RNA polymerase sliding on DNA can couple the transcription of nearby bacterial operons.

DNA transcription initiates after an RNA polymerase (RNAP) molecule binds to the promoter of a gene. In bacteria, the canonical picture is that RNAP comes from the cytoplasmic pool of freely diffusing RNAP molecules. Recent experiments suggest the possible existence of a separate pool of polymerases, competent for initiation, which freely slide on the DNA after having terminated one round of transcription. Promoter-dependent transcription reinitiation from this pool of posttermination RNAP may lead to coupled initiation at nearby operons, but it is unclear whether this can occur over the distance and timescales needed for it to function widely on a bacterial genome in vivo. Here, we mathematically model the hypothesized reinitiation mechanism as a diffusion-to-capture process and compute the distances over which significant interoperon coupling can occur and the time required. These quantities depend on molecular association and dissociation rate constants between DNA, RNAP, and the transcription initiation factor σ70; we measure these rate constants using single-molecule experiments in vitro. Our combined theory/experimental results demonstrate that efficient coupling can occur at physiologically relevant σ70 concentrations and on timescales appropriate for transcript synthesis. Coupling is efficient over terminator-promoter distances up to ∼1,000 bp, which includes the majority of terminator-promoter nearest neighbor pairs in the Escherichia coli genome. The results suggest a generalized mechanism that couples the transcription of nearby operons and breaks the paradigm that each binding of RNAP to DNA can produce at most one messenger RNA.

Stochastic Turing patterns of trichomes in Arabidopsis leaves.

Biological patterns that emerge during the morphogenesis of multicellular organisms can display high precision at large scales, while at cellular scales, cells exhibit large fluctuations stemming from cell-cell differences in molecular copy numbers also called demographic noise. We study the conflicting interplay between high precision and demographic noise in trichome patterns on the epidermis of wild-type Arabidopsis thaliana leaves, as a two-dimensional model system. We carry out a statistical characterization of these patterns and show that their power spectra display fat tails-a signature compatible with noise-driven stochastic Turing patterns-which are absent in power spectra of patterns driven by deterministic instabilities. We then present a theoretical model that includes demographic noise stemming from birth-death processes of genetic regulators which we study analytically and by stochastic simulations. The model captures the observed experimental features of trichome patterns.

A mathematical model of signalling molecule-mediated processes during regeneration of osteochondral defects after chondrocyte implantation.

Treating bone-cartilage defects is a fundamental clinical problem. The ability of damaged cartilage to self-repair is limited due to its avascularity. Left untreated, these defects can lead to osteoarthritis. Details of osteochondral defect repair are elusive, but animal models indicate healing occurs via an endochondral ossification-like process, similar to that in the growth plate. In the growth plate, the signalling molecules parathyroid hormone-related protein (PTHrP) and Indian Hedgehog (Ihh) form a feedback loop regulating chondrocyte hypertrophy, with Ihh inducing and PTHrP suppressing hypertrophy. To better understand this repair process and to explore the regulatory role of signalling molecules on the regeneration process, we formulate a reaction-diffusion mathematical model of osteochondral defect regeneration after chondrocyte implantation. The drivers of healing are assumed to be chondrocytes and osteoblasts, and their interaction via signalling molecules. We model cell proliferation, migration and chondrocyte hypertrophy, and matrix production and conversion, spatially and temporally. We further model nutrient and signalling molecule diffusion and their interaction with the cells. We consider the PTHrP-Ihh feedback loop as the backbone mechanisms but the model is flexible to incorporate extra signalling mechanisms if needed. Our mathematical model is able to represent repair of osteochondral defects, starting with cartilage formation throughout the defect. This is followed by chondrocyte hypertrophy, matrix calcification and bone formation deep inside the defect, while cartilage at the surface is maintained and eventually separated from the deeper bone by a thin layer of calcified cartilage. The complete process requires around 48 months. A key highlight of the model demonstrates that the PTHrP-Ihh loop alone is insufficient and an extra mechanism is required to initiate chondrocyte hypertrophy, represented by a critical cartilage density. A parameter sensitivity study reveals that the timing of the repair process crucially depends on parameters, such as the critical cartilage density, and those describing the actions of PTHrP to suppress hypertrophy, such as its diffusion coefficient, threshold concentration and degradation rate.

How do productivity gradient and diffusion shape patterns in a plant-herbivore grazing system?

Natural systems show heterogeneous patchy distributions of vegetation over large landscapes. Reaction-diffusion systems can demonstrate such heterogeneity of species distributions. Here, we analyse a reaction-diffusion model of plant-herbivore interactions in two-dimensional space to illustrate non-homogeneous distributions of plants and herbivores. The non-spatial system shows bottom-up control, where herbivore density is low under low and high primary productivity but increased at intermediate productivity. In addition, the non-spatial system provides bistability between a dense vegetation state devoid of herbivores and a coexisting state of plants and herbivores. In the spatiotemporal model, we give analytical conditions of occurring diffusion-driven (Turing) instability, where a novel point in our model is the relative dispersal of herbivores, which represents the movement of herbivores from a higher to a lower vegetation state in addition to the self-diffusion of both species. It is shown that heterogeneity in the population distribution does not occur if the relative dispersal of herbivores is low, but it appears in the opposite case. Due to bistability in the underlying non-spatial system, the spatiotemporal model produces initial value-dependent patterns. The two initial values make different patterns despite having the same primary productivity and relative dispersal rate. As productivity increases with a given relative herbivore dispersal, pattern transition occurs from a blend of stripes and spots of low vegetation state to a predominantly low-density vegetation state with smaller patches of densely vegetated states with one initial value. On the contrary, a discernible change in vegetation patterns from cold spots in the dense vegetation to hot stripes in the primarily low-vegetated state is noticed under the other initial population value. Furthermore, the population distributions of plants and herbivores in the entire domain after a long period are heterogeneous for both initial values, provided the relative herbivore dispersal is substantial. We estimated mean population densities to observe species fitness in the whole domain under variable productivity. When productivity is high, the mean population density of plants may go up or down, depending on the herbivore's relative dispersal rate. In contrast to the bottom-up control dynamics of the non-spatial system, the system exhibits a top-down control under high relative dispersal, where the herbivore regulates vegetation growth under high productivity. On the other hand, herbivores are extinct under high productivity if the relative dispersal is low.

Modelling phytoplankton-virus interactions: phytoplankton blooms and lytic virus transmission.

A dynamic reaction-diffusion model of four variables is proposed to describe the spread of lytic viruses among phytoplankton in a poorly mixed aquatic environment. The basic ecological reproductive index for phytoplankton invasion and the basic reproduction number for virus transmission are derived to characterize the phytoplankton growth and virus transmission dynamics. The theoretical and numerical results from the model show that the spread of lytic viruses effectively controls phytoplankton blooms. This validates the observations and experimental results of Emiliana huxleyi-lytic virus interactions. The studies also indicate that the lytic virus transmission cannot occur in a low-light or oligotrophic aquatic environment.

Predator-induced prey dispersal can cause hump-shaped density-area relationships in prey populations.

Predation can both reduce prey abundance directly (through density-dependent effects) and indirectly through prey trait-mediated effects. Over the years, many studies have focused on describing the density-area relationship (DAR). However, the mechanisms responsible for the DAR are not well understood. Loss and fragmentation of habitats, owing to human activities, creates landscape-level spatial heterogeneity wherein patches of varying size, isolation and quality are separated by a human-modified "matrix" of varying degrees of hostility and has been a primary driver of species extinctions and declining biodiversity. How matrix hostility in combination with trait-mediated effects influence DAR, minimum patch size, and species coexistence remains an open question. In this paper, we employ a theoretical spatially explicit predator-prey population model built upon the reaction-diffusion framework to explore effects of predator-induced emigration (trait-mediated emigration) and matrix hostility on DAR, minimum patch size, and species coexistence. Our results show that when trait-mediated response strength is sufficiently strong, ranges of patch size emerge where a nonlinear hump-shaped prey DAR is predicted and other ranges where coexistence is not possible. In a conservation perspective, DAR is crucial not only in deciding whether we should have one large habitat patch or several-small (SLOSS), but for understanding the minimum patch size that can support a viable population. Our study lends more credence to the possibility that predators can alter prey DAR through predator-induced prey dispersal.

Simulation of biochemical dynamics of [Formula: see text] and [Formula: see text] in fibroblast cell.

Calcium dynamics is not only responsible for maintaining the framework and functions of the cell but also plays a role in the dynamics of other biochemical systems in the cell. Phospholipase C-[Formula: see text] l ([Formula: see text]) has a crucial role in the function of fibroblast cells. Experiments have shown that [Formula: see text] and [Formula: see text] have interdependent dynamics in fibroblast cells. However, no reaction-diffusion model exists for the two-way feedback system dynamics of [Formula: see text] and [Formula: see text] in fibroblasts till date. The computational model is designed to investigate the impact of variations in several processes, such as the [Formula: see text] pump, buffer process, source inflow, etc., on the system dynamics of [Formula: see text] and [Formula: see text] in fibroblast cells. The computational findings are obtained using finite element techniques, and the consequences of dysregulation in various processes on the spatiotemporal calcium and [Formula: see text] dynamics in fibroblasts are investigated. The results lead to the conclusion that the effects of buffer, source influx, diffusion, and [Formula: see text] pump can cause fluctuations in the dynamics of [Formula: see text] and [Formula: see text] in fibroblasts. Disruptions in these constitutive processes can result in changes in the dynamics of calcium and [Formula: see text]. Thus, the current model provides new/novel information regarding the precise dysregulatory constitutive systems that regulate calcium and [Formula: see text] kinetics, such as source inflow, diffusion, [Formula: see text], and buffer, can be responsible for excessive calcium and [Formula: see text] concentrations leading to fibrotic illnesses such as cancer and fibrosis.

Spatio-temporal dynamics of the metabolome of climacteric fruit during ripening and post-harvest storage.

Fruit quality traits are determined to a large extent by their metabolome. The metabolite content of climacteric fruit changes drastically during ripening and post-harvest storage, and has been investigated extensively. However, the spatial distribution of metabolites and how it changes in time has received much less attention as fruit are usually considered as homogenous plant organs. Yet, spatio-temporal changes of starch, which is hydrolyzed during ripening, has been used for a long time as a ripening index. As vascular transport of water, and hence convective transport of metabolites, slows down in mature fruit and even stalls after detachment, spatio-temporal changes in their concentration are probably affected by diffusive transport of gaseous molecules that act as substrate (O2), inhibitor (CO2), or regulator (ethylene and NO) of the metabolic pathways that are active during climacteric ripening. In this review, we discuss such spatio-temporal changes of the metabolome and how they are affected by transport of metabolic gases and gaseous hormones. As there are currently no techniques available to measure the metabolite distribution repeatedly by non-destructive means, we introduce reaction-diffusion models as an in silico tool to compute it. We show how the different components of such a model can be integrated and used to better understand the role of spatio-temporal changes of the metabolome in ripening and post-harvest storage of climacteric fruit that is detached from the plant, and discuss future research needs.

Glycolytic Wave Patterns in a Simple Reaction-diffusion System with Inhomogeneous Influx: Dynamic Transitions.

An inhomogeneous profile of chemostatted species generates a rich variety of patterns in glycolytic waves depicted in a Selkov reaction-diffusion framework here. A key role played by diffusion amplitude and symmetry in the chemostatted species profile in dictating the fate of local spatial dynamics involving periodic, quasiperiodic, and chaotic patterns and transitions among them are investigated systematically. More importantly, various dynamic transitions, including wave propagation direction changes, are illustrated in interesting situations. Besides numerical results, our analytical formulation of the amplitude equation connecting complex Ginzburg-Landau and Lambda-omega representation shed light on the phase dynamics of the system. This systematic study of the glycolytic reaction-diffusion wave is in line with previous experimental results in open spatial reactor and will provide a knowledge about the dynamics that shape and control biological information processing and related phenomena.

Ecological release and patch geometry can cause nonlinear density-area relationships.

A primary driver of species extinctions and declining biodiversity is loss and fragmentation of habitats owing to human activities. Many studies spanning a wide diversity of taxa have described the relationship between population density and habitat patch area, i.e., the density-area relationship (DAR), as positive, neutral, negative or some combination of the three. However, the mechanisms responsible for these relationships remain elusive. We employ a theoretical spatially explicit population model built upon the reaction-diffusion framework with absorbing boundary conditions to model a habitat specialist dwelling in islands of habitat surrounded by a hostile matrix. We consider patches with a convex or non-convex geometry. Our results show that a single species following logistic-type population growth exhibits a strictly positive and continuous DAR. However, when multiple asymptotically stable steady states are preset in the system, a discontinuous DAR arises. In the case of two species governed by diffusive Lotka-Volterra growth and competitive interactions, we observe that overall DAR structure can be either (1) positive, (2) positive for small areas and neutral for large, or (3) hump-shaped, i.e., positive for area below a threshold and negative for area above. Patch complexity such as non-convex geometry can cause discontinuities in DAR slope for a single species and create qualitatively different patterns in a competitive system as compared to a convex patch. We also compared our theoretical results with two empirical studies (Anolis lizards on islands and crossbills and pine squirrels in forest fragments) where the pragmatic view of DAR fails to give a mechanistic understanding of what was observed. Close qualitative agreement between theoretical and observed DAR indicates that our model gives a reasonable explanation of the mechanisms underpinning DAR found in those studies. From a conservation perspective, the DAR is crucial to the identification of valuable habitat fragments that favor high abundance and the design of a reserve for a target species. When it comes to protecting a single species, these results suggest that there is unlikely to be a simple solution and that conservation decisions should always be made on a case-by-case basis.

Activation of butterfly eyespots by Distal-less is consistent with a reaction-diffusion process.

Eyespots on the wings of nymphalid butterflies represent colorful examples of pattern formation, yet the developmental origins and mechanisms underlying eyespot center differentiation are still poorly understood. Using CRISPR-Cas9 we re-examine the function of Distal-less (Dll) as an activator or repressor of eyespots, a topic that remains controversial. We show that the phenotypic outcome of CRISPR mutations depends upon which specific exon is targeted. In Bicyclus anynana, exon 2 mutations are associated with both missing and ectopic eyespots, and also exon skipping. Exon 3 mutations, which do not lead to exon skipping, produce only null phenotypes, including missing eyespots, lighter wing coloration and loss of scales. Reaction-diffusion modeling of Dll function, using Wnt and Dpp as candidate morphogens, accurately replicates these complex crispant phenotypes. These results provide new insight into the function of Dll as a potential activator of eyespot development, scale growth and melanization, and suggest that the tuning of Dll expression levels can generate a diversity of eyespot phenotypes, including their appearance on the wing.This article has an associated 'The people behind the papers' interview.

Front-like expansion and arrest of programmed cell death in brown banana spots.

The spot patterns on bananas are a striking case of biological pattern formation and-as a qualitative ripeness indicator-linked to 50 million tons of wasted food per year. Ripening bananas develop these senescent spots as phenolic compounds are enzymatically oxidized and cellular integrity is lost. We characterize the dynamics of the spot expansion and their nucleation rates based on time-lapse movies. Spots nucleate for about 2 days yielding a typical density of 8 spots/cm2. The expansion is initially diffusion controlled and the effective diffusion coefficient decreases with nucleation time from 1.3 to 0.4 mm2d-1. During and after expansion, the browning fronts maintain a steep and constant intensity gradient. We quantitatively reproduce these features by a reaction-diffusion model that considers the local oxygen concentration and browning degree of the peel. All model parameters are based on measurements and front stalling is explained by decreasing oxygen levels in the nucleation sites.

Travelling-Wave and Asymptotic Analysis of a Multiphase Moving Boundary Model for Engineered Tissue Growth.

We derive a multiphase, moving boundary model to represent the development of tissue in vitro in a porous tissue engineering scaffold. We consider a cell, extra-cellular liquid and a rigid scaffold phase, and adopt Darcy's law to relate the velocity of the cell and liquid phases to their respective pressures. Cell-cell and cell-scaffold interactions which can drive cellular motion are accounted for by utilising relevant constitutive assumptions for the pressure in the cell phase. We reduce the model to a nonlinear reaction-diffusion equation for the cell phase, coupled to a moving boundary condition for the tissue edge, the diffusivity being dependent on the cell and scaffold volume fractions, cell and liquid viscosities and parameters that relate to cellular motion. Numerical simulations reveal that the reduced model admits three regimes for the evolution of the tissue edge at large time: linear, logarithmic and stationary. Employing travelling-wave and asymptotic analysis, we characterise these regimes in terms of parameters related to cellular production and motion. The results of our investigation allow us to suggest optimal values for the governing parameters, so as to stimulate tissue growth in an engineering scaffold.

On the global attractivity for a reaction-diffusion malaria model with incubation period in the vector population.

This paper establishes the global attractivity of a positive constant equilibrium of a nonlocal and time-delayed diffusive malaria model in a homogeneous case. The same problem was achieved in a recent paper (Lou and Zhao in J Math Biol 62:543-568, 2011) by using the fluctuation method, but with a sufficient condition that the disease will become stable requires a sufficiently large basic reproduction number [Formula: see text]. The present study is devoted to remove the sufficient condition by utilizing an appropriate Lyapunov functional and shows that the disease will become stable when [Formula: see text] is exactly greater than one, which remarkably improves the known results in Lou and Zhao (2011).

Mathematical modeling of the eyespots in butterfly wings.

Butterfly wing color patterns are a representative model system for studying biological pattern formation, due to their two-dimensional simple structural and high inter- and intra-specific variabilities. Moreover, butterfly color patterns have demonstrated roles in mate choice, thermoregulation, and predator avoidance via disruptive coloration, attack deflection, aposematism, mimicry, and masquerade. Because of the importance of color patterns to many aspects of butterfly biology and their apparent tractability for study, color patterns have been the subjects of many attempts to model their development. Early attempts focused on generalized mechanisms of pattern formation such as reaction-diffusion, diffusion gradient, lateral inhibition, and threshold responses, without reference to any specific gene products. As candidate genes with expression patterns that resembled incipient color patterns were identified, genetic regulatory networks were proposed for color pattern formation based on gene functions inferred from other insects with wings, such as Drosophila. Particularly detailed networks incorporating the gene products, Distal-less (Dll), Engrailed (En), Hedgehog (Hh), Cubitus interruptus (Ci), Transforming growth factor-ß (TGF-ß), and Wingless (Wg), have been proposed for butterfly border ocelli (eyespots) which helps the investigation of the formation of these patterns. Thus, in this work, we develop a mathematical model including the gene products En, Hh, Ci, TGF-ß, and Wg to mimic and investigate the eyespot formation in butterflies. Our simulations show that the level of En has peaks in the inner and outer rings and the level of Ci has peaks in the inner and middle rings. The interactions among these peaks activate cells to produce white, black, and yellow pigments in the inner, middle, and outer rings, respectively, which captures the eyespot pattern of wild type Bicyclus anynana butterflies. Additionally, our simulations suggest that lack of En generates a single black spot and lack of Hh or Ci generates a single white spot, and a deficiency of TGF-ß or Wg will cause the loss of the outer yellow ring. These deficient patterns are similar to those observed in the eyespots of Vanessa atalanta, Vanessa altissima, and Chlosyne nycteis. Thus, our model also provides a hypothesis to explain the mechanism of generating the deficient patterns in these species.

Global Dynamics of a Reaction-Diffusion Model of Zika Virus Transmission with Seasonality.

In this paper, we propose a periodic reaction-diffusion model of Zika virus with seasonal and spatial heterogeneous structure in host and vector population. We introduce the basic reproduction ratio [Formula: see text] for this model and show that the disease-free periodic solution is globally asymptotically stable if [Formula: see text], while the system admits a globally asymptotically stable positive periodic solution if [Formula: see text]. Numerically, we study the Zika transmission in Rio de Janeiro Municipality, Brazil, and investigate the effects of some model parameters on [Formula: see text]. We find that the neglect of seasonality underestimates the value of [Formula: see text] and the maximum carrying capacity affects the spread of Zika virus.

Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law.

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions.

An Efficient Turing-Type Ag2 Se-CoSe2 Multi-Interfacial Oxygen-Evolving Electrocatalyst*.

Although the Turing structures, or stationary reaction-diffusion patterns, have received increasing attention in biology and chemistry, making such unusual patterns on inorganic solids is fundamentally challenging. We report a simple cation exchange approach to produce Turing-type Ag2 Se on CoSe2 nanobelts relied on diffusion-driven instability. The resultant Turing-type Ag2 Se-CoSe2 material is highly effective to catalyze the oxygen evolution reaction (OER) in alkaline electrolytes with an 84.5 % anodic energy efficiency. Electrochemical measurements show that the intrinsic OER activity correlates linearly with the length of Ag2 Se-CoSe2 interfaces, determining that such Turing-type interfaces are more active sites for OER. Combing X-ray absorption and computational simulations, we ascribe the excellent OER performance to the optimized adsorption energies for critical oxygen-containing intermediates at the unconventional interfaces.

In silico study of the role of cell growth factors in photosynthesis using a virtual leaf tissue generator coupled to a microscale photosynthesis gas exchange model.

Computational tools that allow in silico analysis of the role of cell growth and division on photosynthesis are scarce. We present a freely available tool that combines a virtual leaf tissue generator and a two-dimensional microscale model of gas transport during C3 photosynthesis. A total of 270 mesophyll geometries were generated with varying degrees of growth anisotropy, growth extent, and extent of schizogenous airspace formation in the palisade mesophyll. The anatomical properties of the virtual leaf tissue and microscopic cross-sections of actual leaf tissue of tomato (Solanum lycopersicum L.) were statistically compared. Model equations for transport of CO2 in the liquid phase of the leaf tissue were discretized over the geometries. The virtual leaf tissue generator produced a leaf anatomy of tomato that was statistically similar to real tomato leaf tissue. The response of photosynthesis to intercellular CO2 predicted by a model that used the virtual leaf tissue geometry compared well with measured values. The results indicate that the light-saturated rate of photosynthesis was influenced by interactive effects of extent and directionality of cell growth and degree of airspace formation through the exposed surface of mesophyll per leaf area. The tool could be used further in investigations of improving photosynthesis and gas exchange in relation to cell growth and leaf anatomy.

Emergence of spatio-temporal variations in chemotherapeutic drug efficacy: in-vitro and in-Silico 3D tumour spheroid studies.

BACKGROUND: The mechanisms of action and efficacy of cisplatin and paclitaxel at cell population level are well studied and documented, however the localized spatio-temporal effects of the drugs are less well understood. We explore the emergence of spatially preferential drug efficacy resulting from variations in mechanisms of cell-drug interactions. METHODS: 3D spheroids of HeLa-C3 cells were treated with drugs, cisplatin and paclitaxel. This was followed by sectioning and staining of the spheroids to track the spatio-temporal apoptotic effects of the drugs. A mechanistic drug-cell interaction model was developed and simulated to analyse the localized efficacy of these drugs. RESULTS: The outcomes of drug actions on a local cell population was dependant on the interactions between cell repair probability, intracellular drug concentration and cell's mitosis phase. In spheroids treated with cisplatin, drug induced apoptosis is found to be scattered throughout the volume of the spheroids. In contrast, effect of paclitaxel is found to be preferentially localized along the periphery of the spheroids. Combinatorial treatments of cisplatin and paclitaxel result in varying levels of cell apoptosis based on the scheduling strategy. CONCLUSIONS: The preferential action of paclitaxel can be attributed to the cell characteristics of the peripheral population. The model simulations and experimental data show that treatments initiated with paclitaxel are more efficacious due to the cascading of spatial effects of the drugs.