John W. Crawford

Quantification of fungal morphology, gaseous transport and microbial dynamics in soil: an integrated framework utilising fractal geometry

The consequences of heterogeneous structure for nutrient acquisition by soil fungi, microbial dynamics and transport in soil are studied. Fractal geometry provides the unifying theme and forms the basis of a theoretical framework for studying dynamics in heterogenous media. The interpretation of foraging strategies of soil fungi are presented which suggest that the processes governing branching and hyphal mass distribution are independent. Classical diffusion is shown to be inappropriate for the study of diffusion in heterogeneous soil and a new theory is proposed which incorporates heterogeneity and pore tortuosity. The consequences of structure for microbial spatial and temporal dynamics are examined and it is found that an understanding of these and related processes such as nutrient cycling must include the role of soil structure. While stressing the need to appreciate the relevance of the theory to any particular application, it is shown that quantitative fractal geometry can yield insights into the mechanism whereby spatial organisation influences the interaction between structure and biotic processes in the soil.

Towards a general theory of biodiversity

The study of patterns in living diversity is driven by the desire to find the universal rules that underlie the organization of ecosystems. The relative abundance distribution, which characterizes the total number and abundance of species in a community, is arguably the most fundamental measure in ecology. Considerable effort has been expended in striving for a general theory that can explain the form of the distribution. Despite this, a mechanistic understanding of the form in terms of physiological and environmental parameters remains elusive. Recently, it has been proposed that space plays a central role in generating the patterns of diversity. Here we show that an understanding of the observed form of the relative abundance distribution requires a consideration of how individuals pack in time. We present a framework for studying the dynamics of communities which generalizes the prevailing species-based approach to one based on individuals that are characterized by their physiological traits. The observed form of the abundance distribution and its dependence on richness and disturbance are reproduced, and can be understood in terms of the trade-off between time to reproduction and fecundity.

APPLICATIONS OF FRACTALS TO SOIL STUDIES

Publisher Summary This chapter discusses the applications of fractals for soil analysis. The different types of fractal dimensions are discussed that are used in soil science. The mathematical basis of fractal geometry makes it a potentially useful tool to describe the heterogeneity of soil structure quantitatively. A quantitative description of soil structure would ideally be able to be directly related to the processes occurring within the soil. This chapter discusses the relationship between fractal dimensions of the soil structure and soil physical processes. Soil structure has been characterized by fractal dimensions, estimated either directly from images of soil structure or indirectly from bulk density or mercury porosirnetry data, for example. The fragmentation mechanism is also discussed. The main factor that limits the number of applications that the fragmentation fractal dimension (D f ) has is that D f , is estimated from a distribution of aggregates or particles that bear no resemblance to the original soil matrix. In some of the studies reported soil chemical properties and soil mineralogy have been discussed in relation to the fractal dimension estimated for a particular soil, or they have been used to explain apparent changes in fractal scaling. Further research into relating chemical properties and mineralogy of different soil types to their fractal dimensions may help in the understanding the origin of this type of fractal scaling.

Quantification of the fractal nature of colonies of Trichoderma viride

Fractal dimensions of colonies of Trichoderma viride grown on cellophane-coated nutrient agar varied between 1·4 and ca 2. As colonies aged, the fractal dimension increased, i.e. space was explored with increasing effectiveness. The implications of these findings for interpreting the way fungi interact with their environment and nutrient resources are discussed.

A simulation study of crop growth and development under climate change

Climate changes of the order predicted by Global Circulation Models have important implications for arable crop production. We have studied the impact in Scotland using simulation models for three crops of contrasting developmental type: faba or field bean, potato, spring and winter wheat. The models used were the FABEAN, SCRI water-constrained potato model and AFR-CWHEAT2 models respectively. Consideration has been made of the natural year-to-year variation in weather which causes yield variability by using 100 years of input weather data produced by a weather generator. The models were run for four Scottish sites and five Scottish soils. Based on GCM predictions, we used eight scenarios of future climate which combine both temperature and rainfall changes. Current temperature (T0) and rainfall (R0) were used as a baseline, and each of T0 + 1°C, T0 + 2°C, T0 + 3°C were used with rainfall unchanged at R0, and increased by seasonally adjusted amounts ranging from 0 to 1.5 mm per wet day. Possible enhancements due to CO2 fertilisation were not included in the study. Increased temperatures increase crop development rate, which shortens the growing season for wheat and faba bean, but, given a fixed harvest date, lengthens the season for potatoes. Yields of potato increased by up to 33% over all our sites and scenarios, whereas wheat yields decreased by 5–15% and faba bean by 11–41%. Rainfall increases of the amount suggested here do not affect the yield of potatoes or spring wheat, but winter wheat yields are reduced, due to leaching, and faba bean yields increase through alleviation of water shortage. Faba beans also show a reduction in yield variability as a result of increased rainfall. Changes in variability in wheat and potato were less pronounced and tended to reflect the increase in variability which was assumed to accompany the increased rainfall. Predictions for the changes in the frequencies of high and low yields are also presented. The results give an indication of the level of changes in crop production which would be expected in these future climates.

The distribution of anoxic volume in a fractal model of soil

A simple description of soil respiration is combined with a three-dimensional random fractal lattice as a model of soil structure. The lattice consists of gas-filled pores and soil matrix that is a combination of the solid phase and water. A respiration process is assumed to take place in the soil matrix. Oxygen transport occurs by diffusion in the gas-filled pores and, at a much slower rate, in the soil matrix. The stationary state of this process is characterized by the fraction of the matrix that has zero oxygen concentration, i.e., the anoxic fraction. The anoxic fraction of a three-dimensional lattice appears to be largely determined by the presence and distribution of pores that are not connected to the surface of the lattice. Local gradients in connected gas-filled pores play an insignificant role due to the enormous difference in diffusion coefficient between the gas-filled pores and the saturated soil matrix. Analytical and numerical results for the fractal model are compared with calculations for a dual-porosity model comprising spherical aggregates with a lognormal radius distribution. A one-dimensional fractal lattice and the dual-porosity model yield qualitatively similar predictions, suggesting an anoxic fraction that decreases exponentially with the square root of the local oxygen concentration. However, the anoxic fraction of a three-dimensional fractal lattice decreases much faster than exponentially, implying that large clumps of soil matrix are comparatively rare. We propose that this is due to aggregation of soil particles in more than a single dimension, which has important consequences for anaerobic processes in soil. The fractal model accounts for the geometrical implications of three dimensions. A lognormal radius distribution is essentially a one-dimensional structure model.

Context-dependent macroscopic patterns in growing and interacting mycelial networks

Fungal mycelia epitomize, at the cellular level of organization, the growth and pattern-generating properties of a wide variety of indeterminate (indefinitely expandable) living systems. Some of the more important of these properties arise from the capacity of an initially dendritic system of protoplasm filled, apically extending hyphal tubes to anastomose. This integrational process partly restores the symmetry lost during the proliferation of hyphal branches from a germinating spore and so increases the scope for communication and transfer of resources across the system. Growth and pattern generation then depend critically on processes that affect the degree to which resistances to energy transfer within the system are sustained, bypassed or broken down. We use a system of reaction diffusion equations augmented with appropriate initial data to model the processes of expansion and pattern formation within growing mycelia. Such an approach is a test of the feasibility of the hypothesis that radical, adaptive shifts in mycelial pattern can be explained by purely contextual, rather than genetic, changes. Thus we demonstrate that phenotype does not necessarily equate solely to genotype—environment interactions, but may include the physical role in self-organization played by the boundary between the two.

Heterogeneity of the pore and solid volume of soil: distinguishing a fractal space from its non-fractal complement

The scaling properties of fractals are formally defined only in the limit of infinite or infinitessimal length scales, while measurements are necessarily restricted to finite scales. The implications of this limitation are studied from the perspective of characterising the heterogeneity of a fractal and its complement. An examination of the scaling properties of the solid and pore space of soil aggregates is presented as a case study. A relationship is derived between the heterogeneity of the solid volume and that of the pore space, under the assumption that either the solid or pore can be approximated by a fractal. If the solid is fractal, then the pore volume (its space complement) is asymptotically homogeneous as the sample volume increases to infinity i.e. the porosity is effectively scale-independent for large sample sizes. For smaller sample volumes, however, the porosity is scale-dependent, and this dependency is a function of the fractal dimension of the solid. Where the solid has a fractal dimension close to the Euclidean value, the pore volume behaves like a fractal, and this presents a problem from a methodological point of view, in distinguishing between a fractal and its complement. It is shown how misinterpretation can lead to errors in the prediction of moisture distribution and transport processes. A method for making the distinction is presented here, and is tested using image analysis of soil thin sections.

The origins of spatial heterogeneity in vegetative mycelia: a reaction-diffusion model

Fungal mycelia are obviously heterogeneous structures, which is a key feature of their adaptability to grow in complex environments. A mathematical model is presented which facilitates assessment of the role of exogenous and endogenous factors in generating such heterogeneous structures. The model is compared with experimental systems where mycelia were grown in spatially uniform and non-uniform nutrient environments. The fractal nature of mycelia and the response of the structure to a spatially heterogeneous nutrient distribution are reproduced. We show that the nutrient availability plays a decisive role in determining both the fractal growth of the colony and its response to spatially heterogeneous nutrient distribution.

Detailed visualisation of hyphal distribution in fungal mycelia growing in heterogeneous nutritional environments

A technique is described which enables high resolution images of the distribution of individual hyphae within fungal mycelia to be obtained, when the fungi are growing in nutritionally heterogeneous environments. The basis of the technique is to project colonies grown on the surface of cellulose membranes onto high contrast film using a photographic enlarger. Heterogeneity of nutrient resources is controlled by overlaying the membrane onto the surface of a box designed to hold disposable pipette tips, where the holes designed to hold the tips have been filled with gels of contrasting nutrient status. The effect of regular and irregular patterns of relatively high and low nutrient status gels upon development of single mycelia of Trichoderma viride and Rhizoctonia solani is shown. The technique can readily image mycelia up to 30 cm2 whilst still permitting resolution of individual hyphae where nutrient supply is sufficiently low to result in low hyphal densities.