Couple stresses and discrete potentials in the vertex model of cellular monolayers

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

Surfactant amplifies yield-stress effects in the capillary instability of a film coating a tube

To assess how the presence of surfactant in lung airways alters the flow of mucus that leads to plug formation and airway closure, we investigate the effect of insoluble surfactant on the instability of a viscoplastic liquid coating the interior of a cylindrical tube. Evolution equations for the layer thickness using thin-film and long-wave approximations are derived that incorporate yield-stress effects and capillary and Marangoni forces. Using numerical simulations and asymptotic analysis of the thin-film system, we quantify how the presence of surfactant slows growth of the Rayleigh-Plateau instability, increases the size of initial perturbation required to trigger instability and decreases the final peak height of the layer. When the surfactant strength is large, the thin-film dynamics coincide with the dynamics of a surfactant-free layer but with time slowed by a factor of four and the capillary Bingham number, a parameter proportional to the yield stress, exactly doubled. By solving the long-wave equations numerically, we quantify how increasing surfactant strength can increase the critical layer thickness for plug formation to occur and delay plugging. The previously established effect of the yield stress in suppressing plug formation [Shemilt et al., J. Fluid Mech., 2022, vol. 944, A22] is shown to be amplified by introducing surfactant. We discuss the implications of these results for understanding the impact of surfactant deficiency and increased mucus yield stress in obstructive lung diseases.

Modeling collagen fibril self-assembly from extracellular medium in embryonic tendon

Collagen is a key structural component of multicellular organisms and is arranged in a highly organized manner. In structural tissues such as tendons, collagen forms bundles of parallel fibers between cells, which appear within a 24-h window between embryonic day 13.5 (E13.5) and E14.5 during mouse embryonic development. Current models assume that the organized structure of collagen requires direct cellular control, whereby cells actively lay down collagen fibrils from cell surfaces. However, such models appear incompatible with the time and length scales of fibril formation. We propose a phase-transition model to account for the rapid development of ordered fibrils in embryonic tendon, reducing reliance on active cellular processes. We develop phase-field crystal simulations of collagen fibrillogenesis in domains derived from electron micrographs of inter-cellular spaces in embryonic tendon and compare results qualitatively and quantitatively to observed patterns of fibril formation. To test the prediction of this phase-transition model that free protomeric collagen should exist in the inter-cellular spaces before the formation of observable fibrils, we use laser-capture microdissection, coupled with mass spectrometry, which demonstrates steadily increasing free collagen in inter-cellular spaces up to E13.5, followed by a rapid reduction of free collagen that coincides with the appearance of less-soluble collagen fibrils. The model and measurements together provide evidence for extracellular self-assembly of collagen fibrils in embryonic mouse tendon, supporting an additional mechanism for rapid collagen fibril formation during embryonic development.

Red blood cell dynamics in extravascular biological tissues modelled as canonical disordered porous media

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the microhaemodynamics of a vascular network is underpinned by its interconnected structure, and certain structural alterations such as capillary dilation and blockage can substantially change blood flow patterns. However, for extravascular media with disordered microstructure (e.g. the porous intervillous space in the placenta), it remains unclear how the medium’s structure affects the haemodynamics. Here, we simulate cellular blood flow in simple models of canonical porous media representative of extravascular biological tissue, with corroborative microfluidic experiments performed for validation purposes. For the media considered here, we observe three main effects: first, the relative apparent viscosity of blood increases with the structural disorder of the medium; second, the presence of red blood cells (RBCs) dynamically alters the flow distribution in the medium; third, symmetry breaking introduced by moderate structural disorder can promote more homogeneous distribution of RBCs. Our findings contribute to a better understanding of the cell-scale haemodynamics that mediates the relationship linking the function of certain biological tissues to their microstructure.

Model-based Bayesian inference of the ventilation distribution in patients with cystic fibrosis from multiple breath washout, with comparison to ventilation MRI

BACKGROUND

Indices of ventilation heterogeneity (VH) from multiple breath washout (MBW) have been shown to correlate well with VH indices derived from hyperpolarised gas ventilation MRI. Here we report the prediction of ventilation distributions from MBW data using a mathematical model, and the comparison of these predictions with imaging data.

METHODS

We developed computer simulations of the ventilation distribution in the lungs to model MBW measurement with 3 parameters: σ_V, determining the extent of VH; V₀, the lung volume; and V_D, the dead-space volume. These were inferred for each individual from supine MBW data recorded from 25 patients with cystic fibrosis (CF) using approximate Bayesian computation. The fitted models were used to predict the distribution of gas imaged by ³He ventilation MRI measurements collected from the same visit.

RESULTS

The MRI indices measured (I_1/3, the fraction of pixels below one-third of the mean intensity and I_CV, the coefficient of variation of pixel intensity) correlated strongly with those predicted by the MBW model fits (r=0.93,0.88 respectively). There was also good agreement between predicted and measured MRI indices (mean bias ± limits of agreement: I_1/3:-0.003±0.118 and I_CV:-0.004±0.298). Fitted model parameters were robust to truncation of MBW data.

CONCLUSION

We have shown that the ventilation distribution in the lung can be inferred from an MBW signal, and verified this using ventilation MRI. The Bayesian method employed extracts this information with fewer breath cycles than required for LCI, reducing acquisition time required, and gives uncertainty bounds, which are important for clinical decision making.

Advection-dominated transport past isolated disordered sinks: stepping beyond homogenization

We investigate the transport of a solute past isolated sinks in a bounded domain when advection is dominant over diffusion, evaluating the effectiveness of homogenization approximations when sinks are distributed uniformly randomly in space. Corrections to such approximations can be non-local, non-smooth and non-Gaussian, depending on the physical parameters (a Péclet number Pe, assumed large, and a Damköhler number Da) and the compactness of the sinks. In one spatial dimension, solute distributions develop a staircase structure for large Pe , with corrections being better described with credible intervals than with traditional moments. In two and three dimensions, solute distributions are near-singular at each sink (and regularized by sink size), but their moments can be smooth as a result of ensemble averaging over variable sink locations. We approximate corrections to a homogenization approximation using a moment-expansion method, replacing the Green's function by its free-space form, and test predictions against simulation. We show how, in two or three dimensions, the leading-order impact of disorder can be captured in a homogenization approximation for the ensemble mean concentration through a modification to Da that grows with diminishing sink size.

Collagen fibril assembly: New approaches to unanswered questions

Collagen fibrils are essential for metazoan life. They are the largest, most abundant, and most versatile protein polymers in animals, where they occur in the extracellular matrix to form the structural basis of tissues and organs. Collagen fibrils were first observed at the turn of the 20th century. During the last 40 years, the genes that encode the family of collagens have been identified, the structure of the collagen triple helix has been solved, the many enzymes involved in the post-translational modifications of collagens have been identified, mutations in the genes encoding collagen and collagen-associated proteins have been linked to heritable disorders, and changes in collagen levels have been associated with a wide range of diseases, including cancer. Yet despite extensive research, a full understanding of how cells assemble collagen fibrils remains elusive. Here, we review current models of collagen fibril self-assembly, and how cells might exert control over the self-assembly process to define the number, length and organisation of fibrils in tissues.

Spectral graph theory efficiently characterizes ventilation heterogeneity in lung airway networks

This paper introduces a linear operator for the purposes of quantifying the spectral properties of transport within resistive trees, such as airflow in lung airway networks. The operator, which we call the Maury matrix, acts only on the terminal nodes of the tree and is equivalent to the adjacency matrix of a complete graph summarizing the relationships between all pairs of terminal nodes. We show that the eigenmodes of the Maury operator have a direct physical interpretation as the relaxation, or resistive, modes of the network. We apply these findings to both idealized and image-based models of ventilation in lung airway trees and show that the spectral properties of the Maury matrix characterize the flow asymmetry in these networks more concisely than the Laplacian modes, and that eigenvector centrality in the Maury spectrum is closely related to the phenomenon of ventilation heterogeneity caused by airway narrowing or obstruction. This method has applications in dimensionality reduction in simulations of lung mechanics, as well as for characterization of models of the airway tree derived from medical images.

Force networks, torque balance and Airy stress in the planar vertex model of a confluent epithelium

The vertex model is a popular framework for modelling tightly packed biological cells, such as confluent epithelia. Cells are described by convex polygons tiling the plane and their equilibrium is found by minimizing a global mechanical energy, with vertex locations treated as degrees of freedom. Drawing on analogies with granular materials, we describe the force network for a localized monolayer and derive the corresponding discrete Airy stress function, expressed for each N-sided cell as N scalars defined over kites covering the cell. We show how a torque balance (commonly overlooked in implementations of the vertex model) requires each internal vertex to lie at the orthocentre of the triangle formed by neighbouring edge centroids. Torque balance also places a geometric constraint on the stress in the neighbourhood of cellular trijunctions, and requires cell edges to be orthogonal to the links of a dual network that connect neighbouring cell centres and thereby triangulate the monolayer. We show how the Airy stress function depends on cell shape when a standard energy functional is adopted, and discuss implications for computational implementations of the model.

Quantifying the impact of tissue metabolism on solute transport in feto-placental microvascular networks

The primary exchange units in the human placenta are terminal villi, in which fetal capillary networks are surrounded by a thin layer of villous tissue, separating fetal from maternal blood. To understand how the complex spatial structure of villi influences their function, we use an image-based theoretical model to study the effect of tissue metabolism on the transport of solutes from maternal blood into the fetal circulation. For solute that is taken up under first-order kinetics, we show that the transition between flow-limited and diffusion-limited transport depends on two new dimensionless parameters defined in terms of key geometric quantities, with strong solute uptake promoting flow-limited transport conditions. We present a simple algebraic approximation for solute uptake rate as a function of flow conditions, metabolic rate and villous geometry. For oxygen, accounting for nonlinear kinetics using physiological parameter values, our model predicts that villous metabolism does not significantly impact oxygen transfer to fetal blood, although the partitioning of fluxes between the villous tissue and the capillary network depends strongly on the flow regime.

Physical and geometric determinants of transport in fetoplacental microvascular networks

Across mammalian species, solute exchange takes place in complex microvascular networks. In the human placenta, the primary exchange units are terminal villi that contain disordered networks of fetal capillaries and are surrounded externally by maternal blood. We show how the irregular internal structure of a terminal villus determines its exchange capacity for diverse solutes. Distilling geometric features into three parameters, obtained from image analysis and computational fluid dynamics, we capture archetypal features of the structure-function relationship of terminal villi using a simple algebraic approximation, revealing transitions between flow- and diffusion-limited transport at vessel and network levels. Our theory accommodates countercurrent effects, incorporates nonlinear blood rheology, and offers an efficient method for testing network robustness. Our results show how physical estimates of solute transport, based on carefully defined geometrical statistics, provide a viable method for linking placental structure and function and offer a framework for assessing transport in other microvascular systems.

Blood flow and transport in the human placenta

The placenta is a multi-functional organ that exchanges blood gases and nutrients between a mother and her developing fetus. In humans, fetal blood flows through intricate networks of vessels confined within villous trees, the branches of which are bathed in pools of maternal blood. Fluid mechanics and transport processes play a central role in understanding how these elaborate structures contribute to the function of the placenta, and how their disorganization may lead to disease. Recent advances in imaging and computation have spurred significant advances in simulations of fetal and maternal flows within the placenta, across a range of lengthscales. Models describe jets of maternal blood emerging from spiral arteries into a disordered and deformable porous medium, and solute uptake by fetal blood flowing through elaborate three-dimensional capillary networks. We survey recent developments and emerging challenges in modeling flow and transport in this complex organ.

Modelling structural determinants of ventilation heterogeneity: A perturbative approach

We have developed a computational model of gas mixing and ventilation in the human lung represented as a bifurcating network. We have simulated multiple-breath washout (MBW), a clinical test for measuring ventilation heterogeneity (VH) in patients with obstructive lung conditions. By applying airway constrictions inter-regionally, we have predicted the response of MBW indices to obstructions and found that they detect a narrow range of severe constrictions that reduce airway radius to 10%–30% of healthy values. These results help to explain the success of the MBW test to distinguish obstructive lung conditions from healthy controls. Further, we have used a perturbative approach to account for intra-regional airway heterogeneity that avoids modelling each airway individually. We have found, for random airway heterogeneity, that the variance in MBW indices is greater when indices are already elevated due to constrictions. By quantifying this effect, we have shown that variability in lung structure and mechanical properties alone can lead to clinically significant variability in MBW indices (specifically the Lung Clearance Index—LCI, and the gradient of phase-III slopes—S_cond), but only in cases simulating obstructive lung conditions. This method is a computationally efficient way to probe the lung’s sensitivity to structural changes, and to quantify uncertainty in predictions due to random variations in lung mechanical and structural properties.

Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model

Using a popular vertex-based model to describe a spatially disordered planar epithelial monolayer, we examine the relationship between cell shape and mechanical stress at the cell and tissue level. Deriving expressions for stress tensors starting from an energetic formulation of the model, we show that the principal axes of stress for an individual cell align with the principal axes of shape, and we determine the bulk effective tissue pressure when the monolayer is isotropic at the tissue level. Using simulations for a monolayer that is not under peripheral stress, we fit parameters of the model to experimental data for Xenopus embryonic tissue. The model predicts that mechanical interactions can generate mesoscopic patterns within the monolayer that exhibit long-range correlations in cell shape. The model also suggests that the orientation of mechanical and geometric cues for processes such as cell division are likely to be strongly correlated in real epithelia. Some limitations of the model in capturing geometric features of Xenopus epithelial cells are highlighted.

A Mechanosensitive RhoA Pathway that Protects Epithelia against Acute Tensile Stress

Adherens junctions are tensile structures that couple epithelial cells together. Junctional tension can arise from cell-intrinsic application of contractility or from the cell-extrinsic forces of tissue movement. Here, we report a mechanosensitive signaling pathway that activates RhoA at adherens junctions to preserve epithelial integrity in response to acute tensile stress. We identify Myosin VI as the force sensor, whose association with E-cadherin is enhanced when junctional tension is increased by mechanical monolayer stress. Myosin VI promotes recruitment of the heterotrimeric Gα12 protein to E-cadherin, where it signals for p114 RhoGEF to activate RhoA. Despite its potential to stimulate junctional actomyosin and further increase contractility, tension-activated RhoA signaling is necessary to preserve epithelial integrity. This is explained by an increase in tensile strength, especially at the multicellular vertices of junctions, that is due to mDia1-mediated actin assembly.

Mechanical characterization of disordered and anisotropic cellular monolayers

We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue’s anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.

Effect of intermittent inspiratory leaks on measurement of lung clearance index using nitrogen and sulfur hexafluoride

Even small air leaks can have a significant impact on LCI measured using N2. This is particularly the case for leaks that occur towards the end of washout. In contrast, leaks generally have a much smaller impact on LCI measured by SF6. http://ow.ly/az7b30lG5Ku.

Three-dimensional plant architecture and sunlit-shaded patterns: a stochastic model of light dynamics in canopies

Background and Aims

Diurnal changes in solar position and intensity combined with the structural complexity of plant architecture result in highly variable and dynamic light patterns within the plant canopy. This affects productivity through the complex ways that photosynthesis responds to changes in light intensity. Current methods to characterize light dynamics, such as ray-tracing, are able to produce data with excellent spatio-temporal resolution but are computationally intensive and the resulting data are complex and high-dimensional. This necessitates development of more economical models for summarizing the data and for simulating realistic light patterns over the course of a day.

Methods

High-resolution reconstructions of field-grown plants are assembled in various configurations to form canopies, and a forward ray-tracing algorithm is applied to the canopies to compute light dynamics at high (1 min) temporal resolution. From the ray-tracer output, the sunlit or shaded state for each patch on the plants is determined, and these data are used to develop a novel stochastic model for the sunlit-shaded patterns. The model is designed to be straightforward to fit to data using maximum likelihood estimation, and fast to simulate from.

Key Results

For a wide range of contrasting 3-D canopies, the stochastic model is able to summarize, and replicate in simulations, key features of the light dynamics. When light patterns simulated from the stochastic model are used as input to a model of photoinhibition, the predicted reduction in carbon gain is similar to that from calculations based on the (extremely costly) ray-tracer data.

Conclusions

The model provides a way to summarize highly complex data in a small number of parameters, and a cost-effective way to simulate realistic light patterns. Simulations from the model will be particularly useful for feeding into larger-scale photosynthesis models for calculating how light dynamics affects the photosynthetic productivity of canopies.

Human placental oxygenation in late gestation: experimental and theoretical approaches

The placenta is crucial for life. It is an ephemeral but complex organ acting as the barrier interface between maternal and fetal circulations, providing exchange of gases, nutrients, hormones, waste products and immunoglobulins. Many gaps exist in our understanding of the detailed placental structure and function, particularly in relation to oxygen handling and transfer in healthy and pathological states in utero. Measurements to understand oxygen transfer in vivo in the human are limited, with no general agreement on the most appropriate methods. An invasive method for measuring partial pressure of oxygen in the intervillous space through needle electrode insertion at the time of Caesarean sections has been reported. This allows for direct measurements in vivo whilst maintaining near normal placental conditions; however, there are practical and ethical implications in using this method for determination of placental oxygenation. Furthermore, oxygen levels are likely to be highly heterogeneous within the placenta. Emerging non-invasive techniques, such as MRI, and ex vivo research are capable of enhancing and improving current imaging methodology for placental villous structure and increase the precision of oxygen measurement within placental compartments. These techniques, in combination with mathematical modelling, have stimulated novel cross-disciplinary approaches that could advance our understanding of placental oxygenation and its metabolism in normal and pathological pregnancies, improving clinical treatment options and ultimately outcomes for the patient.

Root hydrotropism is controlled via a cortex-specific growth mechanism

Plants can acclimate by using tropisms to link the direction of growth to environmental conditions. Hydrotropism allows roots to forage for water, a process known to depend on abscisic acid (ABA) but whose molecular and cellular basis remains unclear. Here we show that hydrotropism still occurs in roots after laser ablation removed the meristem and root cap. Additionally, targeted expression studies reveal that hydrotropism depends on the ABA signalling kinase SnRK2.2 and the hydrotropism-specific MIZ1, both acting specifically in elongation zone cortical cells. Conversely, hydrotropism, but not gravitropism, is inhibited by preventing differential cell-length increases in the cortex, but not in other cell types. We conclude that root tropic responses to gravity and water are driven by distinct tissue-based mechanisms. In addition, unlike its role in root gravitropism, the elongation zone performs a dual function during a hydrotropic response, both sensing a water potential gradient and subsequently undergoing differential growth.

Root hydrotropism is controlled via a cortex-specific growth mechanism

Plants can acclimate by using tropisms to link the direction of growth to environmental conditions. Hydrotropism allows roots to forage for water, a process known to depend on abscisic acid (ABA) but whose molecular and cellular basis remains unclear. Here we show that hydrotropism still occurs in roots after laser ablation removed the meristem and root cap. Additionally, targeted expression studies reveal that hydrotropism depends on the ABA signalling kinase SnRK2.2 and the hydrotropism-specific MIZ1, both acting specifically in elongation zone cortical cells. Conversely, hydrotropism, but not gravitropism, is inhibited by preventing differential cell-length increases in the cortex, but not in other cell types. We conclude that root tropic responses to gravity and water are driven by distinct tissue-based mechanisms. In addition, unlike its role in root gravitropism, the elongation zone performs a dual function during a hydrotropic response, both sensing a water potential gradient and subsequently undergoing differential growth.

Corrigendum: Airway and Parenchymal Strains during Bronchoconstriction in the Precision Cut Lung Slice

[This corrects the article on p. 309 in vol. 7, PMID: 27559314.].

Image-Based Modeling of Blood Flow and Oxygen Transfer in Feto-Placental Capillaries

During pregnancy, oxygen diffuses from maternal to fetal blood through villous trees in the placenta. In this paper, we simulate blood flow and oxygen transfer in feto-placental capillaries by converting three-dimensional representations of villous and capillary surfaces, reconstructed from confocal laser scanning microscopy, to finite-element meshes, and calculating values of vascular flow resistance and total oxygen transfer. The relationship between the total oxygen transfer rate and the pressure drop through the capillary is shown to be captured across a wide range of pressure drops by physical scaling laws and an upper bound on the oxygen transfer rate. A regression equation is introduced that can be used to estimate the oxygen transfer in a capillary using the vascular resistance. Two techniques for quantifying the effects of statistical variability, experimental uncertainty and pathological placental structure on the calculated properties are then introduced. First, scaling arguments are used to quantify the sensitivity of the model to uncertainties in the geometry and the parameters. Second, the effects of localized dilations in fetal capillaries are investigated using an idealized axisymmetric model, to quantify the possible effect of pathological placental structure on oxygen transfer. The model predicts how, for a fixed pressure drop through a capillary, oxygen transfer is maximized by an optimal width of the dilation. The results could explain the prevalence of fetal hypoxia in cases of delayed villous maturation, a pathology characterized by a lack of the vasculo-syncytial membranes often seen in conjunction with localized capillary dilations.

Hybrid vertex-midline modelling of elongated plant organs

We describe a method for the simulation of the growth of elongated plant organs, such as seedling roots. By combining a midline representation of the organ on a tissue scale and a vertex-based representation on the cell scale, we obtain a multiscale method, which is able to both simulate organ growth and incorporate cell-scale processes. Equations for the evolution of the midline are obtained, which depend on the cell-wall properties of individual cells through appropriate averages over the vertex-based representation. The evolution of the organ midline is used to deform the cellular-scale representation. This permits the investigation of the regulation of organ growth through the cell-scale transport of the plant hormone auxin. The utility of this method is demonstrated in simulating the early stages of the response of a root to gravity, using a vertex-based template acquired from confocal imaging. Asymmetries in the concentrations of auxin between the upper and lower sides of the root lead to bending of the root midline, reflecting a gravitropic response.

Drop spreading with random viscosity

We examine theoretically the spreading of a viscous liquid drop over a thin film of uniform thickness, assuming the liquid’s viscosity is regulated by the concentration of a solute that is carried passively by the spreading flow. The solute is assumed to be initially heterogeneous, having a spatial distribution with prescribed statistical features. To examine how this variability influences the drop’s motion, we investigate spreading in a planar geometry using lubrication theory, combining numerical simulations with asymptotic analysis. We assume diffusion is sufficient to suppress solute concentration gradients across but not along the film. The solute field beneath the bulk of the drop is stretched by the spreading flow, such that the initial solute concentration immediately behind the drop’s effective contact lines has a long-lived influence on the spreading rate. Over long periods, solute swept up from the precursor film accumulates in a short region behind the contact line, allowing patches of elevated viscosity within the precursor film to hinder spreading. A low-order model provides explicit predictions of the variances in spreading rate and drop location, which are validated against simulations.

Stochastic transport in the presence of spatial disorder: Fluctuation-induced corrections to homogenization

Motivated by uncertainty quantification in natural transport systems, we investigate an individual-based transport process involving particles undergoing a random walk along a line of point sinks whose strengths are themselves independent random variables. We assume particles are removed from the system via first-order kinetics. We analyze the system using a hierarchy of approaches when the sinks are sparsely distributed, including a stochastic homogenization approximation that yields explicit predictions for the extrinsic disorder in the stationary state due to sink strength fluctuations. The extrinsic noise induces long-range spatial correlations in the particle concentration, unlike fluctuations due to the intrinsic noise alone. Additionally, the mean concentration profile, averaged over both intrinsic and extrinsic noise, is elevated compared with the corresponding profile from a uniform sink distribution, showing that the classical homogenization approximation can be a biased estimator of the true mean.

The 4-Dimensional Plant: Effects of Wind-Induced Canopy Movement on Light Fluctuations and Photosynthesis

Physical perturbation of a plant canopy brought about by wind is a ubiquitous phenomenon and yet its biological importance has often been overlooked. This is partly due to the complexity of the issue at hand: wind-induced movement (or mechanical excitation) is a stochastic process which is difficult to measure and quantify; plant motion is dependent upon canopy architectural features which, until recently, were difficult to accurately represent and model in 3-dimensions; light patterning throughout a canopy is difficult to compute at high-resolutions, especially when confounded by other environmental variables. Recent studies have reinforced the expectation that canopy architecture is a strong determinant of productivity and yield; however, links between the architectural properties of the plant and its mechanical properties, particularly its response to wind, are relatively unknown. As a result, biologically relevant data relating canopy architecture, light- dynamics, and short-scale photosynthetic responses in the canopy setting are scarce. Here, we hypothesize that wind-induced movement will have large consequences for the photosynthetic productivity of our crops due to its influence on light patterning. To address this issue, in this study we combined high resolution 3D reconstructions of a plant canopy with a simple representation of canopy perturbation as a result of wind using solid body rotation in order to explore the potential effects on light patterning, interception, and photosynthetic productivity. We looked at two different scenarios: firstly a constant distortion where a rice canopy was subject to a permanent distortion throughout the whole day; and secondly, a dynamic distortion, where the canopy was distorted in incremental steps between two extremes at set time points in the day. We find that mechanical canopy excitation substantially alters light dynamics; light distribution and modeled canopy carbon gain. We then discuss methods required for accurate modeling of mechanical canopy excitation (here coined the 4-dimensional plant) and some associated biological and applied implications of such techniques. We hypothesize that biomechanical plant properties are a specific adaptation to achieve wind-induced photosynthetic enhancement and we outline how traits facilitating canopy excitation could be used as a route for improving crop yield.

Patterns of recruitment and injury in a heterogeneous airway network model

In respiratory distress, lung airways become flooded with liquid and may collapse due to surface-tension forces acting on air-liquid interfaces, inhibiting gas exchange. This paper proposes a mathematical multiscale model for the mechanical ventilation of a network of occluded airways, where air is forced into the network at a fixed tidal volume, allowing investigation of optimal recruitment strategies. The temporal response is derived from mechanistic models of individual airway reopening, incorporating feedback on the airway pressure due to recruitment. The model accounts for stochastic variability in airway diameter and stiffness across and between generations. For weak heterogeneity, the network is completely ventilated via one or more avalanches of recruitment (with airways recruited in quick succession), each characterized by a transient decrease in the airway pressure; avalanches become more erratic for airways that are initially more flooded. However, the time taken for complete ventilation of the network increases significantly as the network becomes more heterogeneous, leading to increased stresses on airway walls. The model predicts that the most peripheral airways are most at risk of ventilation-induced damage. A positive-end-expiratory pressure reduces the total recruitment time but at the cost of larger stresses exerted on airway walls.

Airway and Parenchymal Strains during Bronchoconstriction in the Precision Cut Lung Slice

The precision-cut lung slice (PCLS) is a powerful tool for studying airway reactivity, but biomechanical measurements to date have largely focused on changes in airway caliber. Here we describe an image processing tool that reveals the associated spatio-temporal changes in airway and parenchymal strains. Displacements of sub-regions within the PCLS are tracked in phase-contrast movies acquired after addition of contractile and relaxing drugs. From displacement maps, strains are determined across the entire PCLS or along user-specified directions. In a representative mouse PCLS challenged with 10(-4)M methacholine, as lumen area decreased, compressive circumferential strains were highest in the 50 μm closest to the airway lumen while expansive radial strains were highest in the region 50-100 μm from the lumen. However, at any given distance from the airway the strain distribution varied substantially in the vicinity of neighboring small airways and blood vessels. Upon challenge with the relaxant agonist chloroquine, although most strains disappeared, residual positive strains remained a long time after addition of chloroquine, predominantly in the radial direction. Taken together, these findings establish strain mapping as a new tool to elucidate local dynamic mechanical events within the constricting airway and its supporting parenchyma.

Nonlinear compliance modulates dynamic bronchoconstriction in a multiscale airway model

The role of breathing and deep inspirations (DI) in modulating airway hyperresponsiveness remains poorly understood. In particular, DIs are potent bronchodilators of constricted airways in nonasthmatic subjects but not in asthmatic subjects. Additionally, length fluctuations (mimicking DIs) have been shown to reduce mean contractile force when applied to airway smooth muscle (ASM) cells and tissue strips. However, these observations are not recapitulated on application of transmural pressure (PTM) oscillations (that mimic tidal breathing and DIs) in isolated intact airways. To shed light on this paradox, we have developed a biomechanical model of the intact airway, accounting for strain-stiffening due to collagen recruitment (a large component of the extracellular matrix (ECM)), and dynamic actomyosin-driven force generation by ASM cells. In agreement with intact airway studies, our model shows that PTM fluctuations at particular mean transmural pressures can lead to only limited bronchodilation. However, our model predicts that moving the airway to a more compliant point on the static pressure-radius relationship (which may involve reducing mean PTM), before applying pressure fluctuations, can generate greater bronchodilation. This difference arises from competition between passive strain-stiffening of ECM and force generation by ASM yielding a highly nonlinear relationship between effective airway stiffness and PTM, which is modified by the presence of contractile agonist. Effectively, the airway at its most compliant may allow for greater strain to be transmitted to subcellular contractile machinery. The model predictions lead us to hypothesize that the maximum possible bronchodilation of an airway depends on its static compliance at the PTM about which the fluctuations are applied. We suggest the design of additional experimental protocols to test this hypothesis.

Static and dynamic stress heterogeneity in a multiscale model of the asthmatic airway wall

Airway hyperresponsiveness (AHR) is a key characteristic of asthma that remains poorly understood. Tidal breathing and deep inspiration ordinarily cause rapid relaxation of airway smooth muscle (ASM) (as demonstrated via application of length fluctuations to tissue strips) and are therefore implicated in modulation of AHR, but in some cases (such as application of transmural pressure oscillations to isolated intact airways) this mechanism fails. Here we use a multiscale biomechanical model for intact airways that incorporates strain stiffening due to collagen recruitment and dynamic force generation by ASM cells to show that the geometry of the airway, together with interplay between dynamic active and passive forces, gives rise to large stress and compliance heterogeneities across the airway wall that are absent in tissue strips. We show further that these stress heterogeneities result in auxotonic loading conditions that are currently not replicated in tissue-strip experiments; stresses in the strip are similar to hoop stress only at the outer airway wall and are under- or overestimates of stresses at the lumen. Taken together these results suggest that a previously underappreciated factor, stress heterogeneities within the airway wall and consequent ASM cellular response to this micromechanical environment, could contribute to AHR and should be explored further both theoretically and experimentally.

High-Resolution Three-Dimensional Structural Data Quantify the Impact of Photoinhibition on Long-Term Carbon Gain in Wheat Canopies in the Field

Photoinhibition reduces photosynthetic productivity; however, it is difficult to quantify accurately in complex canopies partly because of a lack of high-resolution structural data on plant canopy architecture, which determines complex fluctuations of light in space and time. Here, we evaluate the effects of photoinhibition on long-term carbon gain (over 1 d) in three different wheat (Triticum aestivum) lines, which are architecturally diverse. We use a unique method for accurate digital three-dimensional reconstruction of canopies growing in the field. The reconstruction method captures unique architectural differences between lines, such as leaf angle, curvature, and leaf density, thus providing a sensitive method of evaluating the productivity of actual canopy structures that previously were difficult or impossible to obtain. We show that complex data on light distribution can be automatically obtained without conventional manual measurements. We use a mathematical model of photosynthesis parameterized by field data consisting of chlorophyll fluorescence, light response curves of carbon dioxide assimilation, and manual confirmation of canopy architecture and light attenuation. Model simulations show that photoinhibition alone can result in substantial reduction in carbon gain, but this is highly dependent on exact canopy architecture and the diurnal dynamics of photoinhibition. The use of such highly realistic canopy reconstructions also allows us to conclude that even a moderate change in leaf angle in upper layers of the wheat canopy led to a large increase in the number of leaves in a severely light-limited state.

Multiscale models in the biomechanics of plant growth

Plant growth occurs through the coordinated expansion of tightly adherent cells, driven by regulated softening of cell walls. It is an intrinsically multiscale process, with the integrated properties of multiple cell walls shaping the whole tissue. Multiscale models encode physical relationships to bring new understanding to plant physiology and development.

Exploiting heterogeneous environments: does photosynthetic acclimation optimize carbon gain in fluctuating light?

Plants have evolved complex mechanisms to balance the efficient use of absorbed light energy in photosynthesis with the capacity to use that energy in assimilation, so avoiding potential damage from excess light. This is particularly important under natural light, which can vary according to weather, solar movement and canopy movement. Photosynthetic acclimation is the means by which plants alter their leaf composition and structure over time to enhance photosynthetic efficiency and productivity. However there is no empirical or theoretical basis for understanding how leaves track historic light levels to determine acclimation status, or whether they do this accurately. We hypothesized that in fluctuating light (varying in both intensity and frequency), the light-response characteristics of a leaf should adjust (dynamically acclimate) to maximize daily carbon gain. Using a framework of mathematical modelling based on light-response curves, we have analysed carbon-gain dynamics under various light patterns. The objective was to develop new tools to quantify the precision with which photosynthesis acclimates according to the environment in which plants exist and to test this tool on existing data. We found an inverse relationship between the optimal maximum photosynthetic capacity and the frequency of low to high light transitions. Using experimental data from the literature we were able to show that the observed patterns for acclimation were consistent with a strategy towards maximizing daily carbon gain. Refinement of the model will further determine the precision of acclimation.

A theoretical model to allow prediction of the CSF pressure from observations of the retinal venous pulse

PURPOSE

There is no easy way to estimate the intracranial pressure (ICP) noninvasively. The retinal vein can exhibit large amplitude oscillations at the level of the lamina cribrosa under certain circumstances. The aims of this study were to develop a theoretical understanding of the conditions required to establish this vigorous oscillatory behavior and to determine whether observations of it could lead to a noninvasive estimate of the ICP.

METHODS

A mathematical model was constructed in which the central retinal vein was modeled as 2 Starling resistors in series, 1 located in the eye and the other in the cerebrospinal fluid (CSF) space, separated by a region where it was not collapsible, corresponding to its course within the optic nerve itself. Intraocular pressure (IOP) and ICP were modeled as sinusoidal wave forms.

RESULTS

The model predicted an approximately linear relationship between the IOP and the ICP at the point of onset of oscillatory behavior. The predicted onset IOP also depended weakly on the retinal blood flow rate and on vein diameter and was only mildly sensitive to the phase difference between the two pressure waveforms. The predicted onset curve showed encouraging agreement with measurements in canines.

CONCLUSIONS

The model suggested that it may be possible to estimate the ICP from observations of the retinal venous pulse by using a modified form of ophthalmodynamometry.

Mechanical modelling quantifies the functional importance of outer tissue layers during root elongation and bending

Root elongation and bending require the coordinated expansion of multiple cells of different types. These processes are regulated by the action of hormones that can target distinct cell layers. We use a mathematical model to characterise the influence of the biomechanical properties of individual cell walls on the properties of the whole tissue. Taking a simple constitutive model at the cell scale which characterises cell walls via yield and extensibility parameters, we derive the analogous tissue-level model to describe elongation and bending. To accurately parameterise the model, we take detailed measurements of cell turgor, cell geometries and wall thicknesses. The model demonstrates how cell properties and shapes contribute to tissue-level extensibility and yield. Exploiting the highly organised structure of the elongation zone (EZ) of the Arabidopsis root, we quantify the contributions of different cell layers, using the measured parameters. We show how distributions of material and geometric properties across the root cross-section contribute to the generation of curvature, and relate the angle of a gravitropic bend to the magnitude and duration of asymmetric wall softening. We quantify the geometric factors which lead to the predominant contribution of the outer cell files in driving root elongation and bending.

On the role of stress anisotropy in the growth of stems

We review the role of anisotropic stress in controlling the growth anisotropy of stems. Instead of stress, growth anisotropy is usually considered in terms of compliance. Anisotropic compliance is typical of cell walls, because they contain aligned cellulose microfibrils, and it appears to be sufficient to explain the growth anisotropy of an isolated cell. Nevertheless, a role for anisotropic stress in the growth of stems is indicated by certain growth responses that appear too rapid to be accounted for by changes in cell-wall compliance and because the outer epidermal wall of most growing stems has microfibrils aligned axially, an arrangement that would favour radial expansion based on cell-wall compliance alone. Efforts to quantify stress anisotropy in the stem have found that it is predominantly axial, and large enough in principle to explain the elongation of the epidermis, despite its axial microfibrils. That the epidermis experiences a stress deriving from the inner tissue, the so-called 'tissue stress', has been widely recognized; however, the origin of the dominant axial direction remains obscure. Based on geometry, an isolated cylindrical cell should have an intramural stress anisotropy favouring the transverse direction. Explanations for tissue stress have invoked differential elastic moduli, differential plastic deformation (so-called differential growth), and a phenomenon analogous to the maturation stress generated by secondary cell walls. None of these explanations has been validated. We suggest that understanding the role of stress anisotropy in plant growth requires a deeper understanding of the nature of stress in hierarchical, organic structures.

Buckling of a growing tissue and the emergence of two-dimensional patterns

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We model the growth of gut epithelial cells cultured upon a deformable substrate.

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Growth generates buckling instabilities, contributing to crypt formation in vivo.

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Variations in mechanical properties have little effect on resulting configurations.

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Configurations are controlled by growth patterns & interactions with strata below.

The process of biological growth and the associated generation of residual stress has previously been considered as a driving mechanism for tissue buckling and pattern selection in numerous areas of biology. Here, we develop a two-dimensional thin plate theory to simulate the growth of cultured intestinal epithelial cells on a deformable substrate, with the goal of elucidating how a tissue engineer might best recreate the regular array of invaginations (crypts of Lieberkühn) found in the wall of the mammalian intestine. We extend the standard von Kármán equations to incorporate inhomogeneity in the plate’s mechanical properties and surface stresses applied to the substrate by cell proliferation. We determine numerically the configurations of a homogeneous plate under uniform cell growth, and show how tethering to an underlying elastic foundation can be used to promote higher-order buckled configurations. We then examine the independent effects of localised softening of the substrate and spatial patterning of cellular growth, demonstrating that (within a two-dimensional framework, and contrary to the predictions of one-dimensional models) growth patterning constitutes a more viable mechanism for control of crypt distribution than does material inhomogeneity.

Vertex-element models for anisotropic growth of elongated plant organs

New tools are required to address the challenge of relating plant hormone levels, hormone responses, wall biochemistry and wall mechanical properties to organ-scale growth. Current vertex-based models (applied in other contexts) can be unsuitable for simulating the growth of elongated organs such as roots because of the large aspect ratio of the cells, and these models fail to capture the mechanical properties of cell walls in sufficient detail. We describe a vertex-element model which resolves individual cells and includes anisotropic non-linear viscoelastic mechanical properties of cell walls and cell division whilst still being computationally efficient. We show that detailed consideration of the cell walls in the plane of a 2D simulation is necessary when cells have large aspect ratio, such as those in the root elongation zone of Arabidopsis thaliana, in order to avoid anomalous transverse swelling. We explore how differences in the mechanical properties of cells across an organ can result in bending and how cellulose microfibril orientation affects macroscale growth. We also demonstrate that the model can be used to simulate growth on realistic geometries, for example that of the primary root apex, using moderate computational resources. The model shows how macroscopic root shape can be sensitive to fine-scale cellular geometries.

The role of contractile unit reorganization in force generation in airway smooth muscle

Airway smooth muscle (ASM) cells undergo remodelling and reside in a tissue structure that is subject to heterogenous stress distributions that change dynamically during the breathing cycle. In this paper, we develop a structural model of an ASM cell that consists of contractile units (actin and myosin filaments) in series and parallel, anchored to a nonlinearly elastic cytoskeleton. We mimic a typical experimental protocol that involves isometric force generation through triggering of the contractile machinery, followed by oscillatory length fluctuation of the cell. We use the model to predict the effect of a single instance of rearrangement of the contractile machinery, combined with strain-stiffening of the cytoskeleton, on the force generated by the sarcomeres, and the total force generated by the cell. By linking intra-cellular events to whole-cell behaviour, the model reveals mechanistic relationships between structural properties and cell-level force–length loops. We show how contractile force, shortening velocity and sarcomere operating lengths vary as the internal cell architecture is altered. Additionally, we show how interactions between the internal contractile machinery and cytoskeletal structure play a role in the regulation of force generation and hysteresis of the cell.

Buckling as an origin of ordered cuticular patterns in flower petals

The optical properties of plant surfaces are strongly determined by the shape of epidermal cells and by the patterning of the cuticle on top of the cells. Combinations of particular cell shapes with particular nanoscale structures can generate a wide range of optical effects. Perhaps most notably, the development of ordered ridges of cuticle on top of flat petal cells can produce diffraction-grating-like structures. A diffraction grating is one of a number of mechanisms known to produce 'structural colours', which are more intense and pure than chemical colours and can appear iridescent. We explore the concept that mechanical buckling of the cuticle on the petal epidermis might explain the formation of cuticular ridges, using a theoretical model that accounts for the development of compressive stresses in the cuticle arising from competition between anisotropic expansion of epidermal cells and isotropic cuticle production. Model predictions rationalize cuticle patterns, including those with long-range order having the potential to generate iridescence, for a range of different flower species.

A biomechanical model of anther opening reveals the roles of dehydration and secondary thickening

Understanding the processes that underlie pollen release is a prime target for controlling fertility to enable selective breeding and the efficient production of hybrid crops. Pollen release requires anther opening, which involves changes in the biomechanical properties of the anther wall. In this research, we develop and use a mathematical model to understand how these biomechanical processes lead to anther opening. Our mathematical model describing the biomechanics of anther opening incorporates the bilayer structure of the mature anther wall, which comprises the outer epidermal cell layer, whose turgor pressure is related to its hydration, and the endothecial layer, whose walls contain helical secondary thickening, which resists stretching and bending. The model describes how epidermal dehydration, in association with the thickened endothecial layer, creates forces within the anther wall causing it to bend outwards, resulting in anther opening and pollen release. The model demonstrates that epidermal dehydration can drive anther opening, and suggests why endothecial secondary thickening is essential for this process (explaining the phenotypes presented in the myb26 and nst1nst2 mutants). The research hypothesizes and demonstrates a biomechanical mechanism for anther opening, which appears to be conserved in many other biological situations where tissue movement occurs.

Early gene regulation of osteogenesis in embryonic stem cells

The early gene regulatory networks (GRNs) that mediate stem cell differentiation are complex, and the underlying regulatory associations can be difficult to map accurately. In this study, the expression profiles of the genes Dlx5, Msx2 and Runx2 in mouse embryonic stem cells were monitored over a 48 hour period after exposure to the growth factors BMP2 and TGFβ1. Candidate GRNs of early osteogenesis were constructed based on published experimental findings and simulation results of Boolean and ordinary differential equation models were compared with our experimental data in order to test the validity of these models. Three gene regulatory networks were found to be consistent with the data, one of these networks exhibited sustained oscillation, a behaviour which is consistent with the general view of embryonic stem cell plasticity. The work cycle presented in this paper illustrates how mathematical modelling can be used to elucidate from gene expression profiles GRNs that are consistent with experimental data.

Multiscale systems analysis of root growth and development: modeling beyond the network and cellular scales

Over recent decades, we have gained detailed knowledge of many processes involved in root growth and development. However, with this knowledge come increasing complexity and an increasing need for mechanistic modeling to understand how those individual processes interact. One major challenge is in relating genotypes to phenotypes, requiring us to move beyond the network and cellular scales, to use multiscale modeling to predict emergent dynamics at the tissue and organ levels. In this review, we highlight recent developments in multiscale modeling, illustrating how these are generating new mechanistic insights into the regulation of root growth and development. We consider how these models are motivating new biological data analysis and explore directions for future research. This modeling progress will be crucial as we move from a qualitative to an increasingly quantitative understanding of root biology, generating predictive tools that accelerate the development of improved crop varieties.

Auxin regulates aquaporin function to facilitate lateral root emergence

Aquaporins are membrane channels that facilitate water movement across cell membranes. In plants, aquaporins contribute to water relations. Here, we establish a new link between aquaporin-dependent tissue hydraulics and auxin-regulated root development in Arabidopsis thaliana. We report that most aquaporin genes are repressed during lateral root formation and by exogenous auxin treatment. Auxin reduces root hydraulic conductivity both at the cell and whole-organ levels. The highly expressed aquaporin PIP2;1 is progressively excluded from the site of the auxin response maximum in lateral root primordia (LRP) whilst being maintained at their base and underlying vascular tissues. Modelling predicts that the positive and negative perturbations of PIP2;1 expression alter water flow into LRP, thereby slowing lateral root emergence (LRE). Consistent with this mechanism, pip2;1 mutants and PIP2;1-overexpressing lines exhibit delayed LRE. We conclude that auxin promotes LRE by regulating the spatial and temporal distribution of aquaporin-dependent root tissue water transport.

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.

Growth-induced hormone dilution can explain the dynamics of plant root cell elongation

In the elongation zone of the Arabidopsis thaliana plant root, cells undergo rapid elongation, increasing their length by ∼10-fold over 5 h while maintaining a constant radius. Although progress is being made in understanding how this growth is regulated, little consideration has been given as to how cell elongation affects the distribution of the key regulating hormones. Using a multiscale mathematical model and measurements of growth dynamics, we investigate the distribution of the hormone gibberellin in the root elongation zone. The model quantifies how rapid cell expansion causes gibberellin to dilute, creating a significant gradient in gibberellin levels. By incorporating the gibberellin signaling network, we simulate how gibberellin dilution affects the downstream components, including the growth-repressing DELLA proteins. We predict a gradient in DELLA that provides an explanation of the reduction in growth exhibited as cells move toward the end of the elongation zone. These results are validated at the molecular level by comparing predicted mRNA levels with transcriptomic data. To explore the dynamics further, we simulate perturbed systems in which gibberellin levels are reduced, considering both genetically modified and chemically treated roots. By modeling these cases, we predict how these perturbations affect gibberellin and DELLA levels and thereby provide insight into their altered growth dynamics.

Transport in the placenta: homogenizing haemodynamics in a disordered medium

The placenta is an essential component of the life-support system for the developing foetus, enabling nutrients and waste to be exchanged between the foetal and maternal circulations. Maternal blood flows between the densely packed branches of villous trees, within which are foetal vessels. Here, we explore some of the challenges in modelling maternal haemodynamic transport using homogenization approaches. We first show how two measures can be used to estimate the minimum distance over which the distribution of villous branches appears statistically homogeneous. We then analyse a simplified model problem (solute transport by a unidirectional flow past a distribution of point sinks) to assess the accuracy of homogenization approximations as a function of governing parameters (Péclet and Damköhler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is characterized by large spatial gradients at the scale of individual villi and substantial fluctuations that can be correlated over lengthscales comparable to the whole domain. This study highlights the importance of quantifying errors owing to spatial disorder in multi-scale approximations of physiological systems.