. We first demonstrated the influence of both relative protein content and individual protein synthesis rates on measured synthesis rates over time. When expanded to include 286 individual proteins, the model closely approximated protein synthetic rates measured in vivo. The model revealed a 29% difference in measured synthesis rates from the slowest period of measurement (20 min) to the longest period of measurement (6 wk). In support of these findings, culturing of C2C12 myotubes with isotopic labeling periods of 2, 4, or 7 days revealed up to a doubling of the measured synthesis rate in the shorter labeling period compared with the longer period of labeling. From our model, we conclude that a 4-wk period of labeling is ideal for considering all proteins in a mixed-tissue fraction, while minimizing the slowing effect of fully turned-over proteins. In addition, we advocate that careful consideration must be paid to the period of isotopic labeling when comparing mixed protein synthetic rates between studies.

Producing nanodot arrays with improved hexagonal order by patterning surfaces before ion sputtering.

When the surface of a nominally flat binary material is bombarded with a broad, normally incident ion beam, disordered hexagonal arrays of nanodots can form. Shipman and Bradley have derived equations of motion that govern the coupled dynamics of the height and composition of such a surface [Shipman and Bradley, Phys. Rev. B 84, 085420 (2011)]. We investigate the influence of initial conditions on the hexagonal order yielded by integration of those equations of motion. The initial conditions studied are hexagonal and sinusoidal templates, straight scratches, and nominally flat surfaces. Our simulations indicate that both kinds of templates lead to marked improvements in the hexagonal order if the initial wavelength is approximately equal to or double the linearly selected wavelength. Scratches enhance the hexagonal order in their vicinity if their width is close to or less than the linearly selected wavelength. Our results suggest that prepatterning a binary material can dramatically increase the hexagonal order achieved at large ion fluences.

Modeling the presence probability of invasive plant species with nonlocal dispersal.

Mathematical models for the spread of invading plant organisms typically utilize population growth and dispersal dynamics to predict the time-evolution of a population distribution. In this paper, we revisit a particular class of deterministic contact models obtained from a stochastic birth process for invasive organisms. These models were introduced by Mollison (J R Stat Soc 39(3):283, 1977). We derive the deterministic integro-differential equation of a more general contact model and show that the quantity of interest may be interpreted not as population size, but rather as the probability of species occurrence. We proceed to show how landscape heterogeneity can be included in the model by utilizing the concept of statistical habitat suitability models which condense diverse ecological data into a single statistic. As ecologists often deal with species presence data rather than population size, we argue that a model for probability of occurrence allows for a realistic determination of initial conditions from data. Finally, we present numerical results of our deterministic model and compare them to simulations of the underlying stochastic process.

Towards a continuous population model for natural language vowel shift.

The Great English Vowel Shift of 16th-19th centuries and the current Northern Cities Vowel Shift are two examples of collective language processes characterized by regular phonetic changes, that is, gradual changes in vowel pronunciation over time. Here we develop a structured population approach to modeling such regular changes in the vowel systems of natural languages, taking into account learning patterns and effects such as social trends. We treat vowel pronunciation as a continuous variable in vowel space and allow for a continuous dependence of vowel pronunciation in time and age of the speaker. The theory of mixtures with continuous diversity provides a framework for the model, which extends the McKendrick-von Foerster equation to populations with age and phonetic structures. We develop the general balance equations for such populations and propose explicit expressions for the factors that impact the evolution of the vowel pronunciation distribution. For illustration, we present two examples of numerical simulations. In the first one we study a stationary solution corresponding to a state of phonetic equilibrium, in which speakers of all ages share a similar phonetic profile. We characterize the variance of the phonetic distribution in terms of a parameter measuring a ratio of phonetic attraction to dispersion. In the second example we show how vowel shift occurs upon starting with an initial condition consisting of a majority pronunciation that is affected by an immigrant minority with a different vowel pronunciation distribution. The approach developed here for vowel systems may be applied also to other learning situations and other time-dependent processes of cognition in self-interacting populations, like opinions or perceptions.

Discrete and continuous invariance in phyllotactic tilings.

Phyllotaxis refers to the arrangement of primordia (the first stage in the development of a structure such as a leaf) on plants and phyllotactic planforms refer to the shapes of the primordia in a phyllotactic arrangement. This paper focuses on invariances in phyllotactic planforms as the van Iterson parameter Gamma--a measurement of the ratio of the size of the annular generative region at the plant tip where the patterns form to primordium area-varies. We demonstrate discrete invariance in phyllotactic planforms, by which we mean a similarity in the planform under a scaling Gamma-->Gammaphi(n), where varphi is the golden number and n is an integer. Continuous invariance in planforms is then motivated by examples in which the shapes of primordia are homogeneous as n varies over the real numbers. We also show how continuous invariance results from classical number-theoretical theorems on the approximation of irrational numbers (such as phi) by rational numbers. We define these notions first for the underlying phyllotactic lattice and then for primordium shapes and amplitude equations resulting from partial differential equation (PDE) models.

Spontaneous pattern formation induced by ion bombardment of binary compounds.

A theory is developed that explains the genesis of the strikingly regular hexagonal arrays of nanoscale mounds that can form when a flat surface of a binary compound is subjected to normal-incidence ion bombardment. We find that the species with the higher sputter yield is concentrated at the peaks of the nanodots and that hysteretic switching between the flat and the hexagonally ordered state can occur as the sample temperature is varied. Surface ripples are predicted to emerge for a certain range of the parameters.

The probability of HIV infection in a new host and its reduction with microbicides.

We use a simple mathematical model to estimate the probability and its time dependence that one or more HIV virions successfully infect target cells. For the transfer of a given number of virions to target cells we derive expressions for the probability P(inf), of infection. Thus, in the case of needlestick transfer we determine P(inf) and an approximate time window for post-exposure prophylaxis (PEP). For heterosexual transmission, where the transfer process is more complicated, a parameter gamma is employed which measures the strength of the infection process. For the smaller value of gamma, P(inf) is from 6 x 10(-5) to 0.93 or from 7.82 x 10(-6) to 0.29, where the lower figures are for the transfer of 100 virions and the upper figures are for the transfer of 4.4 million virions. We estimate the reductions in P(inf) which occur with a microbicide of a given efficacy. It is found that reductions may be approximately as stated when the number of virions transferred is less than about 10(5), but declines to zero for viral loads above that number. It is concluded that PEP should always be applied immediately after a needlestick incident. Further, manufacturers of microbicides should be encouraged to investigate and report their effectiveness at various transferred viral burdens.

Phyllotaxis: cooperation and competition between mechanical and biochemical processes.

Current theories and models of the formation of phyllotactic patterns at plant apical meristems center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partial differential equations are similar in form. The implications of this universality in the form of the equations on interpreting the results of simulations are discussed. We develop a combined model that incorporates the coupling of biochemistry and mechanics. The combined model is accessible to analysis by reduction to a set of ordinary differential equations for the amplitudes of shapes associated with both the auxin concentration field and plant surface deformation. Analysis of these amplitude equations reveals the parameter choices under which the two mechanisms may cooperate in determining the pattern, or under which one or the other mechanism may dominate.

Phyllotaxis as an example of the symbiosis of mechanical forces and biochemical processes in living tissue.

Phyllotaxis, the arrangement of a plant's phylla (flowers, bracts, stickers) near its shoot apical meristem (SAM), has intrigued natural scientists for centuries. Even today, the reasons for the observed patterns and their special properties, the physical and chemical mechanisms which give rise to strikingly similar configurations in a wide variety of plants, the almost-constant golden divergence angle, the almost constant plastichrone ratio, the choices of parastichy numbers and the prevalence of Fibonacci sequences to which these numbers belong, are at best only partially understood. Our goals in this Addendum are: To give a brief overview of current thinking on possible mechanisms for primordia (the bumps on the plant surface which eventually mature into fully developed structures such as leaves or florets) formation and give a descriptive narrative of the mathematical models which encode various hypotheses.To emphasize the point that patterns, whether they be phyllotactic configurations on plant surfaces or convection cells on the sun's surface, are macroscopic objects whose behaviors are determined more by symmetries of the proposed model and less by microscopic details. Because of this, the identification of observations with the predications of a particular model can only be made with confidence when the match coincides over a range of circumstances and parameters.To discuss some of the key results of the proposed models and, in particular, introduce the prediction of a new and, in principle, measurable invariant in plant phyllotaxis.To introduce a new model of primordia formation which is more in keeping with the pictures and paradigms of Hofmeister,1 Snow & Snow,2 and Douady and Couder3,4 which see primordia as forming in a fairly narrow annular zone surrounding the plant's SAM separating a region of undifferentiated cells from a fully developed patterned state.To consider the challenge of phyllotaxis in the broader context of pattern formation in biological tissue which responds to both mechanical and biochemical processes.

Phyllotactic patterns on plants.

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the deformation configurations seen on plant surfaces may be understood as the energy-minimizing buckling pattern of a compressed shell (the plant's tunica) on an elastic foundation. The key new idea is that the strain energy is minimized by configurations consisting of special triads of almost periodic deformations. We reproduce a wide spectrum of plant patterns, all with the divergence angles observed in nature, and show how the occurrences of Fibonacci-like sequences and the golden angle are natural consequences.