Theoretical Analyses of Turgor Pressure during Stress Relaxation and Water Uptake, and after Changes in Expansive Growth Rate When Water Uptake Is Normal and Reduced.

Turgor pressure provides the force needed to stress and deform the cell walls of plants, algae, and fungi during expansive growth. However, turgor pressure plays another subtle but equally important role in expansive growth of walled cells: it connects the two biophysical processes of water uptake and wall deformation to ensure that the volumetric rates of water uptake and enlargement of the cell wall chamber are equal. In this study, the role of turgor pressure as a 'connector' is investigated analytically by employing validated and established biophysical equations. The objective is to determine the effect of 'wall loosening' on the magnitude of turgor pressure. It is known that an increase or decrease in turgor pressure and/or wall loosening rate increases or decreases the expansive growth rate, respectively. Interestingly, it is shown that an increase in the wall loosening rate decreases the turgor pressure slightly, thus reducing the effect of wall loosening on increasing the expansive growth rate. Other analyses reveal that reducing the rate of water uptake results in a larger decrease in turgor pressure with the same increase in wall loosening rate, which further reduces the effect of wall loosening on increasing the expansive growth rate.

Biophysical Equations and Pressure Probe Experiments to Determine Altered Growth Processes after Changes in Environment, Development, and Mutations.

Expansive growth is a culmination of many biological processes. It is fundamental to volume growth, development, morphogenesis, sensory responses, and environmental responses of plants, fungi, and algae. Expansive growth of walled cells and plant tissue can be accurately described by a set of three global biophysical equations that model the biophysical processes of water uptake, wall deformation, and turgor pressure. Importantly, these biophysical equations have been validated with the results of pressure probe experiments. Here, a systematic method (scheme) is presented that iterates between analyses with the biophysical equations and experiments conducted with the pressure probe. This iterative scheme is used to determine altered growth processes for four cases; two after changes in the environment, one after a change in development, and another after changes by mutation. It is shown that this iterative scheme can identify which biophysical processes are changed, the magnitude of the changes, and their contribution to the change in expansive growth rate. Dimensionless numbers are employed to determine the magnitude of the changes in the biophysical processes. The biological meaning and implication of the biophysical variables in the biophysical equations are discussed. Further, additional sets of global biophysical equations are presented and discussed.

Helical growth during the phototropic response, avoidance response, and in stiff mutants of Phycomyces blakesleeanus.

The sporangiophores of Phycomyces blakesleeanus have been used as a model system to study sensory transduction, helical growth, and to establish global biophysical equations for expansive growth of walled cells. More recently, local statistical biophysical models of the cell wall are being constructed to better understand the molecular underpinnings of helical growth and its behavior during the many growth responses of the sporangiophores to sensory stimuli. Previous experimental and theoretical findings guide the development of these local models. Future development requires an investigation of explicit and implicit assumptions made in the prior research. Here, experiments are conducted to test three assumptions made in prior research, that (a) elongation rate, (b) rotation rate, and (c) helical growth steepness, R, of the sporangiophore remain constant during the phototropic response (bending toward unilateral light) and the avoidance response (bending away from solid barriers). The experimental results reveal that all three assumptions are incorrect for the phototropic response and probably incorrect for the avoidance response but the results are less conclusive. Generally, the experimental results indicate that the elongation and rotation rates increase during these responses, as does R, indicating that the helical growth steepness become flatter. The implications of these findings on prior research, the "fibril reorientation and slippage" hypothesis, global biophysical equations, and local statistical biophysical models are discussed.

Dimensionless Numbers to Analyze Expansive Growth Processes.

Cells of algae, fungi, and plants have walls and exhibit expansive growth which can increase their volume by as much as 10,000 times. Expansive growth is central to their morphogenesis, development, and sensory responses to environmental stimuli. Equations describing the biophysical processes of the water uptake rate and the wall deformation rate have been derived, validated, and established. A significant amount of research provides insight into the molecular underpinnings of these processes. What is less well known are the relative magnitudes of these processes and how they compare during expansive growth and with walled cells from other species. Here, dimensionless numbers (Π parameters) are used to determine the magnitudes of the biophysical processes involved in the expansive growth rate of cells from algae (Chara corallina), fungi (Phycomyces blakesleeanus), and plants (Pisum satinis L.). It is found for all three species that the cell's capability for the water uptake rate is larger than the wall plastic deformation rate and much larger than the wall elastic deformation rate. Also, the wall plastic deformation rates of all three species are of similar magnitude as their expansive growth rate even though the stress relaxation rates of their walls are very different. It is envisioned that dimensionless numbers can assist in determining how these biophysical processes change during development, morphogenesis, sensory responses, environmental stress, climate change, and after genetic modification.

Dimensionless numbers to study cell wall deformation of stiff mutants of Phycomyces blakesleeanus.

The sporangiophores of Phycomyces blakesleeanus are large cylindrical aerial cells that elongate vertically at rates between 10 µm/min and 60 µm/min. Wild-type sporangiophores grow toward light, opposed to gravitational acceleration and away from solid barriers (tropic responses). Sporangiophores of stiff mutants C149 and C216 exhibit diminished tropic (bending) responses. Originally, it was thought that the altered genes affect the "stiffness" (elastic wall deformation) of the cell wall. Subsequent investigations employing the pressure probe demonstrated that the irreversible (plastic) wall deformation was smaller for the stiff mutants compared to wild type and could account for the diminished tropic responses. However, it was not shown whether the elastic wall deformation was altered in these stiff mutants. Recent theoretical studies have identified dimensionless numbers that can be used to quantitate the magnitudes of biophysical processes involved in expansive growth of walled cells. In this study, dimensionless numbers are used to determine the magnitudes of elastic deformation rate, plastic deformation rate, and stress relaxation rate of the cell wall during expansive growth of the stiff mutant sporangiophores. It is found that the altered genes reduce stress relaxation rates and plastic deformation rates of the wall, but do not significantly alter the magnitude of the elastic deformation rates of the wall. These results indicate that the mutant genes reduce wall loosening chemistry in these sporangiophores and the genetic mutation is not expressed in a change in "wall stiffness," but in "wall viscosity" or "wall extensibility."

A Statistical Model of Expansive Growth in Plant and Fungal Cells: The Case of Phycomyces.

Expansive growth is a process by which walled cells of plants, algae, and fungi use turgor pressure to mediate irreversible wall deformation and regulate their shape and volume. The molecular structure of the primary cell wall must therefore perform multiple functions simultaneously, including providing structural support by combining elastic and irreversible deformation and facilitating the deposition of new material during growth. This is accomplished by a network of microfibrils and tethers composed of complex polysaccharides and proteins that can dynamically mediate the network topology via periodic detachment and reattachment events. Lockhart and Ortega have provided crucial macroscopic understanding of the expansive growth process through global biophysical models, but these models lack the connection to molecular processes that trigger network rearrangements in the wall. Interestingly, the helical growth of the fungal sporangiophores of Phycomyces blakesleeanus is attributed to a limited region (called the growth zone) where microfibrils are deposited, followed by reorientation and slip. Based on past evidence of dominant shear strain between microfibrils (slippage), we propose a mechanistic model of a network of sliding fibrils connected by tethers. A statistical approach is introduced to describe the population behavior of tethers that have elastic properties and the ability to break and reform in time. These properties are responsible for global cell wall mechanics such as creep and stress relaxation. Model predictions are compared with experiments from literature on stress relaxation and turgor pressure step up for the growing cells of P. blakesleeanus, which are later extended to incised pea (Pisum sativus L.) and the algae Chara corallina using the unique dimensionless number Πpe for each species. To our knowledge, this research is the first attempt to use a statistical approach to model the cell wall during expansive growth, and we believe it provides critical insights on cell wall dynamics at a molecular level.

Dimensionless Numbers for Plant Biology.

Dimensionless numbers are ubiquitous in the physical sciences because they provide insight into physical processes, organize large quantities of data, facilitate 'scale analysis' and establish 'similarity'. Here I explore the use of dimensionless numbers in plant biology, focusing on the expansive growth rate of plant, fungal, and algal cells.

Dimensionless number is central to stress relaxation and expansive growth of the cell wall.

Experiments demonstrate that both plastic and elastic deformation of the cell wall are necessary for wall stress relaxation and expansive growth of walled cells. A biophysical equation (Augmented Growth Equation) was previously shown to accurately model the experimentally observed wall stress relaxation and expansive growth rate. Here, dimensional analysis is used to obtain a dimensionless Augmented Growth Equation with dimensionless coefficients (groups of variables, or Π parameters). It is shown that a single Π parameter controls the wall stress relaxation rate. The Π parameter represents the ratio of plastic and elastic deformation rates, and provides an explicit relationship between expansive growth rate and the wall's mechanical properties. Values for Π are calculated for plant, algal, and fungal cells from previously reported experimental results. It is found that the Π values for each cell species are large and very different from each other. Expansive growth rates are calculated using the calculated Π values and are compared to those measured for plant and fungal cells during different growth conditions, after treatment with IAA, and in different developmental stages. The comparison shows good agreement and supports the claim that the Π parameter is central to expansive growth rate of walled cells.

Cell Wall Loosening in the Fungus, Phycomyces blakesleeanus.

A considerable amount of research has been conducted to determine how cell walls are loosened to produce irreversible wall deformation and expansive growth in plant and algal cells. The same cannot be said about fungal cells. Almost nothing is known about how fungal cells loosen their walls to produce irreversible wall deformation and expansive growth. In this study, anoxia is used to chemically isolate the wall from the protoplasm of the sporangiophores of Phycomyces blakesleeanus. The experimental results provide direct evidence of the existence of chemistry within the fungal wall that is responsible for wall loosening, irreversible wall deformation and elongation growth. In addition, constant-tension extension experiments are conducted on frozen-thawed sporangiophore walls to obtain insight into the wall chemistry and wall loosening mechanism. It is found that a decrease in pH to 4.6 produces creep extension in the frozen-thawed sporangiophore wall that is similar, but not identical, to that found in frozen-thawed higher plant cell walls. Experimental results from frozen-thawed and boiled sporangiophore walls suggest that protein activity may be involved in the creep extension.

Stiff mutant genes of phycomyces affect turgor pressure and wall mechanical properties to regulate elongation growth rate.

Regulation of cell growth is paramount to all living organisms. In plants, algae and fungi, regulation of expansive growth of cells is required for development and morphogenesis. Also, many sensory responses of stage IVb sporangiophores of Phycomyces blakesleeanus are produced by regulating elongation growth rate (growth responses) and differential elongation growth rate (tropic responses). "Stiff" mutant sporangiophores exhibit diminished tropic responses and are found to be defective in at least five genes; madD, E, F, G, and J. Prior experimental research suggests that the defective genes affect growth regulation, but this was not verified. All the growth of the single-celled stalk of the stage IVb sporangiophore occurs in a short region termed the "growth zone." Prior experimental and theoretical research indicates that elongation growth rate of the stage IVb sporangiophore can be regulated by controlling the cell wall mechanical properties within the growth zone and the magnitude of the turgor pressure. A quantitative biophysical model for elongation growth rate is required to elucidate the relationship between wall mechanical properties and turgor pressure during growth regulation. In this study, it is hypothesized that the mechanical properties of the wall within the growth zone of stiff mutant sporangiophores are different compared to wild type (WT). A biophysical equation for elongation growth rate is derived for fungal and plant cells with a growth zone. Two strains of stiff mutants are studied, C149 madD120 (-) and C216 geo- (-). Experimental results demonstrate that turgor pressure is larger but irreversible wall deformation rates within the growth zone and growth zone length are smaller for stiff mutant sporangiophores compared to WT. These findings can explain the diminished tropic responses of the stiff mutant sporangiophores. It is speculated that the defective genes affect the amount of wall-building material delivered to the inner cell wall.

Plant cell growth in tissue.

Cell walls are part of the apoplasm pathway that transports water, solutes, and nutrients to cells within plant tissue. Pressures within the apoplasm (cell walls and xylem) are often different from atmospheric pressure during expansive growth of plant cells in tissue. The previously established Augmented Growth Equations are modified to evaluate the turgor pressure, water uptake, and expansive growth of plant cells in tissue when pressures within the apoplasm are lower and higher than atmospheric pressure. Analyses indicate that a step-down and step-up in pressure within the apoplasm will cause an exponential decrease and increase in turgor pressure, respectively, and the rates of water uptake and expansive growth each undergo a rapid decrease and increase, respectively, followed by an exponential return to their initial magnitude. Other analyses indicate that pressure within the apoplasm decreases exponentially to a lower value after a step-down in turgor pressure, which simulates its behavior after an increase in expansive growth rate. Also, analyses indicate that the turgor pressure decays exponentially to a constant value that is the sum of the critical turgor pressure and pressure within the apoplasm during stress relaxation experiments in which pressures within the apoplasm are not atmospheric pressure. Additional analyses indicate that when the turgor pressure is constant (clamped), a decrease in pressure within the apoplasm elicits an increase in elastic expansion followed by an increase in irreversible expansion rate. Some analytical results are supported by prior experimental research, and other analytical results can be verified with existing experimental methods.

Mechanics and modeling of plant cell growth.

Cellular expansive growth is one of the foundations of morphogenesis. In plant and fungal cells, expansive growth is ultimately determined by manipulating the mechanics of the cell wall. Therefore, theoretical and biophysical descriptions of cellular growth processes focus on mathematical models of cell wall biomechanical responses to tensile stresses, produced by the turgor pressure. To capture and explain the biological processes they describe, mathematical models need quantitative information on relevant biophysical parameters, geometry and cellular structure. The increased use of mechanical modeling approaches in plant and fungal cell biology emphasizes the need for the concerted development of both disciplines and underlines the obligation of biologists to understand basic biophysical principles.