Mechanics and modeling of plant cell growth

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1 Online Supplementary Material Mechanics and modeling of plant cell growth Anja Geitmann 1, Joseph K. E. Ortega 2 1 Institut de recherche en biologie végétale, Département de sciences biologiques, Université de Montréal, Québec H1X 2B2, Canada 2 Bioengineering Laboratory, Department of Mechanical Engineering, University of Colorado Denver, Denver, Colorado , USA Corresponding author: Geitmann, A. (anja.geitmann@umontreal.ca) From Lockhart to the augmented growth equation Biophysical equations describing cell wall mechanics and expansive growth Expansive growth of algal, fungal and plant cells is the result of metabolic-mediated biochemical processes in conjunction with interrelated physical processes [1]. From a physical perspective, expansive growth of the cells of all of these evolutionary distant organisms can be viewed the same way. It is defined as a permanent increase in cell volume. Thus the cell wall must increase in surface area, A cw, and volume, V cw (V cw = A cw τ cw, where τ cw is the wall thickness), because the volume of the cell wall chamber, V cwc, must equal the volume of cell contents, V cc, which it encloses. While V cw does not necessarily change at the same rate as A cw during the different developmental stages of a cell [2], the overall tendency is a linear relationship between both parameters, as long as primary growth occurs. During expansive growth, the rate of increase in the volume of cell contents, dv cc /dt, is predominately the result of water uptake, i.e. dv cc /dt dv w /dt. The production and accumulation of active solutes within the cell membrane generates the osmotic pressure that drives the water uptake from the cell exterior to its interior. The driving force for water uptake is typically measured as the difference in the osmotic pressure inside (π i ) and outside (π o ) the cell membrane, i.e., Δπ = π i - π o. The water uptake increases the volume of both the cell contents and the cell wall chamber. When the volume of the cell contents increases, the internal hydrostatic pressure (P i ) increases because the cell wall resists being deformed. As P i increases to magnitudes larger than the external pressure (P o ) it drives water out of the cell. The net water uptake rate is the 1

2 difference between the water flowing into the cell because of the osmotic pressure difference, Δπ, and the water flowing out of the cell because of the turgor pressure, P, which is defined as, P = P i P o, i.e. Δπ - P. Thus an increase in P reduces the net water uptake rate, and at one magnitude of P the net water uptake will stop (P = Δπ). The P produces stresses within the cell wall. The cell wall stretches (deforms) in response to these cell wall stresses and increases its surface area. The deformations are both irreversible (plastic) and reversible (elastic). The assimilation of new cell wall materials into the deforming wall controls cell wall thickness, τ cw and increases the cell wall volume, V cw. The characteristics of the cell wall deformation, or its mechanical behavior, depend on the mechanical properties of the cell wall and its biological state. Equations relating expansive growth of plant cells to the rate of water uptake and cell wall extension were first published by Lockhart [3]. Equation 1 describes the relationship between the rate of increase in water volume and the net rate of water uptake (in relative terms): (dv/dt) w /V w = L pr (Δπ - P) [1] (rate of increase of water volume) = (net rate of water uptake) The relative hydraulic conductance is defined as, L pr = L p A m /V w, and it is assumed that the solute reflection coefficient of the cell membrane is unity. Equation 2 describes the relationship between the rate of increase in the volume of the cell wall chamber and the rate of irreversible cell wall expansion (in relative terms): (dv/dt) cwc /V cwc = φ (P - P c ) [2] (rate of increase of cell wall chamber volume) = (irreversible expansion rate) The irreversible wall extensibility, φ, and the critical turgor pressure, P c, are biophysical variables to be determined for each cell. Equation 2 can be derived from the constitutive equation (equation describing the stress-strain relationship) for a viscous dashpot with a Bingham fluid, i.e. de/dt = (σ - σ c )/μ, where de/dt is the strain rate, σ is the stress, σ c is the critical stress, and μ is the dynamic viscosity of the Bingham fluid [4]. Recent theoretical research indicates that Eq. 2 can be obtained from the diagonal component of a more general tensor equation [5] that can model growth anisotropies such as phototropism and gravitropism [6]. During expansive growth, because the rate of increase in volume of the water is approximately equal to rate of increase in volume of the cell wall chamber, a third equation can be derived for the steady-state turgor pressure: P = (L pr Δπ + φ P c ) / (φ + L pr ) [3] 2

3 Equations 1 3 describe and model expansive growth of cells with walls when the turgor pressure is constant. Over the years, these equations (Eqs 1-3) have been termed the Lockhart Equations. However, it is crucial to note that the Lockhart Equations (Eqs 2 and 3) and its underlying constitutive equations (viscous dashpot with a Bingham fluid) cannot model the behavior exhibited in Figure S1a and S1b, because the underlying constitutive equation cannot model the observed exponential decay in stress or the associated exponential decay in turgor pressure. Subsequently, a more detailed and explicit conceptual model of the expansive growth process emerged, e.g. [7,8]. Experimental evidence demonstrates that during expansive growth, cell wall stresses relax (decrease) because ongoing biochemical processes loosen the cell wall, resulting in a decrease in stress that is observed in Figure S1a. The turgor pressure decreases (Figure S1b) in response to the decrease in cell wall stresses and produces an increase in net water uptake. The increase in net water uptake increases the turgor pressure, which in turn increases the cell wall stresses. Then as before, the cell wall stresses begin to relax because of ongoing cell wall loosening. This process is repeated continuously during expansive growth. The main experimental evidence that supports this iterative pressure relaxation and water uptake concept of expansive growth, is that the turgor pressure decays in growing cells with walls when the water uptake is eliminated and transpiration is suppressed (e.g. [7-10]. The decay in turgor pressure (Figure S1b) results from a decrease in cell wall stresses by ongoing biochemical processes (Figure S1a). The effect of biochemical processes on cell wall stresses can be illustrated by a pretreatment with IAA (auxin), which accelerates the decay of turgor pressure (Figure S1b). This is consistent with the hormone's role in increasing expansive growth rate. Similarly, the stress decay rate increases in the cell wall after a light stimulus, a trigger that is known to increase the elongation growth rate [11]. Importantly, the Lockhart Equations (Eqs 2 and 3) cannot model the decrease in turgor pressure (pressure relaxation) exhibited in Figure S1b, or the iterative pressure relaxation and water uptake process for expansive growth, because Eqs 2 and 3 are only valid for constant turgor pressure, i.e. P = constant or dp/dt = 0. Specifically, Eq. 2 (and its underlying viscoelastic model) cannot model the stress relaxation and pressure relaxation results obtained from growing cells (Figures S1a and b). Some excellent papers and recent reviews correctly describe the iterative pressure relaxation and water uptake conceptual model of expansive growth, and then refer to the Lockhart Equations which cannot model the pressure relaxation, i.e. the decrease in turgor 3

4 pressure. The Lockhart equations form the basis upon which most models are built, however, they have crucial shortcomings of being applicable only in situations when turgor is constant, and when water lost through transpiration is neglected or negligible. To model expansive growth of plant, algal and fungal cells in the more general cases, the Lockhart equations were augmented to account for changing turgor pressure and transpiration [4,12]. These augmented growth equations [13,14] have proven to be very useful in analyzing and interpreting the experimental results of growing algal cells (e.g. [15,16], fungal cells (e.g. [10,12,17-19], and plant organs (e.g. [7,8,20-23]. It is apparent that a slightly more complex viscoelastic model and corresponding biophysical equation is needed. This was achieved by the biophysical equation, Eq. 4 which was derived from the constitutive equation for a Bingham-Maxwell viscoelastic model, i.e. de/dt = (σ - σ c )/μ + (dσ /dt )/E, where E is the elastic modulus [4]. (dv/dt) cwc /V cwc = φ (P - P c ) + (1/ε ) dp/dt [4] (rate of increase of cell wall chamber volume) = (irreversible expansion rate) + (elastic expansion rate) Equation 4 is written in relative terms. The added term, (1/ε ) dp/dt, describes the elastic deformation of the cell wall during expansion. Importantly, this biophysical equation accommodates the condition when the turgor pressure is changing. Hence, Equation 4, and its corresponding constitutive equation for a Bingham-Maxwell viscoelastic model, can model the turgor pressure behavior and stress behavior exhibited by growing plant and fungal cell walls (Figures S1a and b). Another biophysical equation (in relative terms) can be derived to account for the case when water is lost to transpiration [12]. (dv/dt) w /V w = L pr (Δπ - P) - (dt/dt) w /V w [5] (rate of increase of water volume) = (net rate of water uptake) (transpiration rate) The volume of water lost via transpiration is T. Again, because (dv/dt) cwc /V cwc = (dv/dt) w /V w another equation can be obtained for the rate of change of turgor pressure [1,14]. dp/dt = ε { L pr (Δπ - P) - (dt/dt) w /V w - φ (P - P c )} [6] (rate of change of P) (net rate of water uptake) (transpiration rate) (irreversible expansion rate) Equation 6 explicitly shows how the turgor pressure changes when there is a change in the net rate of water uptake, and/or the transpiration rate, and/or the expansive growth rate. As an example of its utility, Eq. 6 explicitly shows that when the water uptake is eliminated, L pr (Δπ - P) = 0, and transpiration is suppressed, (dt/dt) w /V w = 0, the governing biophysical equation for dp/dt and its solution, P(t), are obtained for a pressure relaxation test [4,7]: 4

5 dp/dt = - ε φ (P - P c ) and P(t) = (P i P c ) exp (-ε φ t) + P c [7] The turgor pressure behavior, P(t), describes the exponential pressure decay from P i to P c, and describes the pressure decay obtained experimentally for both plant and fungal cells (Figure S1b). Equations 4, 5, and 6 have been termed the augmented growth equations [1,4,12-14,24]. Some algal, fungal, and plant cells grow predominately in length, i.e. exhibit elongation growth. Equations 4 and 5 can be adapted for elongation growth [1,15]: dl/dt = m (P P c ) + (L o /ε L ) dp/dt [8] (elongation rate) = (irreversible extension rate) + (elastic extension rate) (dl/dt) w A c = L p A m (Δπ P) (dt/dt) w [9] (rate of change in water volume) = (net rate of water uptake) (volumetric transpiration rate) Where L is the length of the cell, m is the longitudinal irreversible wall extensibility, L o is the initial length of the cell, and ε L is the longitudinal component of the volumetric elastic modulus (longitudinal volumetric elastic modulus). It is noted that during expansive growth the rate of increase of the volume of the cell contents is essentially the rate of increase in water volume. Equation 8 was shown to describe the elongation of the internode cells of C. corallina (Figure S1c) and the sporangiophores of P. blakesleeanus before, during, and after a step-down and a step-up in turgor pressure. Again, it is important to note that the Lockhart Equations (Eqs 2 and 3) cannot model the elongation behavior exhibited in Figure S1c. Equation 3 cannot model the step-down and step-up in turgor pressure (Figure S1c, blue curve) and Eq. 2 cannot model the observed recovered elastic extension (nearly instantaneous decrease in length) and elastic extension (nearly instantaneous increase in length) that is produced by the respective step-down and step-up in turgor pressure (Figure S1c, red and green curves). As it is shown in Figure S1c, the second term in the augmented growth equation (Eq. 8) is needed to model the elastic responses. Generally, the augmented growth equations have been used in short-scale time periods (minutes to hours) to analyze and interpret the experimental results of growing algal cells (e.g. [15,16], fungal cells (e.g. [10,12,17-19]), and plant organs (e.g. [7,8,20-23]). More recently, Lewicka [25] has extended their application to large-scale time periods (days) and the underlying viscoelastic models have been reviewed [26]. 5

6 Parameters A cw surface area of the cell wall A m surface area of the cell membrane e strain de/dt strain rate E longitudinal elastic modulus, or Young s modulus L length of the cell (a variable) L o initial length of the cell (a constant) dl/dt elongation rate L pr relative hydraulic conductance of the cell membrane; L = L p A m /V w L p hydraulic conductivity of the cell membrane m longitudinal irreversible wall extensibility P turgor pressure, the difference in pressure inside and outside the cell; P = P i P o P c critical turgor pressure, related to the critical stress, σ c, by geometry of the wall P i hydrostatic pressure inside the cell P o hydrostatic pressure outside the cell dp/dt rate of change of turgor pressure t time T volume of water lost by transpiration (dt/dt) w volumetric transpiration rate (dt/dt) w /V w relative volumetric transpiration rate V cc volume of the cell contents V cw volume of the cell wall V cwc volume of the cell wall chamber V w volume of the water in the cell (dv/dt) cwc /V cwc rate of increase of relative cell wall chamber volume (dv/dt) w /V w rate of increase of relative water volume Y yield threshold, related to the critical turgor pressure, P c ε volumetric elastic modulus ε L longitudinal component of the volumetric elastic modulus μ dynamic viscosity Δπ osmotic pressure difference inside and outside the cell; Δπ = π i - π o π i osmotic pressure inside the cell π o osmotic pressure outside the cell σ longitudinal stress σ c critical longitudinal stress dσ /dt longitudinal stress rate τ cw thickness of the cell wall φ irreversible wall extensibility Glossary Bingham fluid: A fluid that flows like a Newtonian fluid after a critical stress has been exceeded. 6

7 Constitutive equation: A mathematical equation relating the stress and the strain. Maxwell material: A material whose constitutive equation may be obtained from a Maxwell model which is viscous dashpot (damper) filled with a Newtonian fluid and an elastic spring connected in series. Newtonian fluid: Fluid whose stress versus rate of strain curve is linear and passes through the origin. The constant of proportionality is known as the dynamic viscosity. Young's modulus or elastic modulus: Measure of stiffness of an isotropic elastic material. The ratio of the stress and resulting strain, and can be determined experimentally from the slope of a stress-strain curve. References 1 Ortega, J.K.E. (2004) A quantitative biophysical perspective of expansive growth for cells with walls. In Recent Research Developments in Biophysics (Pandalai, S.G., Editor. eds.). pp , 2 Derbyshire, P., Findlay, K., McCann, M.C., and Roberts, K. (2007) Cell elongation in Arabidopsis hypocotyls involves dynamic changes in cell wall thickness. J. Exp. Bot. 58: Lockhart, J.A. (1965) An analysis of irreversible plant cell elongation. J. Theor. Biol. 8: Ortega, J.K.E. (1985) Augmented equation for cell wall expansion. Plant Physiol. 79: Pietruszka, M. (2009) General proof of the validity of a new tensor equation of plant growth. J. Theor. Biol. 256: Pietruszka, M. and Lewicka, S. (2007) Anisotropic plant growth due to phototropism. J. Math. Biol. 54: Cosgrove, D.J. (1985) Cell wall yield properties of growing tissue; evaluation by in vivo stress relaxation. Plant Physiol. 78: Cosgrove, D.J. (1987) Wall relaxation in growing stems: comparison of four species and assessment of measurement techniques. Planta 171: Boyer, J., Cavalieri, A., and Schulze, E. (1985) Control of the rate of cell enlargement: Excision, wall relaxation, and growth-induced water potentials. Planta 163:

8 10 Ortega, J.K.E., Zehr, E.G., and Keanini, R.G. (1989) In vivo creep and stress relaxation experiments to determine the wall extensibility and yield threshold for the sporangiophores of Phycomyces. Biophys. J. 56: Ortega, J.K.E. (1976) Phycomyces: The mechanical and structural dynamics of cell wall growth. PhD thesis. Vol. PhD, Boulder, University of Colorado 12 Ortega, J.K.E., Keanini, R.G., and Manica, K.J. (1988) Pressure probe technique to study transpiration in Phycomyces sporangiophores. Plant Physiol. 87: Ortega, J.K.E. (1990) Governing equations for plant cell growth. Physiologia Plantarum 79: Ortega, J.K.E. (1994) Plant and fungal cell growth: Governing equations for cell wall extension and water transport. Biomimetics 2: Proseus, T.E., Ortega, J.K.E., and Boyer, J.S. (1999) Separating growth from elastic deformation during cell enlargement. Plant Physiol. 119: Proseus, T.E., Zhu, G.L., and Boyer, J.S. (2000) Turgor, temperature and the growth of plant cells: using Chara corallina as a model system. J. Exp. Bot. 51: Ortega, J.K.E., Keanini, R.G., and Manica, K.J. (1988) Phycomyces: Turgor pressure behavior during the light and avoidance growth responses. Photobiology and Photochemistry 48: Ortega, J.K.E., Smith, M.E., Erazo, A.J., Espinosa, M.A., Bell, S.A., and Zehr, E.G. (1991) A comparison of cell-wall-yielding properties for two developmental stages of Phycomyces sporangiophores: Determination by in-vivo creep experiments. Planta 183: Ortega, J.K.E., Smith, M.E., and Espinosa, M.A. (1995) Cell wall extension behavior of Phycomyces sporangiophores during the pressure response. Biophys. J. 68: Serpe, M. and Matthews, M. (1992) Rapid changes in cell wall yielding of elongating Begonia leaves in response to changes in plant water status. Plant Physiol. 100: Serpe, M. and Matthews, M. (2000) Turgor and cell wall yielding in dicot leaf growth in response to changes in relative humidity. Australian Journal of Plant Physiology 27: Murphy, R. and Ortega, J.K.E. (1995) A new pressure probe method to determine the average volumetric elastic modulus of cells in plant tissue. Plant Physiol. 107: Murphy, R. and Ortega, J.K.E. (1996) A study of the stationary volumetric elastic modulus during dehydration and rehydration of stems of pea seedlings. Plant Physiol. 110:

9 24 Ortega, J.K.E. (1993) Pressure probe methods to measure transpiration in single cells. In Water deficits: Plant responses from cell to community (Smith, J. and Griffiths, H., Editors. eds.). pp , BIOS Scientific Publishers LTD, Oxford, Great Britain 25 Lewicka, S. (2006) General and analytic solutions of the Ortega equation. Plant Physiol. 142: Goriely, A., Robertson-Tessi, M., Tabor, M., and Vandiver, R. (2008) Elastic growth models. In Mathematical Modelling of Biosystems (Mondaini, R. and Pardalos, P., Editors. eds.). pp. 1-44, Springer-Verlag, Berlin Heidelberg Figure S1. Schematic graphs of stress relaxation tests. (a) An initial extension (longitudinal strain) and longitudinal load (stress) are imposed on growing sporangiophores of Phycomyces blakesleeanus with a tension-compression machine. At constant strain, stress is monitored as a function of time. Green curve: Sporangiophore has normal turgor pressure. Cell wall stresses are imposed by the longitudinal load and the turgor. Stress eventually decays to zero. Red curve: Immediately prior to load application turgor pressure had been decreased to zero. Stress is only imposed by the load and decays to a constant value. Redrawn from [11]. (b) Pressure relaxation in growing tissues after removal from water source and in an environment of saturating humidity to suppress transpiration. Red curve: The presence of auxin (in the case of pea stem tissue; redrawn from [7] Copyright of the original figure: American Society of Plant Biologists. or the administration of a light stimulus (sporangiophore; redrawn from [11]) accelerate the relaxation. (c) Elongation growth behavior of an internode of Chara corallina before and after a step-down and step-up in P (blue curve) produced with a pressure probe. At room temperature (red curve) cell length represents the sum of irreversible (growth) and elastic deformation, whereas at a colder temperature (green curve), the cell does not grow and the pressure induced deformations are purely elastic. Redrawn from [15] Copyright of the original figure: American Society of Plant Biologists. 9