Lockhart Equation Research Articles

Relative growth rate for trees at the growth stage is coordinated with leaf bulk modulus of elasticity and osmotic potential in a subtropical forest of China

Current theoretical models for tree relative growth rate (RGR) have considered physiological processes based on carbon economy and nutrient-productivity relations, but have not taken water relations into account. According to the ecological energetics and Lockhart equation theories, if more energy is needed to maintain water balance in plants, less energy should be invested towards growth, resulting in a trade-off between growth and drought tolerance. Leaf pressure–volume (PV) curve parameters are the best recognised classical indicators of plant drought tolerance, however, the relationships between tree growth and leaf PV parameters have rarely been investigated in forest communities. In the study, we selected two evergreen (Lithocarpus glaber, Cyclobalanopsis oxyodon) and two deciduous species (Quercus serrata, Platycarya strobilacea) that dominate northern subtropical forests on Mt. Shennongjia, central China. For each species, we selected individuals at different developmental stages (growth vs. mature stage), and measured the relative growth rate of diameter at breast height (RGRDBH) and leaf PV curve parameters, including leaf saturated osmotic potential (Ψsat), maximum bulk elastic modulus (εmax), turgor loss osmotic potential (Ψtlp) and leaf water capacity (Cleaf). Correlation analysis and structural equation models (SEM) were applied to explore the relationships between RGRDBH and PV parameters. The results showed that: 1) at both stages, RGRDBH was positively correlated with Ψsat, but not with Ψtlp and Cleaf; 2) εmax was negatively related to RGRDBH at the growth stage; 3) the relationship between PV parameters and RGRDBH was determined mainly by inter-specific difference; 4) SEM showed that both Ψtlp and Ψsat were affected by εmax; and 5) due to the interaction between the PV parameters, Ψsat and εmax had a direct effect on RGRDBH only during the growth stage. Our results showed that the functional/structural coordination between PV parameters adjusted the RGRDBH of trees, and among different species, there exists a tradeoff between energy allocation for drought tolerance and growth. These findings can be applied to improve RGR models, and show how forests respond to warm-induced drought, which is a major threat to global forests in a future warmer climate.

Water potential control of turgor-driven tracheid enlargement in Scots pine at its xeric distribution edge.

The extent to which water availability can be used to predict the enlargement and final dimensions of xylem conduits remains an open issue. We reconstructed the time course of tracheid enlargement in Pinus sylvestris trees in central Spain by repeated measurements of tracheid diameter on microcores sampled weekly during a 2yr period. We analyzed the role of water availability in these dynamics empirically through time-series correlation analysis and mechanistically by building a model that simulates daily tracheid enlargement rate and duration based on Lockhart's equation and water potential as the sole input. Tracheid enlargement followed a sigmoid-like time course, which varied intra- and interannually. Our empirical analysis showed that final tracheid diameter was strongly related to water availability during tracheid enlargement. The mechanistic model was calibrated and successfully validated (R2 =0.92) against the observed tracheid enlargement time course. The model was also able to reproduce the seasonal variations of tracheid enlargement rate, duration and final diameter (R2 =0.84-0.99). Our results support the hypothesis that tracheid enlargement and final dimensions can be modeled based on the direct effect of water potential on turgor-driven cell expansion. We argue that such a mechanism is consistent with other reported patterns of tracheid dimension variation.

Introducing turgor-driven growth dynamics into functional-structural plant models.

In many scenarios the availability of assimilated carbon is not the constraining factor of plant growth. Rather, organ growth appears driven by sink activity in which water availability plays a determinant role. Current functional-structural plant models (FSPMs) mainly focus on plant-carbon relations and largely disregard the importance of plant water status in organogenesis. Consequently, incorporating a turgor-driven growth concept, coupling carbon and water dynamics in an FSPM, presents a significant improvement towards capturing plant development in a more mechanistic manner. An existing process-based water flow and storage model served as a basis for implementing water control in FSPMs. Its concepts were adjusted to the scale of individual plant organs and interwoven with the basic principles of modelling carbon dynamics to allow evaluation of turgor pressure across the entire plant. This was then linked to plant organ growth by applying the principles of the widely used Lockhart equation. This model successfully integrates a mechanistic understanding of plant water transport dynamics coupled with simple carbon dynamics within a dynamically developing plant architecture. It allows evaluation of turgor pressure on the scale of plant organs, resulting in clear diel and long-term patterns, directly linked to plant organ growth. A conceptual sap flow and turgor-driven growth model was introduced for functional-structural plant modelling. It is applicable to any plant architecture and allows visual exploration of the diel patterns of organ water content and growth. Integrated in existing FSPMs, this new concept fosters an array of possibilities for FSPMs, as it presents a different formulation of growth in terms of local processes, influenced by local and external conditions.

Newly designed organotin(IV) carboxylates with peptide linkage: Synthesis, structural elucidation, physicochemical characterizations and pharmacological investigations

Fourteen new organotin(IV) carboxylate complexes with peptide linkage of (2-(4-methoxy-2-nitrophenylcarbamoyl)benzoic acid) were successfully synthesized and characterized by elemental analyses, FT-IR, NMR (1H, 13C and 119Sn) and single crystal X-ray techniques. FT-IR results of the sodium salt of 2-(4-methoxy-2-nitrophenylcarbamoyl)benzoic acid and complexes showed that the coordination took place via oxygen atoms of the carboxylate group. 1J(119Sn-13C), 2J(119Sn-1H) and θ values calculated from 1H and 13C NMR data using Lockhart's equation reveal a trigonal bipyramidal geometry for triorganotin(IV) derivatives and an octahedral geometry for diorganotin(IV) derivatives. Crystallographic data for three triorganotin(IV) complexes (1–3) showed the tin has distorted trigonal bipyramidal geometry. In vitro anticancer activity against lung carcinoma (H-157) and kidney fibroblast (BHK-21) cell lines as well as antileishmanial activity against the promastigote forms of leishmania major of the synthesized compounds were also studied and the complexes were found biologically active. The in vitro antibacterial activity of most of the synthesized organotin(IV) derivatives against the studied bacterial pathogens is higher than those of the standard 3rd generation antibiotics such as Tetracycline, Penicillin G, Ampicillin, Amoxicillin. This suggest the use of these newly designed organotin(IV) derivatives as potent antibiotics. The synthesized compounds interact with DNA via intercalative mode of interaction. Viscosity measurement results also support the intercalative mode of interaction for the compounds with DNA.

Synthesis, spectroscopic characterization, and biological screening of levofloxacin based organotin(IV) derivatives

Four new organotin (IV) complexes with general formula R3SnL/R2SnL2, where R = CH3, n-C4H9, C6H5 and L = Levofloxacin, were synthesized and characterized by elemental analyses, FT-IR and NMR (1H and 13C) spectroscopy. Spectroscopic data suggested a six-coordinated geometry for diorganotin(IV) derivatives and a five-coordinated geometry for triorganotin(IV) derivatives. The value of Me–Sn–Me bond angle for di- and trimethyltin complexes using the Lockhart equation, were 150° and 116°, respectively, that corresponded to six and five-coordinate geometry, accordingly. The ligand and its complexes were screened for their antibacterial, antifungal, cytotoxic, and free radical scavenging (DPPH) antioxidant activities. The biological data indicated those as potentially bioactive in each field of the study. Accumulated data of DNA interaction with the synthesized complexes based on UV-Vis, cyclic voltammetry and viscometry suggested an intercalative mode of the interaction.

A quantitative report on the impact of chloride on the kinetic coefficients of auxin-induced growth: a numerical contribution to the "acid growth hypothesis".

This work presents the application of several our own novel methods of analysing the kinetics of plant growth, which create, among others, a common platform for the comparison of experimental results. A relatively simple formula is used to parameterize the wide range of data that has been obtained for Zea mays L. in the literature, though it can also be used for different species. A biophysical/biochemical interpretation of the parameters was obtained from a theoretical model that is based on a modified Lockhart equation. The derived formula, which was extended for practical use in Zajdel et al. (Acta Physiol Plant 38:5, 2016), and which was implemented in the attached computer program (ibid.), allowed the data that was obtained from the growth-related problems to be parameterized in a simple way. As a working example that shows the robustness of our approach, we comment in detail on the qualitative assessments of the impact of chloride ions on auxin-induced growth. We note that calculated continuous curves (fits), which are rooted in the growth functional that was introduced by Pietruszka (J Theor Biol 315:119–127, 2012), were in a perfect agreement (R2 ~ 0.99998) with the raw experimental data that was published recently by Burdach et al. (Ann Bot 114:1023–1034, 2014). This fact justified the use of this strict technique, which allows for the determination of kinetic coefficients, to critically evaluate the results and suppositions (claims) therein. Moreover, we calculated the time-delay derivative of elongation growth—pH cross-correlations, and validated the “acid growth hypothesis” in figures by considering, amongst others, the magnitude of the H+-activity of elongation growth (per μm). An empirical constant (field strength), EH+ = Em/(log10 1/aH+ ∙ μm) = 0.157 ± 0.009 [V/mm] was obtained, where Em [mV] is the membrane potential in the perenchymal coleoptile cells of Zea mays L. When this relation is known, the membrane potential can not only be determined for intact growth, but also for different intervening substances exclusively from growth (or growth rate) and pH measurements, i.e. without performing electrophysiological measurements. However, the question of whether this constant is universal remains open.Electronic supplementary materialThe online version of this article (doi:10.1186/s40064-016-3626-y) contains supplementary material, which is available to authorized users.

Limitation of Cell Elongation in Barley (Hordeum vulgare L.) Leaves Through Mechanical and Tissue-Hydraulic Properties.

The aim of the present study was to assess the mechanical and hydraulic limitation of growth in leaf epidermal cells of barley (Hordeum vulgare L.) in response to agents which affect cellular water (mercuric chloride, HgCl(2)) and potassium (cesium chloride, CsCl; tetraethylammonium, TEA) transport, pump activity of plasma membrane H(+)-ATPase and wall acidification (fusicoccin, FC). Cell turgor (P) was measured with the cell pressure probe, and cell osmotic pressure (π) was analyzed through picoliter osmometry of single-cell extracts. A wall extensibility coefficient (M) and tissue hydraulic conductance coefficient (L) were derived using the Lockhart equation. There was a significant positive linear relationship between relative elemental growth rate and P, which fit all treatments, with an overall apparent yield threshold of 0.368 MPa. Differences in growth between treatments could be explained through differences in P. A comparison of L and M showed that growth in all except the FC treatment was co-limited through hydraulic and mechanical properties, though to various extents. This was accompanied by significant (0.17-0.24 MPa) differences in water potential (ΔΨ) between xylem and epidermal cells in the leaf elongation zone. In contrast, FC-treated leaves showed ΔΨ close to zero and a 10-fold increase in L.

Another Brick in the Cell Wall: Biosynthesis Dependent Growth Model

Expansive growth of plant cell is conditioned by the cell wall ability to extend irreversibly. This process is possible if (i) a tensile stress is developed in the cell wall due to the coupling effect between turgor pressure and the modulation of its mechanical properties through enzymatic and physicochemical reactions and if (ii) new cell wall elements can be synthesized and assembled to the existing wall. In other words, expansive growth is the result of coupling effects between mechanical, thermal and chemical energy. To have a better understanding of this process, models must describe the interplay between physical or mechanical variable with biological events. In this paper we propose a general unified and theoretical framework to model growth in function of energy forms and their coupling. This framework is based on irreversible thermodynamics. It is then applied to model growth of the internodal cell of Chara corallina modulated by changes in pressure and temperature. The results describe accurately cell growth in term of length increment but also in term of cell pectate biosynthesis and incorporation to the expanding wall. Moreover, the classical growth model based on Lockhart's equation such as the one proposed by Ortega, appears as a particular and restrictive case of the more general growth equation developed in this paper.

A model of crosslink kinetics in the expanding plant cell wall: Yield stress and enzyme action

The plant primary cell wall is a composite material containing stiff cellulose microfibrils that are embedded within a pectin matrix and crosslinked through a network of hemicellulose polymers. This microstructure endows the wall with nonlinear anisotropic mechanical properties and allows enzymatic regulation of expansive cell growth. We present a mathematical model of hemicellulose crosslink dynamics in an expanding cell wall incorporating strain-enhanced breakage and enzyme-mediated crosslink kinetics. The model predicts the characteristic yielding behaviour in the relationship between stress and strain-rate seen experimentally, and suggests how the effective yield and extensibility of the wall depend on microstructural parameters and on the action of enzymes of the XTH and expansin families. The model suggests that the yielding behaviour encapsulated in the classical Lockhart equation can be explained by the strongly nonlinear dependence of crosslink breakage rate on crosslink elongation. The model also demonstrates how enzymes that target crosslink binding can be effective in softening the wall in its pre-yield state, whereas its post-yield extensibility is determined primarily by the pectin matrix.

USING SWAF, A GENERIC BIOPHYSICAL MODEL OF SUGAR AND WATER ACCUMULATION TO ANALYZE FRUIT DEVELOPMENT

ISHS XXVIII International Horticultural Congress on Science and Horticulture for People (IHC2010): International Symposium on Plant Physiology from Cell to Fruit Production System USING SWAF, A GENERIC BIOPHYSICAL MODEL OF SUGAR AND WATER ACCUMULATION TO ANALYZE FRUIT DEVELOPMENT

Factors controlling plasticity of leaf morphology in Robinia pseudoacacia: III. biophysical constraints on leaf expansion under long‐term water stress

In this article, we measured the relative growth rate (RGR) of leaves of Robinia pseudoacacia seedlings under well-watered and water-stressed conditions (mid-day Ψ(w) = leaf water potential estimated with a pressure bomb of -0.48 and -0.98 MPa, respectively). Pressure-volume (PV) curves were done on growing leaves at 25, 50 and 95% of the mature size (growth stage) in order to compute solute potential (Ψ) and turgor pressure (Ψ(P) ) as a function of Ψ(w) . The PV curves and diurnal measurements of Ψ(w) and RGR allowed us to evaluate the parameters (cell wall extensibility m and growth turgor threshold Y) of the Lockhart equation, RGR = m(Ψ(P)-Y), at each growth stage. Our data showed that m and Y did change with leaf age, but the changes were slow enough to evaluate m and Y on any given day. We believe this is the first study to provide evidence that the Lockhart equation adequately quantifies leaf growth of trees over a range of time domains. The value of m linearly declined and Y linearly increased with growth stage. Also, mild drought stress caused a decline in m and increase in Y relative to controls. Although water stress caused an osmotic adjustment which, in turn, increased Ψ(P) in stressed plants relative to controls, the RGR and final leaf sizes were reduced in water-stressed plants because of the impact of water stress on decreased m and increased Y.

Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth.

Turgor generates the stress that leads to the expansion of plant cell walls during cellular growth. This has been formalized by the Lockhart equation, which can be derived from the physical laws of the deformation of viscoelastic materials. However, the experimental evidence for such a direct correlation between growth rate and turgor is inconclusive. This has led to challenges of the Lockhart model. We model the oscillatory growth of pollen tubes to investigate this relationship. We couple the Lockhart equation to the dynamical equations for the change in material properties. We find that the correct implementation of the Lockhart equation within a feedback loop leading to low amplitude oscillatory growth predicts that in this system changes in the global turgor do not influence the average growth rate in a linear manner, consistent with experimental observations. An analytic analysis of our model demonstrates in which regime the average growth rate becomes uncorrelated from the turgor pressure.

Solutions for a local equation of anisotropic plant cell growth: an analytical study of expansin activity

This paper presents a generalization of the Lockhart equation for plant cell/organ expansion in the anisotropic case. The intent is to take into account the temporal and spatial variation in the cell wall mechanical properties by considering the wall 'extensibility' (Φ), a time- and space-dependent parameter. A dynamic linear differential equation of a second-order tensor is introduced by describing the anisotropic growth process with some key biochemical aspects included. The distortion and expansion of plant cell walls initiated by expansins, a class of proteins known to enhance cell wall 'extensibility', is also described. In this approach, expansin proteins are treated as active agents participating in isotropic/anisotropic growth. Two-parameter models and an equation for describing α- and β-expansin proteins are proposed by delineating the extension of isolated wall samples, allowing turgor-driven polymer creep, where expansins weaken the non-covalent binding between wall polysaccharides. We observe that the calculated halftime (t(1/2) = εΦ(0) log 2) of stress relaxation due to expansin action can be described in mechanical terms.

A fibre-reinforced fluid model of anisotropic plant cell growth

Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility, and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation.

Exact analytic solutions for a global equation of plant cell growth

A generalization of the Lockhart equation for plant cell expansion in isotropic case is presented. The goal is to account for the temporal variation in the wall mechanical properties—in this case by making the wall extensibility a time dependent parameter. We introduce a time-differential equation describing the plant growth process with some key biophysical aspects considered. The aim of this work was to improve prior modeling efforts by taking into account the dynamic character of the plant cell wall with characteristics reminiscent of damped (aperiodic) motion. The equations selected to encapsulate the time evolution of the wall extensibility offer a new insight into the control of cell wall expansion. We find that the solutions to the time dependent second order differential equation reproduce much of the known experimental data for long- and short-time scales. Additionally, in order to support the biomechanical approach, a new growth equation based on the action of expansin proteins is proposed. Remarkably, both methods independently converge to the same kind, sigmoid-shaped, growth description functional V ( t ) ∝ exp ( − exp ( − t ) ) , properly describing the volumetric growth and, consequently, growth rate as its time derivative.

Superposition of the Transpiration-induced Water Potential and the Growth-induced Water Potential Associated with Expanding Tomato Leaves

The growth parameters of Lockhart's equation were determined in expanding leaves of tomato plants (Solanum lycopersicum L.) subjected to different relative humidity in the dark. In tomato leaves, sizes of the growth-induced water potential exhibited linear relations with the relative growth rates (RGR) under various vapor pressure difference (VPD) conditions between leaves and the ambient air. When transpiration rates of tomato plants became significantly larger, lines formed between the growth-induced water potential and RGR had a translational relation as VPD increased. Then, the transpiration-induced water potential became proportionally larger as VPD increased. These results indicated that transpiration flux and growth flux have additive relations, suggesting that water fluxes for growth and transpiration are linearly superimposed.

Effects of Water Flow from the Xylem on the Growth-induced Water Potential and the Growth-effective Turgor Associated with Enlarging Tomato Fruit

Intact tomato (Solanum lycopersicum L.) fruit growth was monitored while heat-girdling treatment was conducted. Heat-girdling treatments revealed that most water flow into enlarging fruit of tomato plants originated from the xylem throughout fruit growth periods. After measuring the water status and growth rates simultaneously, the growth parameters of Lockhart's equation were determined in expanding fruit of tomato plants. During the early stage of fruit growth, when the growth-induced water potential was obtained by calculating the difference between water potentials of the growing region and the water source, it was found that sizes of the growth-induced water potential and the hydraulic conductance were predominant factors to regulate cell expansion. When fruit became almost mature, sizes of the growth-effective turgor and the wall extensibility were primary factors to regulate cell expansion, but not the water potential gradients. Such changes in water flow from the xylem related to enlarging processes of fruit may explain how fruit cracking takes place right before fruit maturation.

The mechanics behind plant development

Morphogenesis in living organisms relies on the integration of both biochemical and mechanical signals. During the last decade, attention has been mainly focused on the role of biochemical signals in patterning and morphogenesis, leaving the contribution of mechanics largely unexplored. Fortunately, the development of new tools and approaches has made it possible to re-examine these processes. In plants, shape is defined by two local variables: growth rate and growth direction. At the level of the cell, these variables depend on both the cell wall and turgor pressure. Multidisciplinary approaches have been used to understand how these cellular processes are integrated in the growing tissues. These include quantitative live imaging to measure growth rate and direction in tissues with cellular resolution. In parallel, stress patterns have been artificially modified and their impact on strain and cell behavior been analysed. Importantly, computational models based on analogies with continuum mechanics systems have been useful in interpreting the results. In this review, we will discuss these issues focusing on the shoot apical meristem, a population of stem cells that is responsible for the initiation of the aerial organs of the plant.

Plant cell walls: supramolecular assembly, signalling and stress

The structure of the primary cell wall in non-graminaceous plants is briefly reviewed and its role in providing mechanical strength to the plant and protecting it from microbial infection are described. A variety of signalling mechanisms involve oligosaccharides released by glycanase enzymes from microbial pathogens, and some of the mechanisms may be implicated in the regulation of metabolism in ripening fruits. There is some evidence that cell walls are able to sense damage or loss of integrity and that signals can accordingly be passed back to the cytoplasm. Primary cell walls must combine the mechanical and other functions with the capacity to grow in a controlled way. A modification of the ‘Molecular Velcro’ model developed originally to describe deformation of wood is used to predict load-deformation curves like those described by the Lockhart equation for the relationship between turgor stress and growth. Predicting a stress threshold for growth does not require the assumption of enzyme activity, although in fact enzyme activity is indeed required to permit growth at the rates normally observed.

General proof of the validity of a new tensor equation of plant growth

Plant cell/organ growth may be partly described by a local tensor equation. We provide a mathematical proof that the Lockhart (global) equation is the diagonal component of this tensor equation.