The role of elastic stresses on leaf venation morphogenesis

Leaf morphogenesis: The multifaceted roles of mechanics

Plants produce a rich diversity of biological forms, and the diversity of leaves is especially notable. Mechanisms of leaf morphogenesis have been studied in the past two decades, with a growing focus on the interactive roles of mechanics in recent years. Growth of plant organs involves feedback by mechanical stress: growth induces stress, and stress affects growth and morphogenesis. Although much attention has been given to potential stress-sensing mechanisms and cellular responses, the mechanical principles guiding morphogenesis have not been well understood. Here we synthesize the overarching roles of mechanics and mechanical stress in multilevel and multiple stages of leaf morphogenesis, encompassing leaf primordium initiation, phyllotaxis and venation patterning, and the establishment of complex mature leaf shapes. Moreover, the roles of mechanics at multiscale levels, from subcellular cytoskeletal molecules to single cells to tissues at the organ scale, are articulated. By highlighting the role of mechanical buckling in the formation of three-dimensional leaf shapes, this review integrates the perspectives of mechanics and biology to provide broader insights into the mechanobiology of leaf morphogenesis.

Combinatorial topology and geometry of fracture networks

A map is proposed from the space of planar surface fracture networks to a four-parameter mathematical space, summarizing the average topological connectivity and geometrical properties of a network idealized as a convex polygonal mesh. The four parameters are identified as the average number of nodes and edges, the angular defect with respect to regular polygons, and the isoperimetric ratio. The map serves as a low-dimensional signature of the fracture network and is visually presented as a pair of three-dimensional graphs. A systematic study is made of a wide collection of real crack networks for various materials, collected from different sources. To identify the characteristics of the real materials, several well-known mathematical models of convex polygonal networks are presented and worked out. These geometric models may correspond to different physical fracturing processes. The proposed map is shown to be discriminative, and the points corresponding to materials of similar properties are found to form closely spaced groups in the parameter space. Results for the real and simulated systems are compared in an attempt to identify crack networks of unknown materials.

Shaping leaf vein pattern by auxin and mechanical feedback

This article comments on: Kneuper I, Teale W, Dawson JE, Tsugeki R, Katifori E, Palme K, Ditengou FA. 2021. Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning. Journal of Experimental Botany 72, 1151–1165.

Optimal Elasticity of Biological Networks

Reinforced elastic sheets surround us in daily life, from concrete shell buildings to biological structures such as the arthropod exoskeleton or the venation network of dicotyledonous plant leaves. Natural structures are often highly optimized through evolution and natural selection, leading to the biologically and practically relevant problem of understanding and applying the principles of their design. Inspired by the hierarchically organized scaffolding networks found in plant leaves, here we model networks of bending beams that capture the discrete and nonuniform nature of natural materials. Using the principle of maximal rigidity under natural resource constraints, we show that optimal discrete beam networks reproduce the structural features of real leaf venation. Thus, in addition to its ability to efficiently transport water and nutrients, the venation network also optimizes leaf rigidity using the same hierarchical reticulated network topology. We study the phase space of optimal mechanical networks, providing concrete guidelines for the construction of elastic structures. We implement these natural design rules by fabricating efficient, biologically inspired metamaterials.

Discontinuous transition to loop formation in optimal supply networks

The structure and design of optimal supply networks is an important topic in complex networks research. A fundamental trait of natural and man-made networks is the emergence of loops and the trade-off governing their formation: adding redundant edges to supply networks is costly, yet beneficial for resilience. Loops typically form when costs for new edges are small or inputs uncertain. Here, we shed further light on the transition to loop formation. We demonstrate that loops emerge discontinuously when decreasing the costs for new edges for both an edge-damage model and a fluctuating sink model. Mathematically, new loops are shown to form through a saddle-node bifurcation. Our analysis allows to heuristically predict the location and cost where the first loop emerges. Finally, we unveil an intimate relationship among betweenness measures and optimal tree networks. Our results can be used to understand the evolution of loop formation in real-world biological networks.

Supply networks with optimal structure do not contain loops but these can arise as a result of damages or fluctuations. Here Kaiser et al. uncover the mechanisms of loop formation, predict their location and draw analogies with loop formation in biological networks such as plants and animal vasculature.

Coordination of tissue cell polarity by auxin transport and signaling

Plants coordinate the polarity of hundreds of cells during vein formation, but how they do so is unclear. The prevailing hypothesis proposes that GNOM, a regulator of membrane trafficking, positions PIN-FORMED auxin transporters to the correct side of the plasma membrane; the resulting cell-to-cell, polar transport of auxin would coordinate tissue cell polarity and induce vein formation. Contrary to predictions of the hypothesis, we find that vein formation occurs in the absence of PIN-FORMED or any other intercellular auxin-transporter; that the residual auxin-transport-independent vein-patterning activity relies on auxin signaling; and that a GNOM-dependent signal acts upstream of both auxin transport and signaling to coordinate tissue cell polarity and induce vein formation. Our results reveal synergism between auxin transport and signaling, and their unsuspected control by GNOM in the coordination of tissue cell polarity during vein patterning, one of the most informative expressions of tissue cell polarization in plants.

Plants, animals and other living things grow and develop over their lifetimes: for example, oak trees come from acorns and chickens begin their lives as eggs. To achieve these transformations, the cells in those living things must grow, divide and change their shape and other features.

Plants and animals specify the directions in which their cells will grow and develop by gathering specific proteins to one side of the cells. This makes one side different from all the other sides, which the cells use as an internal compass that points in one direction. To align their internal compasses, animal cells touch one another and often move around inside the body. Plant cells, on the other hand, are surrounded by a wall that keeps them apart and prevents them from moving around. So how do plant cells align their internal compasses?

Scientists have long thought that a protein called GNOM aligns the internal compasses of plant cells. The hypothesis proposes that GNOM gathers another protein, called PIN1, to one side of a cell. PIN1 would then pump a plant hormone known as auxin out of this first cell and, in doing so, would also drain auxin away from the cell on the opposite side. In this second cell, GNOM would then gather PIN1 to the side facing the first cell, and this process would repeat until all the cells' compasses were aligned.

To test this hypothesis, Verna et al. combined microscopy with genetic approaches to study how cells' compasses are aligned in the leaves of a plant called Arabidopsis thaliana. The experiments revealed that auxin needs to move from cell-to-cell to align the cells’ compasses. However, contrary to the above hypothesis, this movement of auxin was not sufficient: the cells also needed to be able to detect and respond to the auxin that entered them. Along with controlling how auxin moved between the cells, GNOM also regulated how the cells responded to the auxin.

These findings reveal how plants specify which directions their cells grow and develop. In the future, this knowledge may eventually aid efforts to improve crop yields by controlling the growth and development of crop plants.

Leaf vascular architecture in temperate dicotyledons: correlations and link to functional traits

Using 227 dicotyledonous species in temperate region, we found the relationships among densities of different-order veins, creating diversity of leaf vascular architectures. Dicotyledonous angiosperms commonly possess a hierarchical leaf vascular system, wherein veins of different orders have different functions. Minor vein spacing determines leaf hydraulic efficiency, whereas the major veins provide mechanical support. However, there is limited information on the coordination between these vein orders across species, limiting our understanding of how diversity in vein architecture is arrayed. We aimed to examine the (1) relationships between vein densities at two spatial scales (lower- vs. higher-order veins and among minor veins) and (2) relationships of vein densities with plant functional traits. We studied ten traits related to vein densities and three functional traits (leaf dry mass per area [LMA], leaf longevity [LL], and adult plant height [H_adult]) for 227 phylogenetically diverse plant species that occur in temperate regions and examined the vein-vein and vein-functional traits relationships across species. The densities of lower- and higher-order veins were positively correlated across species. The minor vein density was positively correlated with the densities of both areoles and free-ending veins, and vascular networks with higher minor vein density tended to have a lower ratio of free-ending veins to areoles across species. Neither densities of lower- nor higher-order veins were related to LMA and LL. On the other hand, the densities of veins and areoles tended to be positively correlated with H_adult. These results suggest that densities of different-order veins are developmentally coordinated across dicotyledonous angiosperms and form the independent axis in resource use strategies based on the leaf economics spectrum.

Recyclable and Flexible Starch-Ag Networks and Its Application in Joint Sensor

Flexible transparent conductive electrodes are essential component for flexible optoelectronic devices and have been extensively studied in recent years, while most of the researches are focusing on the electrode itself, few topics in material green and recyclability. In this paper, we demonstrate a high-performance transparent conductive electrode (TCE), based on our previous cracking technology, combined with a green and recyclable substrate, a starch film. It not only shows low R_s (less than 1.0 Ω sq^-1), high transparency (> 82%, figure of merit ≈ 10,000), but also provides an ultra-smooth morphology and recyclability. Furthermore, a series of biosensors on human joints are demonstrated, showing great sensitivity and mechanical stability.

A simple developmental model recapitulates complex insect wing venation patterns

Insect wings are typically supported by thickened struts called veins. These veins form diverse geometric patterns across insects. For many insect species, even the left and right wings from the same individual have veins with unique topological arrangements, and little is known about how these patterns form. We present a large-scale quantitative study of the fingerprint-like "secondary veins." We compile a dataset of wings from 232 species and 17 families from the order Odonata (dragonflies and damselflies), a group with particularly elaborate vein patterns. We characterize the geometric arrangements of veins and develop a simple model of secondary vein patterning. We show that our model is capable of recapitulating the vein geometries of species from other, distantly related winged insect clades.

Morphometric analysis of polygonal cracking patterns in desiccated starch slurries

We investigate the geometry of two-dimensional polygonal cracking that forms on the air-exposed surface of dried starch slurries. Two different kinds of starches, made from potato and corn, exhibited distinguished crack evolution, and there were contrasting effects of slurry thickness on the probability distribution of the polygonal cell area. The experimental findings are believed to result from the difference in the shape and size of starch grains, which strongly influence the capillary transport of water and tensile stress field that drives the polygonal cracking.

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

Crack patterns over uneven substrates

Cracks in thin layers are influenced by what lies beneath them. From buried craters to crocodile skin, crack patterns are found over an enormous range of length scales. Regardless of absolute size, their substrates can dramatically influence how cracks form, guiding them in some cases, or shielding regions from them in others. Here we investigate how a substrate's shape affects the appearance of cracks above it, by preparing mud cracks over sinusoidally varying surfaces. We find that as the thickness of the cracking layer increases, the observed crack patterns change from wavy to ladder-like to isotropic. Two order parameters are introduced to measure the relative alignment of these crack networks, and, along with Fourier methods, are used to characterise the transitions between crack pattern types. Finally, we explain these results with a model, based on the Griffith criteria of fracture, that identifies the conditions for which straight or wavy cracks will be seen, and predicts how well-ordered the cracks will be. Our metrics and results can be applied to any situation where connected networks of cracks are expected, or found.

Topological Phenotypes Constitute a New Dimension in the Phenotypic Space of Leaf Venation Networks

The leaves of angiosperms contain highly complex venation networks consisting of recursively nested, hierarchically organized loops. We describe a new phenotypic trait of reticulate vascular networks based on the topology of the nested loops. This phenotypic trait encodes information orthogonal to widely used geometric phenotypic traits, and thus constitutes a new dimension in the leaf venation phenotypic space. We apply our metric to a database of 186 leaves and leaflets representing 137 species, predominantly from the Burseraceae family, revealing diverse topological network traits even within this single family. We show that topological information significantly improves identification of leaves from fragments by calculating a “leaf venation fingerprint” from topology and geometry. Further, we present a phenomenological model suggesting that the topological traits can be explained by noise effects unique to specimen during development of each leaf which leave their imprint on the final network. This work opens the path to new quantitative identification techniques for leaves which go beyond simple geometric traits such as vein density and is directly applicable to other planar or sub-planar networks such as blood vessels in the brain.

Planar reticular networks are ubiquitous in nature and engineering, formed for instance by the arterial vasculature in the mammalian neocortex, urban street grids or the vascular network of plant leaves. We use a topological metric to characterize the way loops are nested in such networks and analyze a large database of 186 leaves and leaflets, revealing for the first time that the nesting of the networks’ cycles constitutes a distinct phenotypic trait orthogonal to previously used geometric features. Furthermore, we demonstrate that the information contained in the leaf topology can significantly improve specimen identification from fragments, and provide an empirical growth model that can explain much of the observed data. Our work can improve understanding of the functional significance of the various leaf vein architectures and their correlation with the environment. It can pave the way for similar analyses in diverse areas of research involving reticulate networks.

Sound Insulation Property of Bionic Thin-Walled Stiffened Plate Based on Plants Venations Growth Mechanism

The leaf can be seen as shin plate structures with stiffener(vein) and the venation distributions are closed related to the external environment load. Leaf venation growing algorithm (VGA) is the abstract description of vein growing process and reflects an ideological of learning from nature. This article concerns the sound insulation property of thin-walled stiffened plates. Numerical method is used to analyze three types of plants: non-stiffened plate, traditional stiffened plate and VGA stiffened plate. The VGA stiffened plate optimized by leaf venations growth algorithm method can reflect the influence of venations layout structure on the noise reduction performance of forest belts. The computational model of sound transmission through a stiffened plate excited by a harmonic oblique incident plane wave and mounted in an infinite baffle using the coupled finite element/boundary element approach is presented. The proposed model also takes the acoustic fluid- structure coupling into account. The results show that the sound transmission losses are closely dependent on the natural frequency. The sound transmission losses of bionic thin-walled stiffened plate are 0.17-2.45dB more than that of traditional stiffened plate in the range of 900-2000Hz. It indicated that the layout of stiffeners is an influence factor for noise reduction for plate structures, just like that of vein layout for tree belts. There is a certain merit to use the method of bionic plant leaves for acoustic optimization.

Adaptation and optimization of biological transport networks

It has been hypothesized that topological structures of biological transport networks are consequences of energy optimization. Motivated by experimental observation, we propose that adaptation dynamics may underlie this optimization. In contrast to the global nature of optimization, our adaptation dynamics responds only to local information and can naturally incorporate fluctuations in flow distributions. The adaptation dynamics minimizes the global energy consumption to produce optimal networks, which may possess hierarchical loop structures in the presence of strong fluctuations in flow distribution. We further show that there may exist a new phase transition as there is a critical open probability of sinks, above which there are only trees for network structures whereas below which loops begin to emerge.

Understanding of Leaf Development-the Science of Complexity

The leaf is the major organ involved in light perception and conversion of solar energy into organic carbon. In order to adapt to different natural habitats, plants have developed a variety of leaf forms, ranging from simple to compound, with various forms of dissection. Due to the enormous cellular complexity of leaves, understanding the mechanisms regulating development of these organs is difficult. In recent years there has been a dramatic increase in the use of technically advanced imaging techniques and computational modeling in studies of leaf development. Additionally, molecular tools for manipulation of morphogenesis were successfully used for in planta verification of developmental models. Results of these interdisciplinary studies show that global growth patterns influencing final leaf form are generated by cooperative action of genetic, biochemical, and biomechanical inputs. This review summarizes recent progress in integrative studies on leaf development and illustrates how intrinsic features of leaves (including their cellular complexity) influence the choice of experimental approach.

Computational models of plant development and form

The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.

Quantitative analysis of venation patterns of Arabidopsis leaves by supervised image analysis

The study of transgenic Arabidopsis lines with altered vascular patterns has revealed key players in the venation process, but details of the vascularization process are still unclear, partly because most lines have only been assessed qualitatively. Therefore, quantitative analyses are required to identify subtle perturbations in the pattern and to test dynamic modeling hypotheses using biological measurements. We developed an online framework, designated Leaf Image Analysis Interface (LIMANI), in which venation patterns are automatically segmented and measured on dark-field images. Image segmentation may be manually corrected through use of an interactive interface, allowing supervision and rectification steps in the automated image analysis pipeline and ensuring high-fidelity analysis. This online approach is advantageous for the user in terms of installation, software updates, computer load and data storage. The framework was used to study vascular differentiation during leaf development and to analyze the venation pattern in transgenic lines with contrasting cellular and leaf size traits. The results show the evolution of vascular traits during leaf development, suggest a self-organizing mechanism for leaf venation patterning, and reveal a tight balance between the number of end-points and branching points within the leaf vascular network that does not depend on the leaf developmental stage and cellular content, but on the leaf position on the rosette. These findings indicate that development of LIMANI improves understanding of the interaction between vascular patterning and leaf growth.

Mechanics in Leaf Venation Morphogenesis and their Biomimetic Inspiration to Construct a 2-Dimensional Reinforcement Layout Model

This paper concerns biomimetic exploration of the leaf rib layout problem. Biological venation of organisms is observed to be similar to reinforced plate/shell systems. Similarity analysis makes it clear that dicotyledonous leaves are an ideal research subject. In this paper, global and local regularities are summarized and existing theories on venation morphogenesis are discussed and compared. An energy hypothesis is proposed to cater for interdisciplinary applications. A venation growing model was then used to construct a two-dimensional reinforcement layout model. The biomechanical expressions developed can be an alternative to describe rib-in-plate or fibre-in-composite materials.

The role of mechanical forces in plant morphogenesis

The shape of an organism relies on a complex network of genetic regulations and on the homeostasis and distribution of growth factors. In parallel to the molecular control of growth, shape changes also involve major changes in structure, which by definition depend on the laws of mechanics. Thus, to understand morphogenesis, scientists have turned to interdisciplinary approaches associating biology and physics to investigate the contribution of mechanical forces in morphogenesis, sometimes re-examining theoretical concepts that were laid out by early physiologists. Major advances in the field have notably been possible thanks to the development of computer simulations and live quantitative imaging protocols in recent years. Here, we present the mechanical basis of shape changes in plants, focusing our discussion on undifferentiated tissues. How can growth be translated into a quantified geometrical output? What is the mechanical basis of cell and tissue growth? What is the contribution of mechanical forces in patterning?

Damage and fluctuations induce loops in optimal transport networks

Leaf venation is a pervasive example of a complex biological network, endowing leaves with a transport system and mechanical resilience. Transport networks optimized for efficiency have been shown to be trees, i.e., loopless. However, dicotyledon leaf venation has a large number of closed loops, which are functional and able to transport fluid in the event of damage to any vein, including the primary veins. Inspired by leaf venation, we study two possible reasons for the existence of a high density of loops in transport networks: resilience to damage and fluctuations in load. In the first case, we seek the optimal transport network in the presence of random damage by averaging over damage to each link. In the second case, we seek the network that optimizes transport when the load is sparsely distributed: at any given time most sinks are closed. We find that both criteria lead to the presence of loops in the optimum state.

Computational morphodynamics: a modeling framework to understand plant growth

Computational morphodynamics utilizes computer modeling to understand the development of living organisms over space and time. Results from biological experiments are used to construct accurate and predictive models of growth. These models are then used to make novel predictions that provide further insight into the processes involved, which can be tested experimentally to either confirm or rule out the validity of the computational models. This review highlights two fundamental challenges: (a) to understand the feedback between mechanics of growth and chemical or molecular signaling, and (b) to design models that span and integrate single cell behavior with tissue development. We review different approaches to model plant growth and discuss a variety of model types that can be implemented to demonstrate how the interplay between computational modeling and experimentation can be used to explore the morphodynamics of plant development.

Describing long-range patterns in leaf vasculature by metaphoric fields

Some patterns in dicotyledonous leaf vasculature depict rather precise, long-range structural features. This work identifies and quantifies these previously unrecognized features in terms of an empirically derived mathematical formalism that generates wave-like spatial patterns referred to as metaphoric fields. These patterns were used to specify regularities in the long-range structure of dicot leaf vasculature, and were found to account significantly for the predominant features of all 27 dicot species studied. The conserved features of these metaphoric fields are discussed in terms of existing models for leaf pattern formation based on efflux-protein mediated auxin transport in a developing cellular field. This work highlights the complex, regular, long-range structures existing in leaf vascular patterns, and provides a means for specifying and identifying the inherent global features of vascular patterns which must be accounted for in functional developmental models.

Localization through surface folding in solid foams under compression

We report a combined experimental and theoretical study of the compression of a solid foam coated with a thin elastic film. Past a critical compression threshold, a pattern of localized folds emerges with a characteristic size that is imposed by an instability of the thin surface film. We perform optical surface measurements of the statistical properties of these localization zones and find that they are characterized by robust exponential tails in the strain distributions. Following a hybrid continuum and statistical approach, we develop a theory that accurately describes the nucleation and length scale of these structures and predicts the characteristic strains associated with the localized regions.

In silico leaf venation networks: growth and reorganization driven by mechanical forces

Development commonly involves an interplay between signaling, genetic expression and biophysical forces. However, the relative importance of these mechanisms during the different stages of development is unclear. Leaf venation networks provide a fitting context for the examination of these questions. In mature leaves, venation patterns are extremely diverse, yet their local structure satisfies a universal property: at junctions between veins, angles and diameters are related by a vectorial equation analogous to a force balance. Using a cell proliferation model, we reproduce in silico the salient features of venation patterns. Provided that vein cells are given different mechanical properties, tensile forces develop along the veins during growth, causing the network to deform progressively. Our results suggest that the local structure of venation networks results from a reorganization driven by mechanical forces, independently of how veins form. This conclusion is supported by recent observations of vein development in young leaves and by the good quantitative agreement between our simulations and data from mature leaves.

Auxin-regulated cell polarity: an inside job?

Auxin is now known to be a key regulator of polar events in plant cells. The mechanism by which auxin conveys a polar signal to the cell is unknown, but one well-known hypothesis is that the auxin flux across the plasma membrane regulates vesicle trafficking. This hypothesis remains controversial because of its reliance on an as-yet-undiscovered membrane flux sensor. In this article I suggest instead that the polar signal is the auxin gradient within the cell cytoplasm. A computer model of vascular development is presented that demonstrates the plausibility of this scenario. The auxin-binding protein ABP1 might be the receptor for the auxin gradient.

Optimizing transient transport in materials having two scales of porosity

Porous materials having multiple scales of porosity afford the opportunity to combine the high surface area and functionality of nanopores with the superior charge/discharge characteristics of wider transport channels. However, the relative volume fractions assigned to nanopores and transport channels must be thoughtfully balanced because the introduction of transport channels reduces the volume available for nanopore functionality. In the present paper, the optimal balance between nanopore capacity and system response time is achieved by adjusting the aperture and spacing of a family of transport channels that provide access to adjacent nanopores during recharge/discharge cycles of materials intended for storage of gas or electric charge. A diffusive transport model is used to describe alternative processes of viscous gas flow, Knudsen gas flow, and ion diffusion or electromigration. The coupled transport equations for the nanopores and transport channels are linearized and solved analytically for a periodic variation in external gas pressure, ion concentration, or electric potential using a separation-of-variables approach in the complex domain. Optimization of these solutions yields closed-form expressions for channel apertures and spacing that provide maximum discharge of gas or electric charge for a fixed system volume and a desired discharge time.