The continuum theory provides a framework in which the growth of a plant root as a dynamic process involving interactions among transport of water and solute, cell division, and the subsequent cell elongation can be described. A plant root is modeled as a one-dimensional, multi-phase, mathematical continuum. The network of cell walls constitute the solid phase of the system. The symplast and the apoplast pathways reside in this network of cell walls. Water and carbohydrates move in opposite directions through the apoplast and symplast pathways within the deforming network of cell walls. The division and elongation of cells depends on the mechanical stress imposed on the cell walls, the rate of metabolic stress relaxation process, and the physical properties of the cell walls. The model consists of five systems of differential equations. The kinematic equations are derived which allow, specifically, the different roles of cell division and elongation in root growth to be considered. These provide the reference system of the model. Equations of water transport in the coupled system of apoplast and symplast pathways are derived from considerations of theories of transport in the porous media and the cellular and membrane properties of the plant root. Equations of solute transport are derived by considering, specifically, the mechanisms involved in solute transport both at the membranes separating individual cells and within the cytoplasm. The rate of cell elongation is described as a function of the mechanical stress in the cell walls, the viscoelastic properties of the cell walls, and a metabolically controlled strain energy relaxation process. Growth in the meristem is modeled as the result of continuous cell elongation and division. The equations of water and solute transport, cell elongation, and meristem growth are solved simultaneously under the reference system provided by the kinematic theory. The model is used to examine the effects of soil water stress, soil resistance to root penetration, and temperature, as well as the carbohydrate supply from the upper part of the plant on the dynamic process of root elongation. The close correspondence between the material coordinate system and the underlying cellular structure of the root allows the comparison between the continuum theory and the results of cell growth studies. Agreement of the model predictions of the pattern of growth along the root axis, as well as the effects of temperature and soil water stress on root growth, with the experimental measurements reported in the literature provides the justification for the theories. / Graduation date: 1991

Thesis (Ph. D.)--Oregon State University, 1996. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

It is postulated that definition of the availability of nutrients in soil t for plant uptake, will not be satisfactory, unless there is consideration of the plant's role as the absorber. The amount of nutrients absorbed by plants from soil can be related to i) the initial concentration in the soil solution, ii) the capacity of the soil to maintain this concentration (the buffer power), iii) the ease of movement of the nutrient to an absorbing root, and iv) the plant demand. These ideas are the foundation of models which successfully describe the nutrient uptake by single roots, growing in an infinite quantity of soil. The aim of the present work was to extend the approach used to explain uptake by single roots, to complete root systems. The requirements for the work are: (i) a theoretical description of nutrient flow to an absorbing, multiple root system; (ii) experiments for examining the proposed hypotheses. When a root absorbs a nutrient, there is a depletion in the nutrient concentration at the root surface. In a system of many roots, the zones of nutrient depletion around each root overlap. This reduces the effective initial concentration in the soil solution. So, in a multiple root system, the soil around each root is limited. The extent of overlap depends on the diffusion coefficient of the nutrient, the plant demand, and the interroot distance. The consequence is a lowering of the concentration at each root surface, below that of a similar root absorbing alone. An electrical analogue (Sanders et al. 1971) of diffusion of solutes to groups of absorbing roots was used to simulate nutrient uptake by plants from soil. The analogue was particularly useful for investigating the general consequences of different plant and soil conditions. To interpret specific plant uptake data, a more flexible computer model of diffusion and mass flow to a root system of variable density, with any specified uptake properties, was developed. For workers interested in an accuracy of ± 20%, an equation for calculating uptake, by systems similar to those which the computer model treats, is presented, which can be solved on a desk calculator. To test the model, experimental data on nutrient uptake, root dimensions and distribution, and soil conditions, during the growth of whole plants in soil was obtained. The computer model predicted the measured plant uptake wall, when values of the plant demand coefficient (which related uptake rate to the external solution concentration) given is the literature, from solution culture work under similar conditions, were used. It is concluded that the theory is an adequate representation of the simple plant-soil system used in the experiments. The expression, relating plant demand to concentration in the soil solution, was simulated on the electrical analogue. The effects of pattern, density, radius and demand coefficient of roots, on the course of uptake of solutes, of varying degrees of mobility, were investigated. Quantitative interrelations between soil and plant characteristics were established, which are discussed in the light of earlier concepts of mobility and availability of nutrients in soil. The uptake of a solute by any root system is roughly determined by the plant demand coefficient and the product of the solute diffusion coefficient, D, the absorption time, t, and the root density, L. The product DtL, for potassium, may often be is the range where root pattern affects uptake. This can be estimated graphically. Theory suggests that the uptake rate of K and K is the plant experiments was reduced by interroot competition. In both experiments, if supply was by physical processes only, the plants were absorbing K from the soil near to a maximum rate, which was set by transport through the soil. In those circumstances, the rate of uptake into a plant is limited by the length of root. Movement through the soil was easily able to supply the plant's requirement of N, until the total quantity was exhausted. It is deduced that this will usually be the case in British arable soils. A major problem, in the experimental determination of plant uptake from soil, results from the inaccessibility of the roots. A technique was devised for estimating the length and pattern of living roots of individual plants, without excessive labour. Radioactive roots are detected by autoradiography as they intersect planes of soil, and the root length par unit volume and rooting pattern follow easily. If two plants are labelled with different isotopes, their root systems can be distinguished. Any spatial interactions between the root systems are detected by the method, and the causes can be inferred. If the roots are extracted from the soil, their length can be automatically measured with an Image Analysing Computer.

The thesis is primarily concerned with the rate of uptake of nutrients by plant roots in relation to the ability of soil to supply nutrients to roots by diffusion and by mass flow. In Chapter One, some current concepts of the chemistry and availability of plant nutrients in soil are briefly discussed. Particular attention is paid to work that stresses the role of the dynamic processes of diffusion, and the mass flow of dissolved nutrients in the transpiration stream, in supplying nutrients to roots. In Chapter Two, diffusion coefficients of nutrients in soil are discussed. In Chapter Three, the mathematical description of nutrient flow to a root system in soil is discussed. An equation is developed which relates the mean nutrient inflow into a root system to the concentration of nutrients initially present in the soil solution, to the mean concentration of nutrients in the soil solution at the surface of the root system,to the mean rate of flow of water to the roots, to the diffusion coefficient of the nutrient in the soil, and to the age of the roots. The mean nutrient inflow into a root system is defined as the mean rate of flow of nutrients into unit length of root. In Chapter Four, an experiment is described in which all the terms in the above mentioned equation were measured apart from the mean root surface concentration of nutrients. In the experiment, the growth and nutrient uptake of leek plants growing outdoors in a moist, fertilised, silty loam soil was followed. The plants were grown in large pots containing an ample reservoir of soil water, so that watering was unnecessary during growth. The growing period was from early May to mid-July. The nutrient uptake rate was followed by harvesting, and analysing for nutrients, samples of ten plants taken at approximately ten day intervals. Root length was measured at each harvest after washing the roots free from the soil. The pots were weighed every few days to determine the water lost by transpiration. At intervals during the growing period soil samples were taken from the pots, and the soil solution was extracted and analysed. From separate experiments, diffusion coefficients for nutrients in the soil were calculated. The data from the experiments yielded all the terms in the above mentioned equation except mean root surface concentration which could be calculated. The nutrients considered were N, P, K, Ca, Na, Mg, S and Cl. It was found that mass flow supplied on average more of all nutrients to the roots than they absorbed apart from K and P, which were supplied mainly by diffusion. For those nutrients that were found to be supplied by mass flow in excess of uptake, calculations indicated that the resultant mean accumulations at the root surface did not give rise to a solution concentration at the root surface greater than 120% of the initial concentration in the soil. In contrast it was calculated that the root surface concentration of K was less than half its initial level in the soil. P had a concentration dependent diffusion coefficient which meant that the quantity of P diffusing to a root had to be calculated numerically using a computer. The results of such a calculation are given in Chapter Five. It was found that even if the roots acted a zero sink for phosphate, the theoretical mean inflow of phosphate was somewhat less than the observed mean inflow. In Chapter Seven an experimental investigation is described into the accumulation of sulphate at the surface of a root in conditions of high mass flow. Autoradiography using S³⁵ was the technique used. In accordance with theory, accumulations became large only when the soil was fairly dry. One of the difficulties in the mathematical analysis of nutrient flow to roots is the uncertainty about the inherent absorbing power of roots of different ages. In Section Three, experiments are described in which the uptake of short segments of leek root of different ages was measured. The leeks for these experiments were grown in solution culture and the mean inflow into the roots was measured by sampling and analysis as described for the pot experiment in soil. Some plants from solution were then selected and short segments of the roots of different ages were sealed into tubes containing two radioactively labelled nutrients, the rest of the root system being grown in unlabelled nutrient solution. P³², and either K⁴² or Sr⁸⁵, Sr⁸⁵ being taken as a label for Ca, were the labelled nutrients used. The plants were harvested after twenty four hours of exposure to labelled nutrients and the quantity of label absorbed was measured. There were no significant differences in the mean absorption of different aged roots, but different root segments varied very widely in their uptake in a way that could not be connected with their age. The possibility of such variation in absorbing power in roots in soil and its consequences in soil are discussed at the end of Chapter Eleven. In Chapter Twelve values for the mean inflow of nutrients into roots from a wide range of published experiments are tabulated. The rates are also given as specific absorption rates, this term meaning the rate of flow of nutrient into unit fresh weight of root. Similar values of mean inflow and specific absorption rate have been measured in widely different conditions, ranging from a few minutes uptake from solution to several weeks uptake from soil. Plant factors which affect mean nutrient inflows are discussed and possible future developments along the lines suggested by the experiment in Chapter Four are considered. In the final Chapter the evidence is stated on which is based current theory of nutrient flow to roots by mass flow and diffusion. In conclusion the insight the theory provides into the concept of nutrient availability in soil, into root competition for nutrients and into ideas about root system efficiency is considered.

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Thesis (doctoral)--Universität Leipzig, 1874. / Cover title. Includes bibliographical references.

Thesis (doctoral)--Universität Leipzig, 1874. / Cover title. Includes bibliographical references.

Thesis (Ph. D.)--University of Michigan, 1920. / "Reprinted from the American journal of botany, vol. VII, November, 1920." Includes bibliographical references (p. 404-406).

Thesis (Ph. D.)--University of Nebraska, 1925. / Cover title. Published also without thesis note. By Frank C. Jean and John E. Weaver. Bibliography: p. 66.

Thesis (Ph. D.)--University of Michigan, 1920. / "Reprinted from the American journal of botany, vol. VII, November, 1920." Includes bibliographical references (p. 404-406).