Karl-Friedrich Fischbach's first publications and their impact in the biological sciences.

Predictions from the theory of transfection, Karl-Friedrich Fischbach's first paper, were confirmed 20 years later. Also a model, proposed already in 1977 by Karl-Friedrich and colleagues, to explain the nonmonotonous dependence on light intensity of phototaxis in Drosophila, finds support in recent studies of functional neuroanatomy.

Oscillation damping in trees.

Oscillation damping is of vital importance for trees to withstand strong gusty winds. Tree adaptation to wind loading takes place over a long time and during a storm only passive damping mechanisms can reduce the impact of the wind on trunk and roots. Structural damping, a phenomenon, which is associated with the conspicuous movements of the branches relative to the trunk is of particular importance. Primary and higher order branches can be seen as multiple tuned mass dampers. Moreover, as the frequency bands overlap within branches and between primary branches and the entire tree, resonance energy transfer can distribute mechanical energy over the entire tree, such that it is dissipated more effectively than in a tree with stiff branches and not so much focused on the tree trunk and the roots. Theoretical studies using modal analysis and finite element methods have supported these assertions. Next to "multiple mass damping" and "multiple resonance damping", both characterized by linear coupling between the elements, a third non linear mode, operative at large amplitudes has been identified: "damping by branching". In all these not mutually exclusive concepts frequency tuning between the elements appears to be a fundamental requisite.

Modes of failure in tubular plant organs.

PREMISE OF STUDY: Hollow tubular organs can bend and deform in one of two ways, i.e., either globally in long-wave deformation or locally in short-wave deformation (i.e., Brazier buckling). Either of these two types of behavior can cause death. Understanding the biophysical advantages and disadvantages of possessing hollow plant organs is important therefore to understanding plant ecology and avoiding damage to private or public property. METHODS: We present computer simulations that successfully predict when a hollow organ experiences different modes of failure as a function of organ length and wall thickness as well as material properties. KEY RESULTS AND CONCLUSIONS: When self-supporting, tubular plant organs are amenable to long-wave buckling and Brazier (short-wave) buckling under gravitational or wind-induced forces. For very slender tubes constructed of isotropic tissues, Brazier buckling depends on the outer wall radius and wall thickness (specifically Rt(2)). Particularly for organs constructed of anisotropic tissues, Brazier buckling becomes a complex phenomenon that depends on a number of geometric parameters (including length of the hollow section) as well as the material properties of tissues. This complexity precludes a definitive (canonical) limit to the relationship between wall thickness and outer radius and the safety limits for Brazier buckling.

Mechanical properties of wood disproportionately increase with increasing density.

PREMISE OF STUDY: Prior work using a large data set has shown that the mechanical properties of wood disproportionately increase with increasing wood density across diverse species, e.g., stems composed of denser wood are stiffer and stronger than stems with equivalent cross-sections composed of less dense wood. However, an alternative approach, introducing the precondition of constant construction cost for the same data set, adduces that for any given construction cost, stems composed of lesser dense woods are stiffer and stronger then stems composed of denser woods. METHODS: We evaluated these two approaches using generic allometric principles and the same large data set. KEY RESULTS: This evaluation shows that construction costs cannot be constant over an entire ensemble of stems composed of different species of wood. For any specified construction cost (denoted by a k-value), only a particular subgroup of stems is addressed. The conclusions derived for this subgroup cannot be generalized to the entire ensemble of stems composed of different species of wood. CONCLUSION: Stems composed of denser wood are, on average as stiff and strong, or stiffer and stronger than stems with equivalent cross-sections composed of less dense wood. Denser wood may have a higher carbon construction cost, but its mechanical benefits likely outweigh the extra cost.

Worldwide correlations of mechanical properties and green wood density.

PREMISE OF THE STUDY: The density of wood is highly correlated with the ability of stems and roots to resist bending or twisting, which is important for evaluating the mechanical behavior of trees. It also provides a measure of carbon storage, which is an important variable in modeling ecosystem processes and tree construction costs. However, most measurements of the density and mechanical properties of wood have little direct bearing on understanding the biomechanics of living plants because they are based on kiln- or air-dried samples. • METHODS: Here, we present and analyze the relationships between four important mechanical properties (Young's modulus, the modulus of rupture, and the maximum strength in shearing and in compression) and the density of green wood (i.e., wood at 50% moisture content) from a worldwide, taxonomically broad spectrum of 161 species. • KEY RESULTS: These data indicate that each of the mechanical properties disproportionately increases across species with increasing green wood density, i.e., stems composed of denser green wood are disproportionately stiffer and stronger than stems with equivalent cross-sections composed of less dense green wood. • CONCLUSIONS: Although denser wood may have a higher carbon construction cost, the mechanical benefits of denser woods likely outweigh the extra cost.

Predicting the allometry of leaf surface area and dry mass.

The manner in which increases in leaf surface area S scale with respect to increases in leaf dry mass M(t) within and across species has important implications to understanding the ability of plants to harvest sunlight, grow, and ultimately reproduce. Thus far, no mechanistic explanation has been advanced to explain why prior work shows that the scaling exponent governing the S to M(t) relationship is generally significantly less than one (i.e., S ∝ M(t)(α < 1.0)) such that increases in M(t) yield diminishing returns with respect to increases in S across most species. Here, we show analytically why this phenomenon occurs and present equations that predict trends observed in the numerical values of scaling exponents for the S vs. M(t) relationships observed across dicot tree species and two aquatic vascular plant species.

Multiple resonance damping or how do trees escape dangerously large oscillations?

To further understand the mechanics of trees under dynamic loads, we recorded damped oscillations of a Douglas fir (Pseudotsuga menziesii) tree and of its stem without branches. Eigenfrequencies of the branches were calculated and compared to the oscillation frequency of the intact tree. The term eigenfrequency is used here to characterize the calculated resonance frequency of a branch fixed at the proximal end to a solid support. All large branches had nearly the same frequency as the tree. This property is a prerequisite for the distribution of mechanical energy between stem and branches and leads to an enhanced efficiency of damping. We propose that trees constitute systems of coupled oscillators tuned to allow optimal energy dissipation.

Plant biomechanics: an overview and prospectus.

We provide a brief overview of the articles appearing in this special issue and place them in the context of the long history of the study of plant biomechanics and what we judge to be the next major intellectual and/or technological challenges in this field.

Allometric theory and the mechanical stability of large trees: proof and conjecture.

Recent allometric theory has postulated that standing leaf mass will scale as the 3/4 power of stem mass and as the 3/4 power of root mass such that stem mass scales isometrically with respect to root mass across very large vascular plant species with self-supporting stems. We show that the isometric scaling of stem mass with respect to root mass (i.e., M(S) ∝ M(R)) can be derived directly from mechanical theory, specifically from the requirement that wind-induced bending moments acting at the base of stems must be balanced by a counter-resisting moment provided by the root system to prevent uprooting. This derivation provides indirect verification of the allometric theory. It also draws attention to the fact that leaf, stem, and root biomass partitioning patterns must accommodate the simultaneous performance of manifold functional obligations.

Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass.

The size-dependent variations of plant height L and mass M with respect to basal stem diameter D are important to the analysis of a broad range of ecological and evolutionary phenomena. Prior examination of some of the world's largest trees suggests that the scaling relationships L alpha D(2/3) and M alpha D(8/3) hold true, ostensibly as functional adaptations for mechanical stability. This concept remains engrained in the literature in the form of null hypotheses (or predictive models), despite numerous examples showing that the 2/3 and 8/3 rules are violated by small and intermediate-sized plants. Here, we present a growth-hydraulic model that provides more accurate and biologically realistic predictions of L and M. This model also sheds light on why L, D, and M scale differently across species and habitats as a result of differences in absolute size.

Damped oscillations of the giant reed Arundo donax (Poaceae).

The slender upright culms of the giant reed (Arundo donax L.) are often exposed to dynamic wind loads causing significant swaying. The giant reed has slightly tapered hollow stems (4-6 m high) with flat leaves and an extensive underground rhizomatous system with solid branches bearing adventitious roots. Quantitative analyses of videorecordings prove that A. donax responds to dynamic deflections of the stem with damped harmonic bending oscillations. The logarithmic decrement can be used to calculate the relative damping, as a measure of the plant's capacity to dissipate vibrational energy. Plants with leaves have a significantly higher damping compared to plants without leaves. A comparison of the relative damping of plants with and without leaves shows that this finding is only partly due to aerodynamic resistance of the leaves. Structural damping also contributes considerably to the overall damping of the foliate A. donax stem. By stepwise removal of the underground plant organs the influence of rhizome, roots, and soil on the vibrational behavior was determined. The data indicate that underground plant organs as well as leaf sheaths covering the nodes have no significant influence on damping.

Oscillations of plants' stems and their damping: theory and experimentation.

Free oscillations of upright plants' stems, or in technical terms slender tapered rods with one end free, can be modelled by considering the equilibrium between bending moments and moments resulting from inertia. For stems with apical loads and negligible mass of the stem and for stems with finite mass but without top loading, analytical solutions of the differential equations with appropriate boundary conditions are available for a finite number of cases. For other cases approximations leading to an upper and a lower estimate of the frequency of oscillation omega can be derived. For the limiting case of omega = 0, the differential equations are identical with Greenhill's equations for the stability against Euler buckling of slender poles. To illustrate, the oscillation frequencies of 25 spruce trees (Picea sitchensis (Bong.) Carr.) were compared with those calculated on the basis of their morphology, their density and their static elasticity modulus. For Arundo donax L. and Cyperus alternifolius L. the observed oscillation frequency was used in turn to calculate the dynamic elasticity modulus, which was compared with that determined in three-point bending. Oscillation damping was observed for A. donax and C. alternifolius for plants' stems with and without leaves or inflorescence. In C. alternifolius the difference can be attributed to the aerodynamic resistance of the leaves, whereas in A. donax structural damping in addition plays a major role.

Micromechanics of plant tissues beyond the linear-elastic range.

We investigated the relation between cell wall structure and the resulting mechanical characteristics of different plant tissues. Special attention was paid to the mechanical behaviour beyond the linear-elastic range, the underlying micromechanical processes and the fracture characteristics. The previously proposed model of reorientation and slippage of the cellulose microfibrils in the cell wall [H.-CH. Spatz et al. (1999) J Exp Biol 202:3269-3272) was supported and is here refined, using measurements of the changes in microfibrillar angle during straining. Our model explains the widespread phenomenon of stress-strain curves with two linear portions of different slope and sheds light on the micromechanical processes involved in viscoelasticity and plastic yield. We also analysed the velocity dependence of viscoelasticity under the perspective of the Kelvin model, resolving the measured viscoelasticity into functions of a velocity-dependent and a velocity-independent friction. The influence of lignin on the above-mentioned mechanical properties was examined by chemical lignin extraction from tissues of Aristolochia macrophylla Lam. and by the use of transgenic plants of Arabidopsis thaliana (L.) Heynh. with reduced lignin content. Additionally, the influence of extraction of hemicelluloses on the mechanical properties was investigated as well as a cell wall mutant of Arabidopsis with an altered configuration of the cellulose microfibrils.

Oscillation frequencies of tapered plant stems.

Free oscillations of upright plant stems, or in technical terms, slender tapered rods with one end free, can be described by considering the equilibrium between bending moments in the form of a differential equation with appropriate boundary conditions. For stems with apical loads, where the mass of the stem is negligible, Mathematica 4.0 returns solutions for tapering modes α = 0, 0.5, and 1. For other values of α, including cases where the modulus of elasticity varies over the length of the stem, approximations leading to an upper and a lower estimate of the frequency of oscillation can be derived. For the limiting case of ω = 0, the differential equation is identical with Greenhill's equation for the stability against Euler buckling of a top-loaded slender pole. For stems without top loads, Mathematica 4.0 returns solutions only for two limiting cases, zero gravity (realized approximately for oscillations in a horizontal orientation of the stem) and for ω = 0 (Greenhill's equation). Approximations can be derived for all other cases. As an example, the oscillation of an Arundo donax plant stem is described.