Self-organization of the vascular system in plant leaves: inter-dependent dynamics of auxin flux and carrier proteins

The vegetative hormone Auxin is involved in vascular tissues formation throughout the plant. Trans-membrane carrier proteins transporting auxin from cell to cell and distributed asymmetrically around each cell give to auxin a polarized movement in tissues, creating streams of auxin that presume future vascular bundles. According to the canalization hypothesis, auxin transport ability of cells is thought to increase with auxin flux, resulting in the self-enhancement of this flux along auxin paths. In this study we evaluate a series of models based on canalization hypothesis using carrier proteins, under different assumptions concerning auxin flux formation and carrier protein dynamics. Simulations are run on a hexagonal lattice with uniform auxin production. A single cell located in the margin of the lattice indicates the petiole, and acts as an auxin sink. The main results are: (1) We obtain branching auxin distribution patterns. (2) The type of self-enhancement described by the functional form of the carrier proteins regulation responding to the auxin flux intensity in different parts of a cell, has a strong effect on the possibility of generating the branching patterns. For response functions with acceleration in the increase of carrier protein numbers compared to the auxin flux, branching patterns are likely to be generated. For linear or decelerating response functions, no branching patterns are formed. (3) When branching patterns are formed, auxin distribution greatly differs between the case in which the number of carrier proteins in different parts of a cell are regulated independently, and the case in which different parts of a cell compete for a limited number of carrier proteins. In the former case, the auxin level is lower in veins than in the surrounding tissue, while in the latter, the auxin is present in greater abundance in veins. These results suggest that canalization is a good candidate for describing plant vein pattern formation.     

A canonical model for production and distribution of root mass in space and time

A mathematical model is presented for the production and distribution of plant root mass in a substrate in space and time. This model is intended to be used for the interpretation of data on root density profiles. It is assumed that a pulse of root apices appears at the surface of the substrate at zero time, emerging from seeds or stems, and that these apices migrate downwards in the substrate, leaving trails of root mass as they go, giving birth to daughter apices, and dying by terminal differentiation. The model has at least 3 parameters, representing, respectively, initial root generating capacity of root axes at zero time, net rate of root apex reproduction (or extinction), and rate of downwards migration of root apices. Two versions of the model are explored, a nondiffusive version and a diffusive version in which there is an additional parameter representing diffusivity of root apices. It is shown that both versions of this theoretical mechanistic model predict that the profile of root density with depth tends to a steady-state exponential form, parameterised by a density of root mass at zero depth and a logarithmic rate of change of root mass with respect to depth. This form has previously appeared in the literature as an empirical model. Data analyses are performed, using this model, on a set of data in which cultures of plants were subjected to treatment with auxin or by genetic manipulation, and the effects of the various treatments on the parameters of the model are described. Interpretations of the experimental results are proposed in the light of the theoretical results derived in this paper.     

Endogenous electromagnetic fields in plant leaves: a new hypothesis for vascular pattern formation

Electromagnetic (EM) phenomena have long been implicated in biological development, but few detailed, practical mechanisms have been put forth to connect electromagnetism with morphogenetic processes. This work describes a new hypothesis for plant leaf veination, whereby an endogenous electric field forming as a result of a coherent Frohlich process, and corresponding to an EM resonant mode of the developing leaf structure, is capable of instigating leaf vascularisation. In order to test the feasibility of this hypothesis, a three-dimensional, EM finite-element model (FEM) of a leaf primordium was constructed to determine if suitable resonant modes were physically possible for geometric and physical parameters similar to those of developing leaf tissue. Using the FEM model, resonant EM modes with patterns of relevance to developing leaf vein modalities were detected. On account of the existence of shared geometric signatures in a leaf's vascular pattern and the electric field component of EM resonant modes supported by a developing leaf structure, further theoretical and experimental investigations are warranted. Significantly, this hypothesis is not limited to leaf vascular patterning, but may be applicable to a variety of morphogenetic phenomena in a number of living systems.     

元江干热河谷植物叶片解剖和养分含量特征

研究了元江干热河谷旱田植物(旱季可浇灌,水分较好)和山坡半萨王纳植被中(自然状况,水分较差)共20种的叶片形态解剖特征,以及7种山坡植物叶片养分含量特征.结果表明,山坡植物叶片比叶重大,气孔密度大,气孔长度小,海绵组织/栅栏组织的值小等.元江干热河谷山坡植物叶片养分含量低,1.3%＞Ca＞N＞K＞1%＞Mg＞P＞S.除氮元素外,其它元素种间差别1～3倍.与热带植物群落叶片养元素含量相比,热带雨林＞元江山坡植物＞东南亚沙地旱生林和巴西矮卡廷加群落,表明元江干热河谷植物叶片具有明显的旱生性形态解剖特征,且叶片养分含量也较低.     

Pattern formation of leaf veins by the positive feedback regulation between auxin flow and auxin efflux carrier

The generation of vascular pattern formation in plants is an interesting process of pattern formation in organisms. It is well known that the plant hormone auxin is involved in plant vascular differentiation and that the PIN1 protein, an auxin efflux carrier, localizes to one side of the cell membrane. Several hypotheses have been proposed to explain the formation of leaf venation. One is the canalization hypothesis that is based on the assumption that a positive feedback regulation exists between the flow of a signal molecule and the capacity of its flow. Here, we attempted to integrate the canalization hypothesis and experimental data. We investigated models of the positive feedback regulation between the auxin flow and PIN1 localization. Model 1, with conserved PIN1 amount in each cell, can generate a branching pattern similar to that of plant leaf venation. We introduced the diffusible enhancer "e" into the model as unknown factor. The obtained patterns show a quasi-periodic distribution of auxin flow paths, when the model dynamics includes domain growth. In order to understand the early initiation process that generates an inhomogeneity from an almost homogeneous distribution, we introduced model 2, a simplified version of model 1. Model 2 can generate inhomogeneity with a parameter dependency similar to that of model 1. To analyse parameter condition required for pattern development, approximated equations are obtained from model 2. The isocline analysis of the equations without spatial structure shows that the inhomogeneous distribution occurs from an almost homogeneous distribution. This parameter condition for generating inhomogeneity is consistent with the results of models 1 and 2.     

A growth and division model for retinoblastoma

A stochastic growth and division model for studying a two hit cancer is developed and applied to retinoblastoma. Retinoblastoma occurs if both genes coding for a tumor suppressor protein on homologous chromosomes become defective. Germinal cases occur when a patient or carrier, born with one defective gene, suffers a second insult to any progeny retinal cell. Somatic cases are far less likely as two hits to the same cell during development are required. Details of the disease, germinal or somatic, unilateral or bilateral, in combination with case data allow for the estimation of the two parameters of the model: mutation rate, estimated at p=7x10(-7) per chromosome per cell division, and carrier frequency, estimated at f=40 per million. The model indicates that carriers of the disease arise from similar mutations to germ cells; in particular, heridary transmission can occur for only a generation or two before dying out. The results show that a stochastic simulation of a multi-hit cancer is feasible and may predict tumor growth dynamics. A simulation run will have to consist of a few million cells in order to observe even a small number of mutations. And several dozens such runs will have to be simulated.     

A mathematical model of auxin-mediated radial growth in trees

A mathematical model is presented to describe the coupling between the concentration of indole-3-acetic acid (IAA) in the cambial region of a tree branch and the radial expansion of the branch during active growth. The main features of the model are (1) the branch cambium is treated as an approximately cylindrical surface of negligible thickness, (2) the rate of radial growth is proportional to the mass of IAA per unit area on the cambial surface, and (3) IAA is transported basipetally through the cambium at a constant speed. We neglect the role of elastic strains in the determination of branch shape, and the effects of IAA synthesis and metabolization in the cambium, so the model is not quantitatively accurate. However, the model does reproduce several important qualitative features of tree growth including the approximate area-preserving property of tree branch junctions and the ability of a branch to maintain its shape despite perturbations due to injury.     

The formation and patterning of leaves: recent advances

Leaves, the plants major photosynthetic organs, form through the activity of groups of pluripotent cells, termed shoot apical meristems (SAMs), located at the growing tips of plants. Leaves develop with a dorso–ventral asymmetry, with the adaxial surface adjacent to the meristem and the abaxial surface developing at a distance from it. Molecular genetic studies have shown that the correct specification of adaxial/abaxial polarity requires communication between the incipient leaf and the meristem, and that the juxtaposition of adaxial/abaxial fates is necessary for lamina outgrowth (Waites and Hudson 1995; McConnell et al. 2001). Over the last few years, a number of factors that control cell fate specification in the apex have been identified. This review will focus on recent advances on distinct but overlapping aspects of leaf development, namely, the transition from meristem to leaf fate and the specification of abaxial/adaxial polarity and its possible role in leaf growth.     

How canalization can make loops: a new model of reticulated leaf vascular pattern formation

Formation of the vascular system in plant leaves can be explained by the canalization hypothesis which states that veins are formed in an initially homogeneous field by a self-organizing process between the plant hormone auxin and auxin carrier proteins. Previous models of canalization can generate vein patterns with branching but fail to generate vein patterns with closed loops. However, closed vein loops are commonly observed in plant leaves and are important in making them robust to herbivore attacks and physical damage. Here we propose a new model which generates a vein system with closed loops. We postulate that the flux bifurcator level is enhanced in cells with a high auxin flux and that it causes reallocation of auxin carriers toward neighbouring cells also having a high bifurcator level. This causes the auxin flux to bifurcate, allowing vein tips to attach to other veins creating vein loops. We explore several alternative functional forms for the flux bifurcator affecting the reallocation of efflux carriers and examine parameter dependence of the resulting vein pattern.     

A comparative study of fluorescent singlet oxygen probes in plant leaves

Four fluorescent singlet oxygen sensors: DanePy, its oxalate salt, Singlet Oxygen Sensor Green and MVP, were infiltrated into tobacco leaves and tested for toxicity, subcellular localization, light sensitivity and capacity to trap the singlet oxygen produced in photoinhibition. For reference, a broad sensitivity free radical probe, TEMPO-9-AC, was also included. Photochemical yield was approximately 15% and 10% inhibited by Singlet Oxygen Sensor Green and MVP, respectively, but was not significantly affected by the other probes. Under photoinhibitory conditions, brought about by irradiating lincomycin-treated leaves with strong photosynthetically active radiation, DanePy and Singlet Oxygen Sensor Green were responsive. Singlet Oxygen Sensor Green was also reactive to low, non-photoinhibitory light exposure of the leaf, which was not characteristic to the other probes. MVP did not respond to singlet oxygen which can partly be explained by a possible attenuation of its blue emission in the leaf, as shown by the example TEMPO-9-AC. DanePy-oxalate did not respond to photosynthetic singlet oxygen due to lack of its penetration into photosynthetic tissue and hence could be useful in detecting any singlet oxygen which escapes from a chloroplast initiation site. DanePy was localized in the chloroplasts, while Singlet Oxygen Sensor Green was mainly found in the epidermal cells preferentially associated with the nucleus.      

A model for water uptake by plant roots

We present a model for water uptake by plant roots from unsaturated soil. The model includes the simultaneous flow of water inside the root network and in the soil. It is constructed by considering first the water uptake by a single root, and then using the parameterized results thereby obtained to build a model for water uptake by the developing root network. We focus our model on annual plants, in particular the model will be applicable to commercial monocultures like maize, wheat, etc. The model is solved numerically, and the results are compared with approximate analytic solutions. The model predicts that as a result of water uptake by plant roots, dry and wet zones will develop in the soil. The wet zone is located near the surface of the soil and the depth of it is determined by a balance between rainfall and the rate of water uptake. The dry zone develops directly beneath the wet zone because the influence of the rainfall at the soil surface does not reach this region, due to the nonlinear nature of the water flow in the partially saturated soil. We develop approximate analytic expressions for the depth of the wet zone and discuss briefly its ecological significance for the plant. Using this model we also address the question of where water uptake sites are concentrated in the root system. The model indicates that the regions near the base of the root system (i.e. close to the ground surface) and near the root tips will take up more water than the middle region of the root system, again due to the highly nonlinear nature of water flow in the soil.     

An individual-based model of hepatitis A transmission

Viral hepatitis A, as other endemic diseases, represents a public health priority worldwide. To study long-time scale human pathogens through individual-based simulations, the development of a dynamic network of contacts is required. In this work, we introduce an individual-based model accounting for the birth and death of the individuals, the generation of new households, and the educational career of the individuals, in order to investigate viral hepatitis A dynamics in the most affected Italian areas. Intervention options such as targeted vaccination, social distancing measures (e.g., closure of day care centers and kindergartens) and improvements in standards of living and hygiene are evaluated. Results show that a very low vaccination coverage is sufficient to control hepatitis A in Italy, while its elimination is not possible since new cases are continuously imported from high endemicity areas outside the country. Finally, the considered social distancing measures can be counterproductive since the fraction of recovered individuals does not decline while the age at infection increases, thus augmenting the probability of developing acute symptoms.     

A mathematical model for water and nutrient uptake by plant root systems

This article deals with modelling the simultaneous uptake of water and highly buffered nutrient, such as phosphate, by root branching structures from partially saturated soil. We use the simultaneous water and nutrient uptake model to investigate the effect that water movement has on nutrient uptake. With the aid of this model we are also able to show that the previous models by Barber and Tinker and Nye systematically underestimated the phosphate uptake, due to the oversimplified approach in dealing with root branching structure. In this article we show how this discrepancy can be remedied and the root branching structure included in the models of plant nutrient uptake. We will also discuss the differences in the results for continuous and spot fertilization combined with variable rainfall.     

Appendix: A model of plaque formation

Equations describing plaque formation in soft agar have been based on certain simplifying assumptions, for which data are presented. The derived equations permit one to calculate (i) average plaque size as a function of the initial density of indicator cells (D_o), (ii) the number of cells lysed per plaque as a function of D_o, and (iii) the cumulative number of cells lysed at various stages of plaque development. The calculated values agree well with those determined experimentally.     

Tracheid analysis and modeling of the minor veins of the coleus and smilax leaves

Tracheid analysis was carried out on the veinlets and minor veins of the coleus (Solenostemon scutellarioides [L.] Codd) leaf. Third- to fifth-order, or minor, veins average 3.4 tracheids in tandem and they bipartition islets when these enclosed islets reach a critical size; both these features of vein length and islet size contribute to a self-similar process of vein pattern generation. An areole was calculated to be initially comprised of about ten cells making the patterning event for vein formation requiring only a few cells. An algorithmic model developed here for minor vein formation includes five production rules, and this computer model explains the 3–4 tracheids per minor vein, presence of isolated tracheids, the structure of veinlets, and the elaborate branching patterns of veinlets in coleus and other plants.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

Light quality regulation of endogenous levels of auxin,abscisic acid and ethylene production in petioles and leaves of wild type and ACC deaminase transgenic <Emphasis Type="Italic">Brassica napus</Emphasis> seedlings

Brassica napus L. seedlings responded to low red to far-red (R/FR) ratio by elongating petioles and decreasing leaf expansion. These typical shade avoidance traits were correlated with significantly decreased endogenous indole-3-acetic acid (IAA) levels and significantly increased endogenous abscisic acid (ABA) levels and ethylene production. The transgenic (T) B. napus line bearing the bacterial ACC deaminase gene, did not respond to low R/FR ratio with altered petiole and leaf growth and less ethylene (especially by petioles) was produced. As with WT seedlings, T seedlings had significantly lower IAA levels in both petioles and leaves under low R/FR ratio. However, ABA levels of low R/FR ratio-grown T seedlings either increased (petioles) or were unaltered (leaves). Our results further suggest that low R/FR ratio regulates endogenous IAA levels independently of ethylene, but there may be an interaction between ABA and ethylene in leaf development.     

A novel feature of structural variegation in leaves of the tropical plant Schismatoglottis calyptrata

We report a novel feature of leaf variegation. As is often the case in tropical forest floor herbs, Schismatoglottis calyptrata leaves feature structural variegation. Examination of leaf anatomy in S. calyptrata revealed a novel feature of structural variegation, which was generated by variation in the spatial arrangement of the adaxial-most tip of the palisade cells. In fully green leaf parts, contact between the adaxial-most tip of the palisade cells and the cone-shaped base of the outer epidermis cells was tight, and palisade cells were arranged radially around each epidermal cell. In dull, grayish-green leaf parts, the contact was loose, and no particular spatial arrangement of palisade cells relative to epidermal cells was observed. This feature of structural variation could be disadvantageous for photosynthesis efficiency in view of the hypothesis that, for rainforest herbs, cone-shaped epidermal cells may function as lenses. However, the high frequency of leaf variegation of S. calyptrata in natural habits suggests that this structural variegation plays an unknown advantageous role.