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

Dimensionless number is central to stress relaxation and expansive growth of the cell wall www.nature.com/scientificreports OPEN Dimensionless number is central to stress relaxation and expansive growth of the cell wall Received: 6 January 2017 Joseph K. E. Ortega Accepted: 19 April 2017 Experiments demonstrate that both plastic and elastic deformation of the cell wall are necessary for Published: xx xx xxxx 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. Expansive growth is defined as a permanent increase in cell volume. Expansive growth of walled cells (e.g. algal, fungal and plant cells) depends on interrelated biochemical and physical processes. Active solutes inside their semi-permeable plasma membrane generate the osmotic potential needed to absorb water from its external envi- ronment and produce turgor pressure that stresses their walls. Wall stresses produce irreversible (plastic) and reversible (elastic) wall deformations in three orthogonal directions (volumetric wall deformation), generally stretching the wall in the two directions parallel to the wall surface (longitudinally and circumferentially) and contracting the wall in the direction perpendicular to the surface, making the wall thinner. New wall materials (polymers, proteins, etc.) are continually added to the inside surface to maintain a nearly constant wall thickness that varies between 0.1 μm and 1.0 μm, depending on the cell species. Researchers have long recognized that plastic deformation of the cell wall is necessary for expansive growth and morphogenesis. It is generally thought that regulation of expansive growth rate of the cell wall chamber that encloses the cell is achieved by controlling the magnitude of plastic deformation of the wall and the magnitude 1–4 5 1 of turgor pressure . Using the constitutive equation (stress-strain relationship) for a Bingham fluid , Lockhart derived a Growth Equation in terms of turgor pressure that describes these concepts, equation (1). Equation (1) describes the relative rate of change in volume of the cell wall chamber, v = (dV/dt)/V, as equal to the relative cw volumetric plastic deformation rate of the cell wall chamber, φ (P − P ), when the turgor pressure, P, exceeds the critical turgor pressure, P . vP =− φ() P (1) cw C e in Th clusive biophysical variables are φ (irreversible wall extensibility), P (turgor pressure), P (critical turgor pressure – a value that must be exceeded before plastic wall deformation occurs). It is noted that nearly all of the research conducted with this equation assumes that changes in volume occur predominately by changing the length of the cell, the diameter is assumed to remain relatively constant. The magnitude of φ has been shown to be 6, 7 biochemically controlled by the cell . Equation (1) explicitly indicates that expansive growth does not depend on Bioengineering Laboratory, Department of Mechanical Engineering, University of Colorado Denver, 1200 Larimer Street, NC-2024-K, P.O. Box 173364, Denver, CO, 80217-3364, USA. Correspondence and requests for materials should be addressed to J.K.E.O. (email: Joseph.Ortega@ucdenver.edu) Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 1 www.nature.com/scientificreports/ elastic wall deformations, but only depends on plastic wall deformations and the inclusive biophysical variables 1, 3, 8 (φ, P, and P ) . Equation (1) is simple, yet demonstrates broad utility in modeling and understanding expansive growth behav- ior of walled cells and is frequently found in many textbooks in plant biology and plant physiology. However, subsequent research demonstrates that equation (1) is not able to describe the results of experiments conducted specifically to reveal the mechanical behavior and constitutive relationship of growing cell walls. Specifically, equation (1) and its underlying constitutive equation for a Bingham uid fl cannot describe the results of in vivo 4, 9, 10 11, stress relaxation experiments and in vivo creep experiments that produce large changes in turgor pressure 12 9 because it does not describe reversible (elastic) wall deformations. Ortega used a constitutive equation derived in linear viscoelastic theory (Maxwell viscoelastic model) to accurately describe the results of stress relaxation experiments conducted on growing sporangiophores of the fungus, Phycomyces blakesleeanus. Subsequently, Ortega modified the Maxwell constitutive equation by replacing the Newtonian fluid inside the dashpot with a 6, 14 Bingham fluid (and later called it a Maxwell-Bingham viscoelastic model ). The Maxwell-Bingham constitutive equation was used as the foundation for a biophysical wall extension equation in terms of turgor pressure, the Augmented Growth Equation , equation (2).  1d  P   vP =− φ() P + cw C    (2) ε dt e addi Th tional variables are ε (volumetric elastic modulus of the wall), and t (time). The Augmented Growth Equation describes the relative rate of change in volume of the cell wall chamber, v = (dV/dt)/V, as the sum of cw the relative volumetric irreversible (plastic) deformation rate, φ (P − P ), and the relative volumetric reversible (elastic) deformation rate, (1/ε) dP/dt, of the cell wall chamber. Again, nearly all of the research conducted with this equation assumes that changes in volume occur predominately by changing the length of the cell, while the diameter is assumed to remain relatively constant. It is noted that equation (1) is recovered from equation (2) when the turgor pressure is constant. Importantly, the Augmented Growth Equation accurately describes the experimental results of in vivo stress relaxation experiments conducted on pea stems and fungal sporangio- 4 11 phores , and in vivo creep experiments conducted on algal internode cells that involve large step changes in tur- gor pressure. The Augmented Growth Equation also demonstrates broad utility in modeling and understanding 6, 14–16 10, 17 17, 18 expansive growth behavior of walled cells , plant tissue and whole plants . 10, 19, 20 Using the results of in vivo stress relaxation experiments, Cosgrove describes a mechanism for expansive growth of the cell wall that is more complicated than the simple wall extension mechanism mod- eled by Equation 1. Cosgrove states: “As a cell absorbs water, the wall extends passively, and polymers in the load-bearing network(s) are distended. In nongrowing cells, wall stress increases as the polymers are stretched like springs. Elastic energy is stored in the strained bonds of these polymers (and also in the increased order of the polymers), and this elastic energy does work on the cell protoplast by compressing it, thereby increasing its turgor pressure and water potential. When the cell water potential increases to the point where it matches that of the external water, net water uptake ceases. In growing cells, this equilibrium is never quite reached because the wall “relaxes,” which means that the load-bearing network breaks, slips, or is cut, and the distended polymers assume a more relaxed condition. Elastic energy of the wall is lost as heat, and a turgor reduction inevitably accompanies the reduced wall stress. Note, however, that this relaxation by itself does not entail a physical expansion of the wall or a change in cell volume. Turgor decreases because the wall simply stops compressing the protoplast. Expansion follows secondarily, as the cell absorbs water in response to the reduced water potential created by the reduction of turgor pressure”. Interestingly, even though this “stress relaxation concept for expansive growth” has been available for a couple of decades, many of the conceptual models that are used to guide experimental investigations and mathematically models of expansive growth, still employ some form of equation (1), which cannot model stress relaxation. A few possible reasons for the continual employment of this concept and equation (1) are (i) it is simple to comprehend and to use, (ii) it is recovered from equation (2) when P is constant (thus elasticity is removed from the model when P is constant), and (iii) it is not apparent how to mathematically relate the stress-relaxation concept to expansive growth. In this paper, dimensional analysis is used to obtain insight into the stress relaxation process and to explic- itly demonstrate a relationship between expansive growth rate and wall elasticity as well as wall plasticity. Dimensional analysis is conducted on the Augmented Growth Equation, equation (2), describing plastic and elastic wall deformation rates. Three Π parameters are identified in the analysis, but only one is shown to control the turgor pressure decay rate that occurs during wall stress relaxation, the Π parameter. Dimensional analysis pe provides a physical interpretation for the Π parameter; it is the ratio of the relative volumetric plastic deforma- pe tion rate of the wall and relative volumetric elastic deformation rate of the wall. This analytical result demonstrates that the ratio of plastic and elastic deformation rates are used to regulate the rate of stress relaxation that occurs during expansive growth of the cell wall chamber. It is shown that the relative magnitude of plastic and elastic deformation rates are important in controlling wall stress relaxation and expansive growth. Importantly, the Π pe parameter provides a mathematical relationship between the relative rate of change in volume of the cell wall chamber, v , and the mechanical properties of the wall, i.e. the irreversible wall extensibility, φ, and the volumetric elastic modulus, ε. It is noted that the turgor pressure, P, and critical turgor pressure, P , do not appear in the Π C pe parameter. Values are calculated for the Π parameter from previously reported experimental results obtained pe from cells in plants, fungi, and algae. It is found that the Π values are large and very different for these different pe cell species. In some cases, data was available to calculate Π values for the same cells when φ and ε changed. pe e r Th esults indicate that Π can be used to calculate the magnitude of v when φ and ε change, because the cal- pe s culated values for v compare very well with measured values for these cells. In addition, it is shown that the Π s pe Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 2 www.nature.com/scientificreports/ parameter can be used to organize data related to cell wall deformation for all walled cells so that the ratio of plastic and elastic deformation rates of cell walls for different cells from plants, algae, and fungi can be compared. Theory, Analyses and Results Dimensional Analysis. The overall process of dimensional analysis consists of obtaining dimensionless groups of variables (Π parameters) involved in the process of interest and exploring the relationship between the Π parameters experimentally and/or analytically. A straightforward method to obtain the Π parameters 21, 22 can be employed when you have governing equations . In this method the variables within the governing equations are made dimensionless with appropriate constant reference quantities and the governing equations are manipulated to produce dimensionless equations and with dimensionless coefficients, or Π parameters. e Th Augmented Growth Equations represent governing equations for expansive growth and can be used for dimen- sional analyses . e exp Th ansive growth rate of a walled cell is determined by the net water uptake rate and cell wall deformation rate. During steady or quasi-steady growth, expansive growth rate is limited and controlled by the smaller rate. e m Th agnitudes of net water uptake rate and wall deformation rate can be evaluated using the Π parameters iden- tified in the dimensionless Augmented Growth Equations . Scale analysis reveals that the net rate of water uptake is approximately eight times greater than the wall deformation rate for cells in growing pea stems (Pisum satinis L.), approximately 100 times greater for growing stage I sporangiophores (P. blakesleeanus), and approximately 12 times greater for growing stage IV sporangiophores (P. blakesleeanus); see ‘scale analysis’ in Calculations, Estimates, and Methods for details. These findings indicate that the smaller ‘wall deformation rate’ limits and controls the magnitude of expansive growth rate for these cells. Therefore, the analysis in this paper focuses on equation (2) that describes the wall deformation rate. Dimensionless Augmented Growth Equation. Here, dimensional analysis will focus on the Augmented Growth Equation that describes the wall deformation rate, equation (2). The analysis is conducted for walled cells growing normally at a quasi-steady rate . Typically P is constant during normal growth and stress relaxa- 6, 10, 12 tion , so P will be treated as a constant in the analysis. The variables (v , t, and P) in equation (2) are made C cw dimensionless (*) with the following constant reference quantities , P (critical turgor pressure) and v (steady or C s average relative volumetric growth rate, i.e. v = v averaged over a time interval = constant). s cw   ∗∗  P =  and PP = P  P     tt == vt and   v     ∗∗ cw  v =  and vv = v cw  cw cs w  v    Substituting the respective expressions that include the dimensionless variable and reference quantity (vari- ables and their respective expressions in the right column above) for each of the variables into equation (2), we get equation (3):  vP  dP ⁎⁎ sC vv =− φPP (1) + scwC  ⁎     ε d( t 3) e va Th riables v *, P*, and t* are dimensionless. Therefore the dimensions, physical characteristics, and rel- cw ative magnitude of each term are represented by their respective dimensional coefficient. Thus, v represents a steady or average magnitude of relative rate of change in volume of the cell wall chamber (constant), φ P repre- sents the magnitude of the relative volumetric irreversible (plastic) deformation rate and (v P /ε) represents the s C magnitude of the relative volumetric reversible (elastic) deformation rate. Dividing equation (3) by the constant reference quantity, v , we get equation (4), the dimensionless Augmented Growth Equation : dP ⁎⁎ vP =Π (1 −+ ) Π cw pv ev (4) dt The Π and Π parameters are dimensionless coefficients defined as follows : pv ev     φP relative volumetric plasticdeformationrateofthe wall C     Π=  =    pv      v  relative volumetric growth rate    and    P  relative volumetric elasticdeformationrateofthe wall C    Π= = . ev         ε relative volumetric growth rate    The Π parameters represents ratios of the dimensional coefficients shown in equation (3), where the first sub- script of the Π parameter refers to the numerator and the second subscript refers to the denominator. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 3 www.nature.com/scientificreports/ Wall Stress Relaxation and Pressure Relaxation. Wall stress and pressure relaxation are the founda- 10, 19, 20 tion of the “stress relaxation concept of expansive growth” . Insight can be obtained by solving the dimen- sionless governing equation for stress relaxation. Typically, an in vivo stress relaxation experiment is conduct by isolating a growing walled cell from its water supply and eliminating the loss of water via transpiration. Then 4, 10 the decay in turgor pressure is measured as a function of time, usually with a pressure probe . It is essential to prevent water uptake and water loss from the cell (water volume and volume of the cell wall chamber must remain constant), so the decay in wall stress and concurrent decay in turgor pressure is the result of relaxation alone. u Th s, experimentally and analytically, the condition that must be satisfied for a stress relaxation experiment is that 10, 13 the volume of the cell remains constant during the experiment, or v = 0 . Imposing the condition, v * = 0, cw cw the dimensionless governing equation for a stress relaxation experiment is obtained from equation (4): dP 0( =Π P −+ 1) Π. pv ev dt Rearranging, equation (5) is obtained where the dimensionless parameter, Π , is equal to the ratio, Π /Π : pe pv ev dP =−Π− (1 P ) pe (5) dt Using the dimensionless initial condition, P* = P * = P /P , equation (5) is integrated and the dimensionless i i C solution is obtained: ⁎⁎ ⁎ PP =− (1)exp −Π t +. 1 i () pe (6) Where     εφ relative volumetric plasticdeformationrateofthe wall    Π=  = .   pe      v relative volumetric elasticdeformationrateofthe wall      (7) Equation (5) demonstrates that the rate of decay of the dimensionless pressure, P*, during stress and pressure relaxation is directly related to the magnitude of Π . Equation (6) shows that the decay of the dimensionless pres- pe −1 sure, P*, is exponential and the dimensionless time constant for the exponential decay is, t * = (Π ) . Equation c pe (7) reveals that Π is the ratio of relative volumetric plastic and elastic deformation rates of the wall. pe Magnitudes of Π from in vivo stress relaxation and in vivo creep experiments. e m Th agnitude pe of Π can be calculated from results obtained from in vivo stress relaxation and in vivo creep experiments con- pe ducted with the pressure probe. In an in vivo stress relaxation experiment, the growing cell is removed from its 10, 13 10 water supply and prevented from transpiring . Usually this requires the tested cells to be incised from tissue or removed from mycelium (plucked) . Then the exponential decay in P is measured with a pressure probe and −1 the time constant for the exponential decay is, t = (εφ) . The determination of Π , employing the in vivo stress c pe relaxation method has the theoretical advantage that the product, εφ, can be calculated directly from the halftime, T , of the exponential decay of the turgor pressure; εφ = ln 2/T and Π = (ln 2/v T ). 1/2 1/2 pe s 1/2 Sometimes it is not possible to conduct in vivo stress relaxation experiments on walled cells. Then the magni- tude of Π can be calculated using equation (7) and the values of ε, φ, and v obtained from in vivo creep exper- pe s 4, 6, 10, 11 iments . In an in vivo creep experiment the P in a growing cell is increased (stepped-up) with a pressure probe and the increase in elongation growth rate is measured. The magnitude of φ can be determined from the increase in elongation growth rate . The magnitude of ε is determined separately from experiments during which a pulse-up in P is produced with the pressure probe . Theoretically, in vivo creep and in vivo stress relaxation experiments should produce the same values for φ and (εφ). Figure 1 compares Π values determined from in pe vivo creep and in vivo stress relaxation experiments conducted on incised pea stem sections (P. satinis L.) , incised 11, 12 6 algal internodes cells (Chara corallina) , intact stage I sporangiophores (P. blakesleeanus) , plucked stage IV 4 6 sporangiophores (P. blakesleeanus) and intact stage IV sporangiophores (P. blakesleeanus) ; see Calculations and Estimates for the calculations of Π values. It should be noted that the vertical scale for Π is logarithmic. pe pe Figure 1 demonstrates that the Π values for pea cells of P. satinis L., internode algal cells of C. corallina, and pe fungal sporangiophores of P. blakesleeanus are large and very different, by an order of magnitude or more. The different magnitudes of the Π values presented in Fig. 1 provide a comparison of the ratio of plastic and elastic pe deformation rates of growing walls from plant, algal, and fungal cells. Expansive growth rates calculated from a constant Π . The steady or average relative volumetric pe growth rate, v , can be calculated from Π when ε and φ are measured. Rearranging equation (7) we get: s pe εφ v = pe (8) Importantly, equation (8) provides an explicit mathematically relationship between expansive growth rate, v , and stress relaxation of the cell wall, as characterized by Π . Also, equation (8) demonstrates that v can be s pe s determined without measuring P and P , if Π is known. C pe It is noted that the magnitude of Π for pea cells of P. satinis L. calculated from in vivo stress relaxation pe experiments is the same as those calculated from in vivo creep experiments, even though the measured relative volumetric growth rates, v , of the cells during the in vivo creep experiments are much larger than for the in sM Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 4 www.nature.com/scientificreports/ Figure 1. A comparison of the Π values calculated for growing cells of pea stems (Pea) , growing internode pe 11, 12 4, 6 cells of C. coralline (Chara) , and growing sporangiophores of P. blakesleeanus (Stage I and Stage IV) . Note that the vertical scale for Π is logarithmic. Also compared are the values obtained from in vivo stress relaxation pe (SR) and in vivo creep (CR) experiments for pea cells and stage IV sporangiophores. Only the in vivo creep experiments conducted on Stage I (CR) and Stage IV (CR) sporangiophores used intact cells (growing on the mycelium). The confidence intervals represent the maximum and minimum values of Π that can be calculated pe 4, 6, 10 from the statistical data (standard deviation and standard errors) presented in the respective papers (see Calculations, Estimates, and Methods for details). vivo stress relaxation experiments (~3 times larger). Based on maximum and minimum values of Π that can be pe calculated from the statistical data, the magnitude of Π for intact stage IV sporangiophores of P. blakesleeanus pe calculated from in vivo creep experiments is only slightly larger than the value calculated from in vivo stress relax- ation experiments conducted on plucked stage IV sporangiophore, even though the measured relative volumetric growth rates, v , are much larger (~6 times larger). Again, based on maximum and minimum values of Π that sM pe can be calculated from the statistical data, the magnitudes of Π for stage IV and stage I sporangiophores are pe the same, even though the average value for v of stage IV sporangiophores is more than four times larger than sM those of stage I sporangiophores. es Th e results suggests that the ratios of plastic and elastic deformation rates (i.e. the magnitude of Π ) of cell walls from a specific cell species are the same. It appears that the ratio of plastic and pe elastic deformation rates of cell walls from pea stems of P. satinis L. is 32 (i.e. Π = 32). Similarly, it appears that pe the ratio of plastic and elastic deformation rates of cell walls from sporangiophores of P. blakesleeanus is approx- imately 1524 (i.e. Π = 1524, the average value of intact stage I and stage IV from in vivo creep experiments). If pe this suggestion is correct, then a constant value for the Π parameter can be used to calculate the steady relative pe volumetric growth rate, v , for cells from pea stems of P. satinis L. and for sporangiophores of P. blakesleeanus when φ and/or ε change because of changes growth conditions (incised and growing in water and just cut from the plant), addition of growth hormone IAA, changes in development (stage I and stage IV), and intact versus plucked sporangiophores (see Figs 2 and 3). Specifically, v can be calculated for growing cells from pea stems by employing equation (9). εφ vP (L .. satinis ) = (9) Equation (9) is employed to calculate v values using values for ε and φ, and compare them to measured values for pea stems of P. satinis L. (Fig. 2). Equation (10) is employed to calculate v values using values for ε and φ, and compare them to measured values for stage I and stage IV sporangiophores of P. blakesleeanus (Fig. 3). εφ vP () .= blakesleeanus (10) Discussion Dimensional analysis reveals that the rate of wall stress relaxation is directly related to the magnitude of a sin- gle dimensionless parameter, Π ; see equation (5). Equation (6) demonstrates that the dimensionless pressure pe −1 (and stress) relaxation is exponential and the dimensionless time constant is, t * = (Π ) . Dimensional analysis c pe provides a physical interpretation for the Π parameter, it is the ratio of relative volumetric plastic and elastic pe deformation rates of the cell wall; see equation (7). The Π parameter explicitly reveals a relationship between the pe steady or average relative volumetric growth rate of the cell, v , and the wall stress relaxation as characterized by Π ; see equation (7). In fact, v can be calculated using equation (8) when Π , φ, and ε are known; see equations pe s pe Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 5 www.nature.com/scientificreports/ Figure 2. A comparison of v values calculated (dark grey) and measured (light grey) for cells in pea stems (P. satinis L.). Equation (9) is employed to calculate v values (dark grey) using a constant value for Π (Π = 32) s pe pe and values for ε and φ taken from Cosgrove ; see Calculations, Estimates, and Methods for details. The confidence intervals for the measured values (light grey) are standard errors taken from Cosgrove . The confidence intervals for the calculated values (dark grey) represents the maximum and minimum values that can be calculated from maximum and minimum values obtained from standard errors of φ . Figure 3. A comparison of v values calculated (dark grey) and measured (light grey) for stage I and stage IV sporangiophores of P. blakesleeanus. Equation (10) is employed to calculate v values (dark grey) using a 4, 6 constant value for Π (Π = 1524) and using values for ε and φ taken from Ortega et al. ; see Calculations, pe pe Estimates, and Methods for details. The confidence intervals for the measured values (light grey) are standard errors taken from Ortega et al. . The confidence intervals for the calculated values (dark grey) represents the maximum and minimum values that can be calculated from maximum and minimum values obtained from standard errors of ε and φ . The measured v value for the plucked (p) stage IV sporangiophore (light grey) is for a single experiment. (9) and (10). Equation (8) represents a method to calculate v when P or P are not known. Good agreement is s C obtained for calculated values of v compared to measured values, for cells from pea stems and sporangiophores (stage I and stage IV) when constant Π values are used; see Figs 2 and 3. pe Figure 1 compares magnitudes of Π for cells from growing pea stems of P. satinis L. (Pea), growing internode pe cells of C. coralline (Chara), and growing sporangiophores of P. blakesleeanus (Stage I and Stage IV). The results indicate that the magnitude of Π is very large (Π ≫ 1) for all three species of walled cells. The large values pe pe for Π indicate that the relative volumetric plastic wall deformation rate is very much larger than the relative pe volumetric elastic wall deformation rate during steady and quasi-steady expansive growth. From the perspective of a constitutive relationship (stress-strain relationship), the strain rate of the wall is proportional to the stress for plastic (irreversible) deformation, but proportional to the stress rate for elastic deformation. Therefore the Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 6 www.nature.com/scientificreports/ constitutive relationship for plastic and elastic deformation are different. From an energy perspective, wall stress relaxation represents the dissipation of mechanical energy stored by elastic wall deformation. Plastic and elastic wall deformation rates represent the rates of two very different processes. An increase in elastic wall deformation rate represents an increase in the rate that mechanical energy is stored in the wall. This is a reversible process, and the stored mechanical energy can be recovered. An increase in plastic wall deformation rate represents an increase in rate of dissipation of stored mechanical energy in the wall. This is an irreversible process, and the lost mechanical energy cannot be recovered. The large values of Π for the walled cells indicates that plastic wall pe deformation rate dominates wall deformation during steady and quasi-steady expansive growth. From the per- spectives of constitutive relationships and energy, this finding draws into question the use of elastic models and elastic wall deformations to study expansive growth and morphogenesis. Figure 1 also compares values for Π determined from the results of in vivo creep and in vivo stress relax- pe ation experiments. The results indicate that the Π values are essentially identical for cells from pea stems and pe nearly statistically identical for stage IV sporangiophores. These experimental results are consistent with the- oretical prediction that the Π values obtained from in vivo creep and in vivo stress relaxation experiments pe should be the same. However, the results are also interesting because the growth rate for test specimens dur- ing in vivo stress relaxation experiments are three (for pea) to six (for stage IV) times smaller than during in vivo creep experiments. One wonders if the magnitude of Π is invariant and is determined by a chemical wall pe loosening-hardening mechanism (chemorheology) that is fundamental to each species of cell, so that the same ratio of plastic and elastic wall deformation rates are produced for each cell species. If this suggestion were correct, it would make two predictions. First, it would predict that the chemorheology for cells in pea stems (P. satinis L.), algal internode cells (C. corallina) and sporangiophores (P. blakesleeanus) are die ff rent from each other because the magnitude of their respective values of Π are very different from each other, by an order of magnitude or pe more. Second, it would predict that a single value for Π represents a cell species and can be used for that cell pe species to calculate v when φ and/or ε change because of alterations in growth conditions, addition of growth hormone, and development. er Th e is evidence to support the first prediction. In a variety of higher plant cells within tissue, a pH-dependent 7, 25 protein (expansin) is found to loosen the wall by disrupting the hydrogen bonds between microfibrils . In large internode algal cells of C. corallina, experimental evidence indicates that making and breaking calcium 26, 27 bridges between pectin polymers loosens and hardens the wall . It is not known how the sporangiophore of 28–30 P. blakesleeanus loosens and hardens its wall , but it is predicted that its chemorheology will be different from those used by either cells in pea stems of P. satinis L. or internode algal cells of C. coralline, because the magnitude of Π for the sporangiophores is orders of magnitude larger than those of either pea or Chara. pe e s Th econd prediction was evaluated here by using a single value for Π to calculate v for cells in pea stems pe s (Π = 32) and sporangiophores (Π = 1524) when φ and/or ε change. Figure 2 compares calculated and meas- pe pe ured v values for cells in pea stems (P. satinis L.) when φ changes because of changing growth conditions (‘incised and growing in water’ and ‘just cut from the plant’), and the addition of the growth hormone IAA. Figure 3 com- pares calculated and measured v values for the sporangiophores of P. blakesleeanus when φ and ε change because of development (stage I and stage IV) and when it is removed from the mycelium (plucked) and its base is put in pure water. The respective calculated and measured v values for cells in pea stems and sporangiophores compare very well and appear to be the statistically the same. It is concluded that the respective calculated and measured values of v for pea stems, stage IV, and stage I sporangiophores are the same. Therefore, it is also concluded that the results support the second prediction. e Th results presented in Fig.  1 indicate that the magnitude of Π for the sporangiophores of P. blakesleeanus is pe orders of magnitude larger, and significantly larger, than the magnitude of Π for cells from pea stems. This find- pe ing indicates that wall deformation behavior can be significantly different for walled cells from different species. In the future, the Π parameter can be determined for different species of walled cells. Then the Π parameters pe pe can be used to provide insight into similarities and differences in the ratio of plastic and elastic wall deformation rates during expansive growth of different cell species. Future research may also provide more support for the idea that the magnitude of Π is related to the chemorheology that regulates wall deformation when the wall is pe stressed. In future research, it would be interesting and important to determine Π for ‘growth mutant’ cells pe and ‘wall mutant’ cells , and compare their magnitudes to Π values obtained from wild type cells of the same pe species. This could reveal whether the mutation produced a change in the ratio of plastic and elastic wall defor - mation rates. e dim Th ensionless parameters obtained and studied here are related to expansive growth rate and growth rate regulation. However, other studies concerning cell size and cell morphology have introduced different dimen- sionless parameters that include the elastic modulus of the wall, E, and the turgor pressure, P. Boudaoud intro- duced the dimensionless parameter, E/P, to study the size and shape of isolated walled cells. The cells are modeled as elastic shells where growth is driven by turgor pressure. Mechanical equilibrium is used to obtain scaling laws for cell size (cell radius) that employ the dimensionless parameter, E/P. The magnitude of the radius of the sporangiophores of P. blakesleeanus can generally be predicted with the published scaling equation. Goriely et al. introduced the dimensionless parameter, P w/(h E), to characterize the overall wall deformation (P is a eff eff measure of the normal stresses acting on the wall, w is width of the apical tip, and h is the thickness of the shell or wall). This research used nonlinear elastic models to get insight into how the shape of apical hyphae are produced. e g Th eneral shape of apical hyphae produced with the dimensionless parameter is similar to those seen in stage I sporangiophores of P. blakesleeanus. Ortega obtained additional dimensionless Π parameters describing water uptake rates and transpiration rate (also see the ‘scale analysis’ in the Calculations, Estimates, and Methods) that may be used in future research to study expansive growth rate, water uptake rate, and transpiration rate of water-stressed cells in crop plants and may provide insight into how plants adapt to drought conditions. In Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 7 www.nature.com/scientificreports/ Figure 4. e r Th esults of a modified constant-tension extension experiment is presented. At ten minutes a weight (1.24 g) was used to apply constant tension to a five millimeter long section of a frozen-thawed wall that includes the growth zone , and an initial extension of approximately 810 µm is observed (see Calculations, Estimates, and Methods – ‘Constant-tension extension experiment’ for more details). At 15 min (marked by the downward pointing arrow), 50 mM potassium-acetate buffer, pH = 4.5, replaces the bathing water (pH = 7.0) in the test apparatus . Immediately, creep extension is observed until 22 min on the time scale. At 25 min (marked by the upward pointing arrow) the tension is released and the extension decreases (recovered elastic extension) until 28 min on the time scale. general, different dimensionless parameters can be employed to obtain insight and to investigate different aspects of size, shape, and growth of walled cells. In the physical sciences, Π parameters have been used to establish similarity between uid flo fl ws, heat flows, 21, 22, 35 and other transport processes by ensuring that the magnitudes of relevant Π parameters are the identical . In the future, we plan to investigate the use of wall deformation similarity (by matching the magnitudes of the Π pe parameter) to guide and valid local mathematical models that are making strides towards describing spatial and 36–42 temporal wall deformation of the cell wall during expansive growth . Also, future research will investigate 30, 32 the use of wall deformation similarity to augment experimental assays that are used to investigate cell wall loosen-hardening mechanisms and chemorheology of cell walls. For example, constant-tension extension exper- iments were conducted on frozen and thawed walls of stage IV sporangiophore and showed that lowering the pH in the wall produces creep extension . The creep extension is transient (~10 min in duration) and the creep −1 extension rate is large for the first couple of minutes (~110 µ m min ), then decreases for the next eight minutes −1 30 (~8 µ m min ) before it stops . The pH mediated creep extension rates cover the range of the elongation rates −1 exhibited by the stage IV sporangiophore (20–60 µ m min ). This result might suggest that lowering the pH in the living and growing cell wall is the mechanism employed by the sporangiophore to produce expansive growth of the wall and possibly regulate its growth rate. However, if the constant-tension extension protocol is modified so that the results can be used to calculate the magnitude of Π , then the magnitude of Π for pH mediated creep pe pe extension can be compared to that obtained for natural growing stage IV sporangiophores. Figure 4 shows the results of a modified constant-tension extension experiment. e Th magnitude of Π is calculated from the experimental results (see Calculation, Estimates and Methods – pe ‘Constant-tension extension experiment’ for details). e Th calculated magnitude for Π is 130 (Π ≈ 130), which pe pe is an order of magnitude smaller than the magnitude obtained for normal growing stage IV sporangiophores (Π ≈ 1500). This result suggests that decreasing the pH alone cannot account for the wall deformation rate pe that occurs in natural growing stage IV sporangiophores because it does not produce the same ratio of relative volumetric plastic and elastic deformation rate that occurs during normal growing stage IV sporangiophores. It is envisioned that the Π parameter may be used in a similar way to assess the results of other experimental assays pe 19, 32 developed for higher plant cells . Calculations, Estimates, and Methods Scale analysis for ‘net water uptake rate’ and ‘wall deformation rate’. e f Th ollowing two dimen - sionless Augmented Growth Equations describe the net water uptake rate and the wall deformation rate . ⁎⁎ ⁎⁎ vP =Π () Δ− π −Π vN (o et fwater uptake rate) wwvTvT Net water uptake rate = water uptake – water loss due to transpiration (in relative terms) dP ⁎⁎ vP =Π (1 −+ ) Π (Walldeformation rate) cw pv ev dt Wall deformation rate = plastic deformation + elastic deformation (in relative terms) e dim Th ensionless Π parameters are defined as follows : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 8 www.nature.com/scientificreports/     LP relative volumetric wateruptakerate  C   Π=  =  wv     v relative volumetric growth rate          v relative volumetric transpirationrate sT     Π=  =    Tv        v relative volumetric growth rate         φP relative volumetric plasticdeformation rate of thewall C     Π=  =   pv       v   relative volumetric growth rate     P  relative volumetric elasticdeformation rate of thewall   Π= =  ev          ε relative volumetric growth rate   e Th magnitude of each term in the dimensionless Augmented Growth Equations is determined by the magni- tude of the dimensionless Π parameter associated with the term. So the magnitude of the ratio of net water uptake rate and wall deformation rate can be determine with the following Π parameter: wd   Π− Π  wv Tv Π= wd  Π+ Π  pv ev  Estimates of Π for steady or quasi-steady growing cells in pea stem (P. satinis L.). e va Th lues wd for the biophysical variables are obtained from Cosgrove . 20 .   (0.3) MPa ()  hMPa  LP v  C sT   − 0 0.0751         v v  s s h   Π= (pea) = ≈. 78 wd    02 . 5 φP P  C C   (0.3) MPa   +  () (0.3) MPa   hMPa    v ε   +   0.0751  (9.5) MPa     Estimates of Π for steady or quasi-steady growing sporangiophores (P. blakesleeanus). The wd 6 43 44 values for the biophysical variables are obtained from Ortega , Cosgrove et al. , and Ortega et al. 83.0   (0.4) MPa () 01 . 2   hMPa LP v    C sT  − 0.021  −   h   v v     s s h   Π= (stage I) = ≈ 102   wd   φP P 08 . 1 C C   (0.4) MPa   +  () (0.4) MPa   hMPa    v ε   +  0.021   (69) MPa     23.5   (0.26MPa) () 01 . 2   hMPa LP v    C sT  − 0.068  −   h   v v     s s h   Π= (stage IV)  = ≈ 11.7 wd   φP P 20 .   C C   (0.26MPa) + ()   (0.26MPa)   hMPa    v ε   +  0.068   (61) MPa     Estimates for Π from in vivo stress relaxation experiments. e m Th agnitude of Π =  ln 2/(v T ) pe pe s 1/2 is estimated for pea stems and fungal sporangiophores. The following data was obtained from incised pea stems sections (P. satinis L.) that include the growth zone (for Fig. 1). The stems sections were incised so that they can be removed from their water supply and transpiration eliminated: Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 9 www.nature.com/scientificreports/ −1 vh = 0.± 0249 0.0022 () SE −− 11 vh (largest)( =. 0 0249 +. 0 0022) =. 0 0271 h −− 11 vh (smallest)( =. 0 0249 −. 0 0022) =. 0 0227 h −1 T5 =± 22 .. 37 () SE min0 =± .. 87 006S () Eh 12 / −1 εφ(largest)2 == ln /2 Tlnh /0.= 81 08 . 6h 1/2 −1 εφ(smallest)2 == ln /2 Tlnh /0.= 93 07 . 5h 1/2 −1     εφ(largest) (0.8557) h    Π (largest) =  =  ≅. 37 7   pe   −1    v (smallest)     (0.0227 h )   −1     εφ(smallest) (0.7453) h     Π (smallest) =  =  ≅. 27 5 pe   −1      v (largest)  (0.0271 h )     Π. () pea3 =± 26 51 . pe The following data was obtained for plucked stage IV sporangiophores of P. blakesleeanus . The sporangi- ophores were plucked (carefully removed from the mycelium) so that they could be put in an environmental chamber where the water is removed and the air is maintained at 100% RH to eliminate transpiration. lm == 40 mm 41 × 0 µ dl −− 11 =. 55 µµ mmin = 330 mh dt   dl −1   v = =. 0 0083 h s   lt d  T4 == minh 0.067 1/2   ln2  ln2   −1   εφ =  = =. 10 3 h         T 0.067 h   1/2       εφ ln2 (0.693)       Π =  =  =  ≅ 1246    pe   −1         v  v T  (0.. 0083 hh )(0 067 )       ss 1/2 Estimates for Π from in vivo creep experiments. The magnitude of Π = (εφ/v ) is estimated for pe pe s pea stems, internode algal cells, and fungal sporangiophores (for Fig. 1). The following values were obtained for incised pea stems of Pisum satinis L. :   dl −1   v = = 0.± 0751 0.0043 () SE h s   lt d  −− 11 vh (largest)( =. 0 0751 +. 0 0043) =. 0 0794 h −− 11 vh (smallest)( =. 0 0751 −. 0 0043) =. 0 0667 h −− 11 φ =± 02 .. 50 017 hMPa −− 11 φ(largest)0 =.25 +. 0 017hMPa =. 0 267 −− 11 φ(smallest)0 =.25 −. 0 017 hMPa =. 0 233 ε =. 95 MPa −−     εφ (9.. 5 MPah )(0 25 MPa )    Π. (mean) =  =  ≈ 31 6   pe   −1    v     (0.0751 h )   −− 11     εφ(largest) (9.. 5 MPah )(0 267 MPa )    Π (largest) =  = ≈ 38   pe  −1     v (smallest)  (0.0667 h )     −− 11     εφ(smallest) (9.. 5 MPah )(0 233 MPa )     Π (smallest) =  =  ≈. 27 9   pe  −       v (largest)     (0.0794 h )  Π () pea3 ≈± 26 pe 11, 12 e f Th ollowing values were used for incised internode algal cells of C. corallina : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 10 www.nature.com/scientificreports/ lm == 13 mm 13 .× 10 µ dl −− 11 =. 0 014 µµ ms =. 50 4 mh dt   dl −− 61 −1   v = =. 11 ×= 10 sh 0.0039 s     lt d −1     dl 50.4 µmh −− 11     m =  =  = 285 µmh MPa         − () PP dt  (0.− 527 0.35) MPa     −− 11    m  285 µmh MPa −− 11   φ =  =  =. 0 022 hMPa  4       l  13 .× 10 µm   ε = 100 MPa −− 11     (100 MPah )(0.022 MPa ) εφ    Π =  =  ≅ 564   pe   −1      v     (0.0039 h )  4, 6 e f Th ollowing values were obtained for intact stage IV sporangiophores of Phycomyces blakesleeanus : lm == 30 mm 30 .× 10 µ dl −1 =± 34 31 . () SE μmmin () n2 = 0 dt    dl  34.+ 13.1m µ /min −− 11   v (largest) = =  =. 0 0012 minh =. 0 074 s         lt d  30000µ m      dl 34.− 31 . µ m/min −− 11   v (smallest) =  = =. 0 0010 minh =. 0 062   s       lt d  30000 µ m  −− 11 mm =± 997 160 () SE μm2 in MPan () = 0 −− 11    m  997 + 160 µmmin MPa   φ(largest) =  =   4      l   30 .× 10 µm  −− 11 −− 11 =. 0 039 minMPa =. 23 hMPa −− 11   m − mmin MPa   997 160 µ   φ(smallest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 028 minMPa =. 17 hMPa ε =± 60.. 95 1S () E2 MPan () = 7 −− 11     εφ(largest) (66 MPah )(2.3) MPa    Π (largest) =  =  ≈ 2448   pe  −1     v (smallest)  (0.062 h )     −− 11     εφ (smallest) (558 .. MPah )(1 7) MPa    Π (smallest) =  =  ≈ 1282   pe   −1    v (largest)     (0.074 h )   Π () stageIV =± 1865 583 pe e f Th ollowing values were obtained for intact stage I sporangiophores of Phycomyces blakesleeanus : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 11 www.nature.com/scientificreports/ lm == 20 mm 20 .× 10 µ dl −1 =± 71 .. 07 () SE μmmin () n1 = 7 dt     dl 71 .+ 07 . µ m/min −− 11     v (largest) = = =. 0 00039 minh =. 0 023   s     lt d  20000µ m       dl  71 .− 07 . µ m/min −− 11   v (smallest) = =  =. 0 00032 minh =. 0 019 s        lt d  20000µ m   −− 11 mm =± 116 18.8S () E1 μ minMPa () n = 7 −− 11    m  116 +. 18 8 µmmin MPa   φ(largest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00674 minMPa =. 0 4044 hMPa −− 11    m  116 −. 18 8 µmmin MPa φ(smallest) =   =   4      l   20 .× 10 µm  −− 11 −− 11 =. 0 00486 minMPa =. 029 hMPa ε =± 68.. 95 6S () E2 MPan () = 7 −− 11     .. εφ() largest (745 MPah )(0 40 MPa )    Π () largest =  =  ≈ 1568   pe   −1    vs () mallest     (0.019 h )   −− 11     εφ() smallest (633 .. MPah )(0 29 MPa )     Π () smallest =  =  ≈ 798 pe  −1     vl (argest)  (0.023 h )     Π () stageI =± 1183 385 pe v values for cells from pea stems using Π = 32. This data for pea stems was taken directly from s pe Cosgrove , for data labeled “Water” (for Fig. 2). The value for ε is located in the text. v is the measured relative sM volumetric growth rate and v is the calculated relative volumetric growth rate. sC −1 vh =± 0.. 0249 0 0022 () SE () n1 = 0 sM ε =. 95 MPa −− 1 1 φ =± 0.. 074 0 007hMPa   −− 11   εφ (9.. 5 MPaM )(0 074 +. 0 007 Pa h )   −1   vh (largest) = =  ≅ 0.024 sC     Π  32     pe  −− 11     εφ (9.. 5 MPaM )(0 074 −. 0 007 Pa h )   −1   vh (smallest) =  =  ≅ 0.020 sC     Π 32      pe  −1 vh =± 0.. 022 0 002 sC This data for pea stems was taken directly from Cosgrove , for data labeled “IAA”. The value for ε that is located in the text. −1 vh =± 0.. 067 0 004 () SE () n8 = sM ε =. 95 MPa −− 11 φ =± 02 .. 20 014 hMPa   −− 11   εφ (9.. 5 MPaM )(0 22 +. 0 014 Pa h )   −1   vh (largest) = = ≅ 0.070   sC    Π  32     pe    −− 11   εφ (9.. 5 MPaM )(0 22 −. 0 0147 Pa h )   −1   vh (smallest) =  =  ≅ 0.061 sC     Π   32    pe  −1 vh =± 0.. 0655 0 0045 sC This data for pea stems was taken directly from Cosgrove , for data labeled “Cut directly form plant”. The value for ε is located in the text. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 12 www.nature.com/scientificreports/ −1 vh =± 0.. 075 0 004 () SE () n1 = 0 sM ε =. 95MPa −− 11 φ =± 02 .. 50 017hMPa   −− 11   εφ (9.. 5 MPaM )(0 25 +. 0 017 Pa h )   −1 vh (largest) =  =  ≅ 0.079 sC     Π   32    pe  −− 11     (9.. 5 MPaM )(0 25 −. 0 017 Pa h ) εφ   −1   vh (smallest) =  =  ≅ 0.069 sC     Π 32      pe  −1 vh =± 0.. 074 0 005 sC v values for sporangiophores using Π = 1524. The data for plucked stage IV sporangiophore was sC pe 4 6 taken from Ortega et al. and Ortega . v is the measured relative volumetric growth rate and v is the calculated sM sC relative volumetric growth rate (for Fig. 3). For v , dL/dt is measured and then divided by l. sM lm == 40 mm 41 × 0 µ dl −− 11 =. 55 µµ mmin = 330 mh dt   dl −1   vh = = 0.008 sM   lt d  ε = 61MPa −1     εφ 10.3 h −− 11   φ = =  =. 0 1688 MPah        ε   61 MPa    −− 11   εφ (61 MPaM )(0.17 Pa h )   −1   vh = =  ≅ 0.007 sC     Π  1524     pe  4 6 e d Th ata for intact stage IV sporangiophores were taken from Ortega et al. and Ortega . lm == 30 mm 31 × 0 µ dl −− 11 =. 34 13 ±.1(SE) µµ mmin =± 2046 186 (SE) mh (n = 20) dt   dl −1   v = =± 0.. 068 0 006 () SE h () n2 = 0 sM   lt d  ε =± 60.. 95 1S () E2 MPan () = 7 −− 11    m  997 + 160 mmin MPa   φ(largest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 039 minMPa =. 23 hMPa −− 11    m  997 − 160 µmmin MPa   φ(smallest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 028 minMPa =. 17 hMPa −− 11     (66 MPaM )(2.3) Pa h εφ   −1   vh (largest) =  =  ≅ 0.099 sC     Π 1524      pe  −− 11     εφ (558 .. MPaM )(1 7) Pa h   −1 vh (smallest) =  =  ≅ 0.062 sC       Π  1524     pe  −1 vh =± 0.. 0805 0 0185 sC e d Th ata for intact stage I sporangiophores were taken from Ortega . Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 13 www.nature.com/scientificreports/ lm == 20 mm 21 × 0 µ dl −− 11 =. 71 ±. 07 (SE) µµ mmin =± 426 42 (SE) mh (n = 20) dt   dl −1   v = =± 0.. 021 0 002 () SE h () n1 = 7 sM     lt d ε =± 68.. 95 6S () E2 MPan () = 7 −− 11    m  116 µmmin MPa −− 11 −1   φ =  = (60min hM )0 =.348 Pa h  4       l  21 × 0 µm   −− 11    m  116 +. 18 8 µmmin MPa   φ(largest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00674 minMPa =. 0 4044 hMPa −− 11    m  116 −. 18 8 µmmin MPa   φ(smallest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00486 minMPa =. 029 hMPa −− 11     εφ (745 .. MPaM )(0 40 Pa h )   −1 vh (largest) =  =  ≅ 0.020 sC     Π  1524     pe    −− 11   εφ (633 .. MPaM )(0 29 Pa h )   −1   vh (smallest) = =  ≅ 0.012 sC     Π  1524     pe  −1 vh =± 0.. 016 0 004 sC Constant-tension extension experiment - load removed. Unidirectional constant-tension extension experiments were conducted on frozen and then thawed walls of the stage IV sporangiophore using the same method and experimental apparatus used and described by Ortega et al. (for Fig. 4). A five-millimeter long sec- tion of the wall from a stage IV sporangiophore, that includes the growth zone, was adapted to a bathing solution of pure water for 20 min in a test apparatus. Then a tensile load of 1.24 gram was applied to the wall. The tensile load produces longitudinal stress and longitudinal extension of the wall (810 μm). Aer ft the extension, the length remained constant for four minutes (11–15 min). At 15 min (marked by the downward pointing arrow), 50 mM potassium-acetate buffer, pH 4.5, replaces the bathing water in the test apparatus. It can be seen that “creep” extension begins immediately and continues for seven minutes. The creep extension behavior produced by the 50 mM potassium-acetate buffer, pH 4.5, is very similar to that obtained when the pH of the solution was lower to 4.6 by adding a predetermined amount of pH Red 4.0 buffer to the bathing water . Aer ft wards (22–25 min), the wall section length remains constant for three minutes. At 25 min, the tensile load of 1.24 grams (σ = 42 MPa; see Ortega et al. for detailed stress calculations) is slowly removed and the wall section decreases in length over the next three minutes. The decrease in length is “recovered” elastic extension. e d Th ata obtained from this experiment is used for the following calculations. When σ is constant (11–25 min) the change in length (creep) stimulated by the decrease in pH is governed by; v = (dl/dt)/l = φ (σ − σ ), and s C φ = v /(σ − σ ). When the stress is removed (at 25 min), then the decrease in length is governed by; −(dl/dt)/l = − s C (1/ε) dσ/dt, and ε = Δσ/(Δl/l). From the experiment and Fig. 4 we get: l =. 50 mm == 5000 µmlengthofcellwallbeing tested −1 dl/1 dt == 7m µ minaverage creep extensionrateduring7 mininterval −− 11 vd == (/ ldtl )/ 17 µµ mmin /5000 m0 =.0034 min σσ −= 42 MPa −. 16 3MPa =. 25 7MPa = effective stress during creep extension −− 41 −1 φσ =− v /( σ )1 =.31 × 0min MPa sC εσ =Δ /(Δ= ll /) (4 −− 2MPa)/(62/ µµ mm 5000 ) = 3387 MPa εφ Π =≈ 130 pe References 1. Lockhart, J. A. An analysis of irreversible plant cell elongation. J Theor Biol. 8, 264–275, doi:10.1016/0022-5193(65)90077-9 (1965). 2. Green, P. B. Cell Morphogenesis. Ann Rev Plant Physiol. 20, 365–394, doi:10.1146/annurev.pp.20.060169.002053 (1969). 3. Green, P. 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E., Bell, S. A. & Erazo, A. J. Pressure clamp method to measure transpiration in growing single plant cells: Demonstration with sporangiophores of Phycomyces. Plant Physiol 100, 1036–1041, doi:10.1104/pp.100.2.1036 (1992). Acknowledgements This research was support by NSF Grant, MCB-0948921 to J.K.E. Ortega. The author acknowledges and thanks Cindy M. Munoz and Jason T. Truong for conducting the experiments whose results are presented in Figure 4. Author Contributions J.K.E.O. conceived the ideas, conducted the analyses, conducted the calculations and estimates, and wrote the manuscript. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 15 www.nature.com/scientificreports/ Additional Information Supplementary information accompanies this paper at doi:10.1038/s41598-017-03002-6 Competing Interests: The authors declare that they have no competing interests. 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Dimensionless number is central to stress relaxation and expansive growth of the cell wall

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www.nature.com/scientificreports OPEN Dimensionless number is central to stress relaxation and expansive growth of the cell wall Received: 6 January 2017 Joseph K. E. Ortega Accepted: 19 April 2017 Experiments demonstrate that both plastic and elastic deformation of the cell wall are necessary for Published: xx xx xxxx 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. Expansive growth is defined as a permanent increase in cell volume. Expansive growth of walled cells (e.g. algal, fungal and plant cells) depends on interrelated biochemical and physical processes. Active solutes inside their semi-permeable plasma membrane generate the osmotic potential needed to absorb water from its external envi- ronment and produce turgor pressure that stresses their walls. Wall stresses produce irreversible (plastic) and reversible (elastic) wall deformations in three orthogonal directions (volumetric wall deformation), generally stretching the wall in the two directions parallel to the wall surface (longitudinally and circumferentially) and contracting the wall in the direction perpendicular to the surface, making the wall thinner. New wall materials (polymers, proteins, etc.) are continually added to the inside surface to maintain a nearly constant wall thickness that varies between 0.1 μm and 1.0 μm, depending on the cell species. Researchers have long recognized that plastic deformation of the cell wall is necessary for expansive growth and morphogenesis. It is generally thought that regulation of expansive growth rate of the cell wall chamber that encloses the cell is achieved by controlling the magnitude of plastic deformation of the wall and the magnitude 1–4 5 1 of turgor pressure . Using the constitutive equation (stress-strain relationship) for a Bingham fluid , Lockhart derived a Growth Equation in terms of turgor pressure that describes these concepts, equation (1). Equation (1) describes the relative rate of change in volume of the cell wall chamber, v = (dV/dt)/V, as equal to the relative cw volumetric plastic deformation rate of the cell wall chamber, φ (P − P ), when the turgor pressure, P, exceeds the critical turgor pressure, P . vP =− φ() P (1) cw C e in Th clusive biophysical variables are φ (irreversible wall extensibility), P (turgor pressure), P (critical turgor pressure – a value that must be exceeded before plastic wall deformation occurs). It is noted that nearly all of the research conducted with this equation assumes that changes in volume occur predominately by changing the length of the cell, the diameter is assumed to remain relatively constant. The magnitude of φ has been shown to be 6, 7 biochemically controlled by the cell . Equation (1) explicitly indicates that expansive growth does not depend on Bioengineering Laboratory, Department of Mechanical Engineering, University of Colorado Denver, 1200 Larimer Street, NC-2024-K, P.O. Box 173364, Denver, CO, 80217-3364, USA. Correspondence and requests for materials should be addressed to J.K.E.O. (email: Joseph.Ortega@ucdenver.edu) Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 1 www.nature.com/scientificreports/ elastic wall deformations, but only depends on plastic wall deformations and the inclusive biophysical variables 1, 3, 8 (φ, P, and P ) . Equation (1) is simple, yet demonstrates broad utility in modeling and understanding expansive growth behav- ior of walled cells and is frequently found in many textbooks in plant biology and plant physiology. However, subsequent research demonstrates that equation (1) is not able to describe the results of experiments conducted specifically to reveal the mechanical behavior and constitutive relationship of growing cell walls. Specifically, equation (1) and its underlying constitutive equation for a Bingham uid fl cannot describe the results of in vivo 4, 9, 10 11, stress relaxation experiments and in vivo creep experiments that produce large changes in turgor pressure 12 9 because it does not describe reversible (elastic) wall deformations. Ortega used a constitutive equation derived in linear viscoelastic theory (Maxwell viscoelastic model) to accurately describe the results of stress relaxation experiments conducted on growing sporangiophores of the fungus, Phycomyces blakesleeanus. Subsequently, Ortega modified the Maxwell constitutive equation by replacing the Newtonian fluid inside the dashpot with a 6, 14 Bingham fluid (and later called it a Maxwell-Bingham viscoelastic model ). The Maxwell-Bingham constitutive equation was used as the foundation for a biophysical wall extension equation in terms of turgor pressure, the Augmented Growth Equation , equation (2).  1d  P   vP =− φ() P + cw C    (2) ε dt e addi Th tional variables are ε (volumetric elastic modulus of the wall), and t (time). The Augmented Growth Equation describes the relative rate of change in volume of the cell wall chamber, v = (dV/dt)/V, as the sum of cw the relative volumetric irreversible (plastic) deformation rate, φ (P − P ), and the relative volumetric reversible (elastic) deformation rate, (1/ε) dP/dt, of the cell wall chamber. Again, nearly all of the research conducted with this equation assumes that changes in volume occur predominately by changing the length of the cell, while the diameter is assumed to remain relatively constant. It is noted that equation (1) is recovered from equation (2) when the turgor pressure is constant. Importantly, the Augmented Growth Equation accurately describes the experimental results of in vivo stress relaxation experiments conducted on pea stems and fungal sporangio- 4 11 phores , and in vivo creep experiments conducted on algal internode cells that involve large step changes in tur- gor pressure. The Augmented Growth Equation also demonstrates broad utility in modeling and understanding 6, 14–16 10, 17 17, 18 expansive growth behavior of walled cells , plant tissue and whole plants . 10, 19, 20 Using the results of in vivo stress relaxation experiments, Cosgrove describes a mechanism for expansive growth of the cell wall that is more complicated than the simple wall extension mechanism mod- eled by Equation 1. Cosgrove states: “As a cell absorbs water, the wall extends passively, and polymers in the load-bearing network(s) are distended. In nongrowing cells, wall stress increases as the polymers are stretched like springs. Elastic energy is stored in the strained bonds of these polymers (and also in the increased order of the polymers), and this elastic energy does work on the cell protoplast by compressing it, thereby increasing its turgor pressure and water potential. When the cell water potential increases to the point where it matches that of the external water, net water uptake ceases. In growing cells, this equilibrium is never quite reached because the wall “relaxes,” which means that the load-bearing network breaks, slips, or is cut, and the distended polymers assume a more relaxed condition. Elastic energy of the wall is lost as heat, and a turgor reduction inevitably accompanies the reduced wall stress. Note, however, that this relaxation by itself does not entail a physical expansion of the wall or a change in cell volume. Turgor decreases because the wall simply stops compressing the protoplast. Expansion follows secondarily, as the cell absorbs water in response to the reduced water potential created by the reduction of turgor pressure”. Interestingly, even though this “stress relaxation concept for expansive growth” has been available for a couple of decades, many of the conceptual models that are used to guide experimental investigations and mathematically models of expansive growth, still employ some form of equation (1), which cannot model stress relaxation. A few possible reasons for the continual employment of this concept and equation (1) are (i) it is simple to comprehend and to use, (ii) it is recovered from equation (2) when P is constant (thus elasticity is removed from the model when P is constant), and (iii) it is not apparent how to mathematically relate the stress-relaxation concept to expansive growth. In this paper, dimensional analysis is used to obtain insight into the stress relaxation process and to explic- itly demonstrate a relationship between expansive growth rate and wall elasticity as well as wall plasticity. Dimensional analysis is conducted on the Augmented Growth Equation, equation (2), describing plastic and elastic wall deformation rates. Three Π parameters are identified in the analysis, but only one is shown to control the turgor pressure decay rate that occurs during wall stress relaxation, the Π parameter. Dimensional analysis pe provides a physical interpretation for the Π parameter; it is the ratio of the relative volumetric plastic deforma- pe tion rate of the wall and relative volumetric elastic deformation rate of the wall. This analytical result demonstrates that the ratio of plastic and elastic deformation rates are used to regulate the rate of stress relaxation that occurs during expansive growth of the cell wall chamber. It is shown that the relative magnitude of plastic and elastic deformation rates are important in controlling wall stress relaxation and expansive growth. Importantly, the Π pe parameter provides a mathematical relationship between the relative rate of change in volume of the cell wall chamber, v , and the mechanical properties of the wall, i.e. the irreversible wall extensibility, φ, and the volumetric elastic modulus, ε. It is noted that the turgor pressure, P, and critical turgor pressure, P , do not appear in the Π C pe parameter. Values are calculated for the Π parameter from previously reported experimental results obtained pe from cells in plants, fungi, and algae. It is found that the Π values are large and very different for these different pe cell species. In some cases, data was available to calculate Π values for the same cells when φ and ε changed. pe e r Th esults indicate that Π can be used to calculate the magnitude of v when φ and ε change, because the cal- pe s culated values for v compare very well with measured values for these cells. In addition, it is shown that the Π s pe Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 2 www.nature.com/scientificreports/ parameter can be used to organize data related to cell wall deformation for all walled cells so that the ratio of plastic and elastic deformation rates of cell walls for different cells from plants, algae, and fungi can be compared. Theory, Analyses and Results Dimensional Analysis. The overall process of dimensional analysis consists of obtaining dimensionless groups of variables (Π parameters) involved in the process of interest and exploring the relationship between the Π parameters experimentally and/or analytically. A straightforward method to obtain the Π parameters 21, 22 can be employed when you have governing equations . In this method the variables within the governing equations are made dimensionless with appropriate constant reference quantities and the governing equations are manipulated to produce dimensionless equations and with dimensionless coefficients, or Π parameters. e Th Augmented Growth Equations represent governing equations for expansive growth and can be used for dimen- sional analyses . e exp Th ansive growth rate of a walled cell is determined by the net water uptake rate and cell wall deformation rate. During steady or quasi-steady growth, expansive growth rate is limited and controlled by the smaller rate. e m Th agnitudes of net water uptake rate and wall deformation rate can be evaluated using the Π parameters iden- tified in the dimensionless Augmented Growth Equations . Scale analysis reveals that the net rate of water uptake is approximately eight times greater than the wall deformation rate for cells in growing pea stems (Pisum satinis L.), approximately 100 times greater for growing stage I sporangiophores (P. blakesleeanus), and approximately 12 times greater for growing stage IV sporangiophores (P. blakesleeanus); see ‘scale analysis’ in Calculations, Estimates, and Methods for details. These findings indicate that the smaller ‘wall deformation rate’ limits and controls the magnitude of expansive growth rate for these cells. Therefore, the analysis in this paper focuses on equation (2) that describes the wall deformation rate. Dimensionless Augmented Growth Equation. Here, dimensional analysis will focus on the Augmented Growth Equation that describes the wall deformation rate, equation (2). The analysis is conducted for walled cells growing normally at a quasi-steady rate . Typically P is constant during normal growth and stress relaxa- 6, 10, 12 tion , so P will be treated as a constant in the analysis. The variables (v , t, and P) in equation (2) are made C cw dimensionless (*) with the following constant reference quantities , P (critical turgor pressure) and v (steady or C s average relative volumetric growth rate, i.e. v = v averaged over a time interval = constant). s cw   ∗∗  P =  and PP = P  P     tt == vt and   v     ∗∗ cw  v =  and vv = v cw  cw cs w  v    Substituting the respective expressions that include the dimensionless variable and reference quantity (vari- ables and their respective expressions in the right column above) for each of the variables into equation (2), we get equation (3):  vP  dP ⁎⁎ sC vv =− φPP (1) + scwC  ⁎     ε d( t 3) e va Th riables v *, P*, and t* are dimensionless. Therefore the dimensions, physical characteristics, and rel- cw ative magnitude of each term are represented by their respective dimensional coefficient. Thus, v represents a steady or average magnitude of relative rate of change in volume of the cell wall chamber (constant), φ P repre- sents the magnitude of the relative volumetric irreversible (plastic) deformation rate and (v P /ε) represents the s C magnitude of the relative volumetric reversible (elastic) deformation rate. Dividing equation (3) by the constant reference quantity, v , we get equation (4), the dimensionless Augmented Growth Equation : dP ⁎⁎ vP =Π (1 −+ ) Π cw pv ev (4) dt The Π and Π parameters are dimensionless coefficients defined as follows : pv ev     φP relative volumetric plasticdeformationrateofthe wall C     Π=  =    pv      v  relative volumetric growth rate    and    P  relative volumetric elasticdeformationrateofthe wall C    Π= = . ev         ε relative volumetric growth rate    The Π parameters represents ratios of the dimensional coefficients shown in equation (3), where the first sub- script of the Π parameter refers to the numerator and the second subscript refers to the denominator. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 3 www.nature.com/scientificreports/ Wall Stress Relaxation and Pressure Relaxation. Wall stress and pressure relaxation are the founda- 10, 19, 20 tion of the “stress relaxation concept of expansive growth” . Insight can be obtained by solving the dimen- sionless governing equation for stress relaxation. Typically, an in vivo stress relaxation experiment is conduct by isolating a growing walled cell from its water supply and eliminating the loss of water via transpiration. Then 4, 10 the decay in turgor pressure is measured as a function of time, usually with a pressure probe . It is essential to prevent water uptake and water loss from the cell (water volume and volume of the cell wall chamber must remain constant), so the decay in wall stress and concurrent decay in turgor pressure is the result of relaxation alone. u Th s, experimentally and analytically, the condition that must be satisfied for a stress relaxation experiment is that 10, 13 the volume of the cell remains constant during the experiment, or v = 0 . Imposing the condition, v * = 0, cw cw the dimensionless governing equation for a stress relaxation experiment is obtained from equation (4): dP 0( =Π P −+ 1) Π. pv ev dt Rearranging, equation (5) is obtained where the dimensionless parameter, Π , is equal to the ratio, Π /Π : pe pv ev dP =−Π− (1 P ) pe (5) dt Using the dimensionless initial condition, P* = P * = P /P , equation (5) is integrated and the dimensionless i i C solution is obtained: ⁎⁎ ⁎ PP =− (1)exp −Π t +. 1 i () pe (6) Where     εφ relative volumetric plasticdeformationrateofthe wall    Π=  = .   pe      v relative volumetric elasticdeformationrateofthe wall      (7) Equation (5) demonstrates that the rate of decay of the dimensionless pressure, P*, during stress and pressure relaxation is directly related to the magnitude of Π . Equation (6) shows that the decay of the dimensionless pres- pe −1 sure, P*, is exponential and the dimensionless time constant for the exponential decay is, t * = (Π ) . Equation c pe (7) reveals that Π is the ratio of relative volumetric plastic and elastic deformation rates of the wall. pe Magnitudes of Π from in vivo stress relaxation and in vivo creep experiments. e m Th agnitude pe of Π can be calculated from results obtained from in vivo stress relaxation and in vivo creep experiments con- pe ducted with the pressure probe. In an in vivo stress relaxation experiment, the growing cell is removed from its 10, 13 10 water supply and prevented from transpiring . Usually this requires the tested cells to be incised from tissue or removed from mycelium (plucked) . Then the exponential decay in P is measured with a pressure probe and −1 the time constant for the exponential decay is, t = (εφ) . The determination of Π , employing the in vivo stress c pe relaxation method has the theoretical advantage that the product, εφ, can be calculated directly from the halftime, T , of the exponential decay of the turgor pressure; εφ = ln 2/T and Π = (ln 2/v T ). 1/2 1/2 pe s 1/2 Sometimes it is not possible to conduct in vivo stress relaxation experiments on walled cells. Then the magni- tude of Π can be calculated using equation (7) and the values of ε, φ, and v obtained from in vivo creep exper- pe s 4, 6, 10, 11 iments . In an in vivo creep experiment the P in a growing cell is increased (stepped-up) with a pressure probe and the increase in elongation growth rate is measured. The magnitude of φ can be determined from the increase in elongation growth rate . The magnitude of ε is determined separately from experiments during which a pulse-up in P is produced with the pressure probe . Theoretically, in vivo creep and in vivo stress relaxation experiments should produce the same values for φ and (εφ). Figure 1 compares Π values determined from in pe vivo creep and in vivo stress relaxation experiments conducted on incised pea stem sections (P. satinis L.) , incised 11, 12 6 algal internodes cells (Chara corallina) , intact stage I sporangiophores (P. blakesleeanus) , plucked stage IV 4 6 sporangiophores (P. blakesleeanus) and intact stage IV sporangiophores (P. blakesleeanus) ; see Calculations and Estimates for the calculations of Π values. It should be noted that the vertical scale for Π is logarithmic. pe pe Figure 1 demonstrates that the Π values for pea cells of P. satinis L., internode algal cells of C. corallina, and pe fungal sporangiophores of P. blakesleeanus are large and very different, by an order of magnitude or more. The different magnitudes of the Π values presented in Fig. 1 provide a comparison of the ratio of plastic and elastic pe deformation rates of growing walls from plant, algal, and fungal cells. Expansive growth rates calculated from a constant Π . The steady or average relative volumetric pe growth rate, v , can be calculated from Π when ε and φ are measured. Rearranging equation (7) we get: s pe εφ v = pe (8) Importantly, equation (8) provides an explicit mathematically relationship between expansive growth rate, v , and stress relaxation of the cell wall, as characterized by Π . Also, equation (8) demonstrates that v can be s pe s determined without measuring P and P , if Π is known. C pe It is noted that the magnitude of Π for pea cells of P. satinis L. calculated from in vivo stress relaxation pe experiments is the same as those calculated from in vivo creep experiments, even though the measured relative volumetric growth rates, v , of the cells during the in vivo creep experiments are much larger than for the in sM Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 4 www.nature.com/scientificreports/ Figure 1. A comparison of the Π values calculated for growing cells of pea stems (Pea) , growing internode pe 11, 12 4, 6 cells of C. coralline (Chara) , and growing sporangiophores of P. blakesleeanus (Stage I and Stage IV) . Note that the vertical scale for Π is logarithmic. Also compared are the values obtained from in vivo stress relaxation pe (SR) and in vivo creep (CR) experiments for pea cells and stage IV sporangiophores. Only the in vivo creep experiments conducted on Stage I (CR) and Stage IV (CR) sporangiophores used intact cells (growing on the mycelium). The confidence intervals represent the maximum and minimum values of Π that can be calculated pe 4, 6, 10 from the statistical data (standard deviation and standard errors) presented in the respective papers (see Calculations, Estimates, and Methods for details). vivo stress relaxation experiments (~3 times larger). Based on maximum and minimum values of Π that can be pe calculated from the statistical data, the magnitude of Π for intact stage IV sporangiophores of P. blakesleeanus pe calculated from in vivo creep experiments is only slightly larger than the value calculated from in vivo stress relax- ation experiments conducted on plucked stage IV sporangiophore, even though the measured relative volumetric growth rates, v , are much larger (~6 times larger). Again, based on maximum and minimum values of Π that sM pe can be calculated from the statistical data, the magnitudes of Π for stage IV and stage I sporangiophores are pe the same, even though the average value for v of stage IV sporangiophores is more than four times larger than sM those of stage I sporangiophores. es Th e results suggests that the ratios of plastic and elastic deformation rates (i.e. the magnitude of Π ) of cell walls from a specific cell species are the same. It appears that the ratio of plastic and pe elastic deformation rates of cell walls from pea stems of P. satinis L. is 32 (i.e. Π = 32). Similarly, it appears that pe the ratio of plastic and elastic deformation rates of cell walls from sporangiophores of P. blakesleeanus is approx- imately 1524 (i.e. Π = 1524, the average value of intact stage I and stage IV from in vivo creep experiments). If pe this suggestion is correct, then a constant value for the Π parameter can be used to calculate the steady relative pe volumetric growth rate, v , for cells from pea stems of P. satinis L. and for sporangiophores of P. blakesleeanus when φ and/or ε change because of changes growth conditions (incised and growing in water and just cut from the plant), addition of growth hormone IAA, changes in development (stage I and stage IV), and intact versus plucked sporangiophores (see Figs 2 and 3). Specifically, v can be calculated for growing cells from pea stems by employing equation (9). εφ vP (L .. satinis ) = (9) Equation (9) is employed to calculate v values using values for ε and φ, and compare them to measured values for pea stems of P. satinis L. (Fig. 2). Equation (10) is employed to calculate v values using values for ε and φ, and compare them to measured values for stage I and stage IV sporangiophores of P. blakesleeanus (Fig. 3). εφ vP () .= blakesleeanus (10) Discussion Dimensional analysis reveals that the rate of wall stress relaxation is directly related to the magnitude of a sin- gle dimensionless parameter, Π ; see equation (5). Equation (6) demonstrates that the dimensionless pressure pe −1 (and stress) relaxation is exponential and the dimensionless time constant is, t * = (Π ) . Dimensional analysis c pe provides a physical interpretation for the Π parameter, it is the ratio of relative volumetric plastic and elastic pe deformation rates of the cell wall; see equation (7). The Π parameter explicitly reveals a relationship between the pe steady or average relative volumetric growth rate of the cell, v , and the wall stress relaxation as characterized by Π ; see equation (7). In fact, v can be calculated using equation (8) when Π , φ, and ε are known; see equations pe s pe Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 5 www.nature.com/scientificreports/ Figure 2. A comparison of v values calculated (dark grey) and measured (light grey) for cells in pea stems (P. satinis L.). Equation (9) is employed to calculate v values (dark grey) using a constant value for Π (Π = 32) s pe pe and values for ε and φ taken from Cosgrove ; see Calculations, Estimates, and Methods for details. The confidence intervals for the measured values (light grey) are standard errors taken from Cosgrove . The confidence intervals for the calculated values (dark grey) represents the maximum and minimum values that can be calculated from maximum and minimum values obtained from standard errors of φ . Figure 3. A comparison of v values calculated (dark grey) and measured (light grey) for stage I and stage IV sporangiophores of P. blakesleeanus. Equation (10) is employed to calculate v values (dark grey) using a 4, 6 constant value for Π (Π = 1524) and using values for ε and φ taken from Ortega et al. ; see Calculations, pe pe Estimates, and Methods for details. The confidence intervals for the measured values (light grey) are standard errors taken from Ortega et al. . The confidence intervals for the calculated values (dark grey) represents the maximum and minimum values that can be calculated from maximum and minimum values obtained from standard errors of ε and φ . The measured v value for the plucked (p) stage IV sporangiophore (light grey) is for a single experiment. (9) and (10). Equation (8) represents a method to calculate v when P or P are not known. Good agreement is s C obtained for calculated values of v compared to measured values, for cells from pea stems and sporangiophores (stage I and stage IV) when constant Π values are used; see Figs 2 and 3. pe Figure 1 compares magnitudes of Π for cells from growing pea stems of P. satinis L. (Pea), growing internode pe cells of C. coralline (Chara), and growing sporangiophores of P. blakesleeanus (Stage I and Stage IV). The results indicate that the magnitude of Π is very large (Π ≫ 1) for all three species of walled cells. The large values pe pe for Π indicate that the relative volumetric plastic wall deformation rate is very much larger than the relative pe volumetric elastic wall deformation rate during steady and quasi-steady expansive growth. From the perspective of a constitutive relationship (stress-strain relationship), the strain rate of the wall is proportional to the stress for plastic (irreversible) deformation, but proportional to the stress rate for elastic deformation. Therefore the Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 6 www.nature.com/scientificreports/ constitutive relationship for plastic and elastic deformation are different. From an energy perspective, wall stress relaxation represents the dissipation of mechanical energy stored by elastic wall deformation. Plastic and elastic wall deformation rates represent the rates of two very different processes. An increase in elastic wall deformation rate represents an increase in the rate that mechanical energy is stored in the wall. This is a reversible process, and the stored mechanical energy can be recovered. An increase in plastic wall deformation rate represents an increase in rate of dissipation of stored mechanical energy in the wall. This is an irreversible process, and the lost mechanical energy cannot be recovered. The large values of Π for the walled cells indicates that plastic wall pe deformation rate dominates wall deformation during steady and quasi-steady expansive growth. From the per- spectives of constitutive relationships and energy, this finding draws into question the use of elastic models and elastic wall deformations to study expansive growth and morphogenesis. Figure 1 also compares values for Π determined from the results of in vivo creep and in vivo stress relax- pe ation experiments. The results indicate that the Π values are essentially identical for cells from pea stems and pe nearly statistically identical for stage IV sporangiophores. These experimental results are consistent with the- oretical prediction that the Π values obtained from in vivo creep and in vivo stress relaxation experiments pe should be the same. However, the results are also interesting because the growth rate for test specimens dur- ing in vivo stress relaxation experiments are three (for pea) to six (for stage IV) times smaller than during in vivo creep experiments. One wonders if the magnitude of Π is invariant and is determined by a chemical wall pe loosening-hardening mechanism (chemorheology) that is fundamental to each species of cell, so that the same ratio of plastic and elastic wall deformation rates are produced for each cell species. If this suggestion were correct, it would make two predictions. First, it would predict that the chemorheology for cells in pea stems (P. satinis L.), algal internode cells (C. corallina) and sporangiophores (P. blakesleeanus) are die ff rent from each other because the magnitude of their respective values of Π are very different from each other, by an order of magnitude or pe more. Second, it would predict that a single value for Π represents a cell species and can be used for that cell pe species to calculate v when φ and/or ε change because of alterations in growth conditions, addition of growth hormone, and development. er Th e is evidence to support the first prediction. In a variety of higher plant cells within tissue, a pH-dependent 7, 25 protein (expansin) is found to loosen the wall by disrupting the hydrogen bonds between microfibrils . In large internode algal cells of C. corallina, experimental evidence indicates that making and breaking calcium 26, 27 bridges between pectin polymers loosens and hardens the wall . It is not known how the sporangiophore of 28–30 P. blakesleeanus loosens and hardens its wall , but it is predicted that its chemorheology will be different from those used by either cells in pea stems of P. satinis L. or internode algal cells of C. coralline, because the magnitude of Π for the sporangiophores is orders of magnitude larger than those of either pea or Chara. pe e s Th econd prediction was evaluated here by using a single value for Π to calculate v for cells in pea stems pe s (Π = 32) and sporangiophores (Π = 1524) when φ and/or ε change. Figure 2 compares calculated and meas- pe pe ured v values for cells in pea stems (P. satinis L.) when φ changes because of changing growth conditions (‘incised and growing in water’ and ‘just cut from the plant’), and the addition of the growth hormone IAA. Figure 3 com- pares calculated and measured v values for the sporangiophores of P. blakesleeanus when φ and ε change because of development (stage I and stage IV) and when it is removed from the mycelium (plucked) and its base is put in pure water. The respective calculated and measured v values for cells in pea stems and sporangiophores compare very well and appear to be the statistically the same. It is concluded that the respective calculated and measured values of v for pea stems, stage IV, and stage I sporangiophores are the same. Therefore, it is also concluded that the results support the second prediction. e Th results presented in Fig.  1 indicate that the magnitude of Π for the sporangiophores of P. blakesleeanus is pe orders of magnitude larger, and significantly larger, than the magnitude of Π for cells from pea stems. This find- pe ing indicates that wall deformation behavior can be significantly different for walled cells from different species. In the future, the Π parameter can be determined for different species of walled cells. Then the Π parameters pe pe can be used to provide insight into similarities and differences in the ratio of plastic and elastic wall deformation rates during expansive growth of different cell species. Future research may also provide more support for the idea that the magnitude of Π is related to the chemorheology that regulates wall deformation when the wall is pe stressed. In future research, it would be interesting and important to determine Π for ‘growth mutant’ cells pe and ‘wall mutant’ cells , and compare their magnitudes to Π values obtained from wild type cells of the same pe species. This could reveal whether the mutation produced a change in the ratio of plastic and elastic wall defor - mation rates. e dim Th ensionless parameters obtained and studied here are related to expansive growth rate and growth rate regulation. However, other studies concerning cell size and cell morphology have introduced different dimen- sionless parameters that include the elastic modulus of the wall, E, and the turgor pressure, P. Boudaoud intro- duced the dimensionless parameter, E/P, to study the size and shape of isolated walled cells. The cells are modeled as elastic shells where growth is driven by turgor pressure. Mechanical equilibrium is used to obtain scaling laws for cell size (cell radius) that employ the dimensionless parameter, E/P. The magnitude of the radius of the sporangiophores of P. blakesleeanus can generally be predicted with the published scaling equation. Goriely et al. introduced the dimensionless parameter, P w/(h E), to characterize the overall wall deformation (P is a eff eff measure of the normal stresses acting on the wall, w is width of the apical tip, and h is the thickness of the shell or wall). This research used nonlinear elastic models to get insight into how the shape of apical hyphae are produced. e g Th eneral shape of apical hyphae produced with the dimensionless parameter is similar to those seen in stage I sporangiophores of P. blakesleeanus. Ortega obtained additional dimensionless Π parameters describing water uptake rates and transpiration rate (also see the ‘scale analysis’ in the Calculations, Estimates, and Methods) that may be used in future research to study expansive growth rate, water uptake rate, and transpiration rate of water-stressed cells in crop plants and may provide insight into how plants adapt to drought conditions. In Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 7 www.nature.com/scientificreports/ Figure 4. e r Th esults of a modified constant-tension extension experiment is presented. At ten minutes a weight (1.24 g) was used to apply constant tension to a five millimeter long section of a frozen-thawed wall that includes the growth zone , and an initial extension of approximately 810 µm is observed (see Calculations, Estimates, and Methods – ‘Constant-tension extension experiment’ for more details). At 15 min (marked by the downward pointing arrow), 50 mM potassium-acetate buffer, pH = 4.5, replaces the bathing water (pH = 7.0) in the test apparatus . Immediately, creep extension is observed until 22 min on the time scale. At 25 min (marked by the upward pointing arrow) the tension is released and the extension decreases (recovered elastic extension) until 28 min on the time scale. general, different dimensionless parameters can be employed to obtain insight and to investigate different aspects of size, shape, and growth of walled cells. In the physical sciences, Π parameters have been used to establish similarity between uid flo fl ws, heat flows, 21, 22, 35 and other transport processes by ensuring that the magnitudes of relevant Π parameters are the identical . In the future, we plan to investigate the use of wall deformation similarity (by matching the magnitudes of the Π pe parameter) to guide and valid local mathematical models that are making strides towards describing spatial and 36–42 temporal wall deformation of the cell wall during expansive growth . Also, future research will investigate 30, 32 the use of wall deformation similarity to augment experimental assays that are used to investigate cell wall loosen-hardening mechanisms and chemorheology of cell walls. For example, constant-tension extension exper- iments were conducted on frozen and thawed walls of stage IV sporangiophore and showed that lowering the pH in the wall produces creep extension . The creep extension is transient (~10 min in duration) and the creep −1 extension rate is large for the first couple of minutes (~110 µ m min ), then decreases for the next eight minutes −1 30 (~8 µ m min ) before it stops . The pH mediated creep extension rates cover the range of the elongation rates −1 exhibited by the stage IV sporangiophore (20–60 µ m min ). This result might suggest that lowering the pH in the living and growing cell wall is the mechanism employed by the sporangiophore to produce expansive growth of the wall and possibly regulate its growth rate. However, if the constant-tension extension protocol is modified so that the results can be used to calculate the magnitude of Π , then the magnitude of Π for pH mediated creep pe pe extension can be compared to that obtained for natural growing stage IV sporangiophores. Figure 4 shows the results of a modified constant-tension extension experiment. e Th magnitude of Π is calculated from the experimental results (see Calculation, Estimates and Methods – pe ‘Constant-tension extension experiment’ for details). e Th calculated magnitude for Π is 130 (Π ≈ 130), which pe pe is an order of magnitude smaller than the magnitude obtained for normal growing stage IV sporangiophores (Π ≈ 1500). This result suggests that decreasing the pH alone cannot account for the wall deformation rate pe that occurs in natural growing stage IV sporangiophores because it does not produce the same ratio of relative volumetric plastic and elastic deformation rate that occurs during normal growing stage IV sporangiophores. It is envisioned that the Π parameter may be used in a similar way to assess the results of other experimental assays pe 19, 32 developed for higher plant cells . Calculations, Estimates, and Methods Scale analysis for ‘net water uptake rate’ and ‘wall deformation rate’. e f Th ollowing two dimen - sionless Augmented Growth Equations describe the net water uptake rate and the wall deformation rate . ⁎⁎ ⁎⁎ vP =Π () Δ− π −Π vN (o et fwater uptake rate) wwvTvT Net water uptake rate = water uptake – water loss due to transpiration (in relative terms) dP ⁎⁎ vP =Π (1 −+ ) Π (Walldeformation rate) cw pv ev dt Wall deformation rate = plastic deformation + elastic deformation (in relative terms) e dim Th ensionless Π parameters are defined as follows : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 8 www.nature.com/scientificreports/     LP relative volumetric wateruptakerate  C   Π=  =  wv     v relative volumetric growth rate          v relative volumetric transpirationrate sT     Π=  =    Tv        v relative volumetric growth rate         φP relative volumetric plasticdeformation rate of thewall C     Π=  =   pv       v   relative volumetric growth rate     P  relative volumetric elasticdeformation rate of thewall   Π= =  ev          ε relative volumetric growth rate   e Th magnitude of each term in the dimensionless Augmented Growth Equations is determined by the magni- tude of the dimensionless Π parameter associated with the term. So the magnitude of the ratio of net water uptake rate and wall deformation rate can be determine with the following Π parameter: wd   Π− Π  wv Tv Π= wd  Π+ Π  pv ev  Estimates of Π for steady or quasi-steady growing cells in pea stem (P. satinis L.). e va Th lues wd for the biophysical variables are obtained from Cosgrove . 20 .   (0.3) MPa ()  hMPa  LP v  C sT   − 0 0.0751         v v  s s h   Π= (pea) = ≈. 78 wd    02 . 5 φP P  C C   (0.3) MPa   +  () (0.3) MPa   hMPa    v ε   +   0.0751  (9.5) MPa     Estimates of Π for steady or quasi-steady growing sporangiophores (P. blakesleeanus). The wd 6 43 44 values for the biophysical variables are obtained from Ortega , Cosgrove et al. , and Ortega et al. 83.0   (0.4) MPa () 01 . 2   hMPa LP v    C sT  − 0.021  −   h   v v     s s h   Π= (stage I) = ≈ 102   wd   φP P 08 . 1 C C   (0.4) MPa   +  () (0.4) MPa   hMPa    v ε   +  0.021   (69) MPa     23.5   (0.26MPa) () 01 . 2   hMPa LP v    C sT  − 0.068  −   h   v v     s s h   Π= (stage IV)  = ≈ 11.7 wd   φP P 20 .   C C   (0.26MPa) + ()   (0.26MPa)   hMPa    v ε   +  0.068   (61) MPa     Estimates for Π from in vivo stress relaxation experiments. e m Th agnitude of Π =  ln 2/(v T ) pe pe s 1/2 is estimated for pea stems and fungal sporangiophores. The following data was obtained from incised pea stems sections (P. satinis L.) that include the growth zone (for Fig. 1). The stems sections were incised so that they can be removed from their water supply and transpiration eliminated: Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 9 www.nature.com/scientificreports/ −1 vh = 0.± 0249 0.0022 () SE −− 11 vh (largest)( =. 0 0249 +. 0 0022) =. 0 0271 h −− 11 vh (smallest)( =. 0 0249 −. 0 0022) =. 0 0227 h −1 T5 =± 22 .. 37 () SE min0 =± .. 87 006S () Eh 12 / −1 εφ(largest)2 == ln /2 Tlnh /0.= 81 08 . 6h 1/2 −1 εφ(smallest)2 == ln /2 Tlnh /0.= 93 07 . 5h 1/2 −1     εφ(largest) (0.8557) h    Π (largest) =  =  ≅. 37 7   pe   −1    v (smallest)     (0.0227 h )   −1     εφ(smallest) (0.7453) h     Π (smallest) =  =  ≅. 27 5 pe   −1      v (largest)  (0.0271 h )     Π. () pea3 =± 26 51 . pe The following data was obtained for plucked stage IV sporangiophores of P. blakesleeanus . The sporangi- ophores were plucked (carefully removed from the mycelium) so that they could be put in an environmental chamber where the water is removed and the air is maintained at 100% RH to eliminate transpiration. lm == 40 mm 41 × 0 µ dl −− 11 =. 55 µµ mmin = 330 mh dt   dl −1   v = =. 0 0083 h s   lt d  T4 == minh 0.067 1/2   ln2  ln2   −1   εφ =  = =. 10 3 h         T 0.067 h   1/2       εφ ln2 (0.693)       Π =  =  =  ≅ 1246    pe   −1         v  v T  (0.. 0083 hh )(0 067 )       ss 1/2 Estimates for Π from in vivo creep experiments. The magnitude of Π = (εφ/v ) is estimated for pe pe s pea stems, internode algal cells, and fungal sporangiophores (for Fig. 1). The following values were obtained for incised pea stems of Pisum satinis L. :   dl −1   v = = 0.± 0751 0.0043 () SE h s   lt d  −− 11 vh (largest)( =. 0 0751 +. 0 0043) =. 0 0794 h −− 11 vh (smallest)( =. 0 0751 −. 0 0043) =. 0 0667 h −− 11 φ =± 02 .. 50 017 hMPa −− 11 φ(largest)0 =.25 +. 0 017hMPa =. 0 267 −− 11 φ(smallest)0 =.25 −. 0 017 hMPa =. 0 233 ε =. 95 MPa −−     εφ (9.. 5 MPah )(0 25 MPa )    Π. (mean) =  =  ≈ 31 6   pe   −1    v     (0.0751 h )   −− 11     εφ(largest) (9.. 5 MPah )(0 267 MPa )    Π (largest) =  = ≈ 38   pe  −1     v (smallest)  (0.0667 h )     −− 11     εφ(smallest) (9.. 5 MPah )(0 233 MPa )     Π (smallest) =  =  ≈. 27 9   pe  −       v (largest)     (0.0794 h )  Π () pea3 ≈± 26 pe 11, 12 e f Th ollowing values were used for incised internode algal cells of C. corallina : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 10 www.nature.com/scientificreports/ lm == 13 mm 13 .× 10 µ dl −− 11 =. 0 014 µµ ms =. 50 4 mh dt   dl −− 61 −1   v = =. 11 ×= 10 sh 0.0039 s     lt d −1     dl 50.4 µmh −− 11     m =  =  = 285 µmh MPa         − () PP dt  (0.− 527 0.35) MPa     −− 11    m  285 µmh MPa −− 11   φ =  =  =. 0 022 hMPa  4       l  13 .× 10 µm   ε = 100 MPa −− 11     (100 MPah )(0.022 MPa ) εφ    Π =  =  ≅ 564   pe   −1      v     (0.0039 h )  4, 6 e f Th ollowing values were obtained for intact stage IV sporangiophores of Phycomyces blakesleeanus : lm == 30 mm 30 .× 10 µ dl −1 =± 34 31 . () SE μmmin () n2 = 0 dt    dl  34.+ 13.1m µ /min −− 11   v (largest) = =  =. 0 0012 minh =. 0 074 s         lt d  30000µ m      dl 34.− 31 . µ m/min −− 11   v (smallest) =  = =. 0 0010 minh =. 0 062   s       lt d  30000 µ m  −− 11 mm =± 997 160 () SE μm2 in MPan () = 0 −− 11    m  997 + 160 µmmin MPa   φ(largest) =  =   4      l   30 .× 10 µm  −− 11 −− 11 =. 0 039 minMPa =. 23 hMPa −− 11   m − mmin MPa   997 160 µ   φ(smallest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 028 minMPa =. 17 hMPa ε =± 60.. 95 1S () E2 MPan () = 7 −− 11     εφ(largest) (66 MPah )(2.3) MPa    Π (largest) =  =  ≈ 2448   pe  −1     v (smallest)  (0.062 h )     −− 11     εφ (smallest) (558 .. MPah )(1 7) MPa    Π (smallest) =  =  ≈ 1282   pe   −1    v (largest)     (0.074 h )   Π () stageIV =± 1865 583 pe e f Th ollowing values were obtained for intact stage I sporangiophores of Phycomyces blakesleeanus : Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 11 www.nature.com/scientificreports/ lm == 20 mm 20 .× 10 µ dl −1 =± 71 .. 07 () SE μmmin () n1 = 7 dt     dl 71 .+ 07 . µ m/min −− 11     v (largest) = = =. 0 00039 minh =. 0 023   s     lt d  20000µ m       dl  71 .− 07 . µ m/min −− 11   v (smallest) = =  =. 0 00032 minh =. 0 019 s        lt d  20000µ m   −− 11 mm =± 116 18.8S () E1 μ minMPa () n = 7 −− 11    m  116 +. 18 8 µmmin MPa   φ(largest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00674 minMPa =. 0 4044 hMPa −− 11    m  116 −. 18 8 µmmin MPa φ(smallest) =   =   4      l   20 .× 10 µm  −− 11 −− 11 =. 0 00486 minMPa =. 029 hMPa ε =± 68.. 95 6S () E2 MPan () = 7 −− 11     .. εφ() largest (745 MPah )(0 40 MPa )    Π () largest =  =  ≈ 1568   pe   −1    vs () mallest     (0.019 h )   −− 11     εφ() smallest (633 .. MPah )(0 29 MPa )     Π () smallest =  =  ≈ 798 pe  −1     vl (argest)  (0.023 h )     Π () stageI =± 1183 385 pe v values for cells from pea stems using Π = 32. This data for pea stems was taken directly from s pe Cosgrove , for data labeled “Water” (for Fig. 2). The value for ε is located in the text. v is the measured relative sM volumetric growth rate and v is the calculated relative volumetric growth rate. sC −1 vh =± 0.. 0249 0 0022 () SE () n1 = 0 sM ε =. 95 MPa −− 1 1 φ =± 0.. 074 0 007hMPa   −− 11   εφ (9.. 5 MPaM )(0 074 +. 0 007 Pa h )   −1   vh (largest) = =  ≅ 0.024 sC     Π  32     pe  −− 11     εφ (9.. 5 MPaM )(0 074 −. 0 007 Pa h )   −1   vh (smallest) =  =  ≅ 0.020 sC     Π 32      pe  −1 vh =± 0.. 022 0 002 sC This data for pea stems was taken directly from Cosgrove , for data labeled “IAA”. The value for ε that is located in the text. −1 vh =± 0.. 067 0 004 () SE () n8 = sM ε =. 95 MPa −− 11 φ =± 02 .. 20 014 hMPa   −− 11   εφ (9.. 5 MPaM )(0 22 +. 0 014 Pa h )   −1   vh (largest) = = ≅ 0.070   sC    Π  32     pe    −− 11   εφ (9.. 5 MPaM )(0 22 −. 0 0147 Pa h )   −1   vh (smallest) =  =  ≅ 0.061 sC     Π   32    pe  −1 vh =± 0.. 0655 0 0045 sC This data for pea stems was taken directly from Cosgrove , for data labeled “Cut directly form plant”. The value for ε is located in the text. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 12 www.nature.com/scientificreports/ −1 vh =± 0.. 075 0 004 () SE () n1 = 0 sM ε =. 95MPa −− 11 φ =± 02 .. 50 017hMPa   −− 11   εφ (9.. 5 MPaM )(0 25 +. 0 017 Pa h )   −1 vh (largest) =  =  ≅ 0.079 sC     Π   32    pe  −− 11     (9.. 5 MPaM )(0 25 −. 0 017 Pa h ) εφ   −1   vh (smallest) =  =  ≅ 0.069 sC     Π 32      pe  −1 vh =± 0.. 074 0 005 sC v values for sporangiophores using Π = 1524. The data for plucked stage IV sporangiophore was sC pe 4 6 taken from Ortega et al. and Ortega . v is the measured relative volumetric growth rate and v is the calculated sM sC relative volumetric growth rate (for Fig. 3). For v , dL/dt is measured and then divided by l. sM lm == 40 mm 41 × 0 µ dl −− 11 =. 55 µµ mmin = 330 mh dt   dl −1   vh = = 0.008 sM   lt d  ε = 61MPa −1     εφ 10.3 h −− 11   φ = =  =. 0 1688 MPah        ε   61 MPa    −− 11   εφ (61 MPaM )(0.17 Pa h )   −1   vh = =  ≅ 0.007 sC     Π  1524     pe  4 6 e d Th ata for intact stage IV sporangiophores were taken from Ortega et al. and Ortega . lm == 30 mm 31 × 0 µ dl −− 11 =. 34 13 ±.1(SE) µµ mmin =± 2046 186 (SE) mh (n = 20) dt   dl −1   v = =± 0.. 068 0 006 () SE h () n2 = 0 sM   lt d  ε =± 60.. 95 1S () E2 MPan () = 7 −− 11    m  997 + 160 mmin MPa   φ(largest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 039 minMPa =. 23 hMPa −− 11    m  997 − 160 µmmin MPa   φ(smallest) =  =   4       l  30 .× 10 µm   −− 11 −− 11 =. 0 028 minMPa =. 17 hMPa −− 11     (66 MPaM )(2.3) Pa h εφ   −1   vh (largest) =  =  ≅ 0.099 sC     Π 1524      pe  −− 11     εφ (558 .. MPaM )(1 7) Pa h   −1 vh (smallest) =  =  ≅ 0.062 sC       Π  1524     pe  −1 vh =± 0.. 0805 0 0185 sC e d Th ata for intact stage I sporangiophores were taken from Ortega . Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 13 www.nature.com/scientificreports/ lm == 20 mm 21 × 0 µ dl −− 11 =. 71 ±. 07 (SE) µµ mmin =± 426 42 (SE) mh (n = 20) dt   dl −1   v = =± 0.. 021 0 002 () SE h () n1 = 7 sM     lt d ε =± 68.. 95 6S () E2 MPan () = 7 −− 11    m  116 µmmin MPa −− 11 −1   φ =  = (60min hM )0 =.348 Pa h  4       l  21 × 0 µm   −− 11    m  116 +. 18 8 µmmin MPa   φ(largest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00674 minMPa =. 0 4044 hMPa −− 11    m  116 −. 18 8 µmmin MPa   φ(smallest) = =   4      l  20 .× 10 µm   −− 11 −− 11 =. 0 00486 minMPa =. 029 hMPa −− 11     εφ (745 .. MPaM )(0 40 Pa h )   −1 vh (largest) =  =  ≅ 0.020 sC     Π  1524     pe    −− 11   εφ (633 .. MPaM )(0 29 Pa h )   −1   vh (smallest) = =  ≅ 0.012 sC     Π  1524     pe  −1 vh =± 0.. 016 0 004 sC Constant-tension extension experiment - load removed. Unidirectional constant-tension extension experiments were conducted on frozen and then thawed walls of the stage IV sporangiophore using the same method and experimental apparatus used and described by Ortega et al. (for Fig. 4). A five-millimeter long sec- tion of the wall from a stage IV sporangiophore, that includes the growth zone, was adapted to a bathing solution of pure water for 20 min in a test apparatus. Then a tensile load of 1.24 gram was applied to the wall. The tensile load produces longitudinal stress and longitudinal extension of the wall (810 μm). Aer ft the extension, the length remained constant for four minutes (11–15 min). At 15 min (marked by the downward pointing arrow), 50 mM potassium-acetate buffer, pH 4.5, replaces the bathing water in the test apparatus. It can be seen that “creep” extension begins immediately and continues for seven minutes. The creep extension behavior produced by the 50 mM potassium-acetate buffer, pH 4.5, is very similar to that obtained when the pH of the solution was lower to 4.6 by adding a predetermined amount of pH Red 4.0 buffer to the bathing water . Aer ft wards (22–25 min), the wall section length remains constant for three minutes. At 25 min, the tensile load of 1.24 grams (σ = 42 MPa; see Ortega et al. for detailed stress calculations) is slowly removed and the wall section decreases in length over the next three minutes. The decrease in length is “recovered” elastic extension. e d Th ata obtained from this experiment is used for the following calculations. When σ is constant (11–25 min) the change in length (creep) stimulated by the decrease in pH is governed by; v = (dl/dt)/l = φ (σ − σ ), and s C φ = v /(σ − σ ). When the stress is removed (at 25 min), then the decrease in length is governed by; −(dl/dt)/l = − s C (1/ε) dσ/dt, and ε = Δσ/(Δl/l). From the experiment and Fig. 4 we get: l =. 50 mm == 5000 µmlengthofcellwallbeing tested −1 dl/1 dt == 7m µ minaverage creep extensionrateduring7 mininterval −− 11 vd == (/ ldtl )/ 17 µµ mmin /5000 m0 =.0034 min σσ −= 42 MPa −. 16 3MPa =. 25 7MPa = effective stress during creep extension −− 41 −1 φσ =− v /( σ )1 =.31 × 0min MPa sC εσ =Δ /(Δ= ll /) (4 −− 2MPa)/(62/ µµ mm 5000 ) = 3387 MPa εφ Π =≈ 130 pe References 1. Lockhart, J. A. An analysis of irreversible plant cell elongation. J Theor Biol. 8, 264–275, doi:10.1016/0022-5193(65)90077-9 (1965). 2. Green, P. B. Cell Morphogenesis. Ann Rev Plant Physiol. 20, 365–394, doi:10.1146/annurev.pp.20.060169.002053 (1969). 3. Green, P. 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E., Bell, S. A. & Erazo, A. J. Pressure clamp method to measure transpiration in growing single plant cells: Demonstration with sporangiophores of Phycomyces. Plant Physiol 100, 1036–1041, doi:10.1104/pp.100.2.1036 (1992). Acknowledgements This research was support by NSF Grant, MCB-0948921 to J.K.E. Ortega. The author acknowledges and thanks Cindy M. Munoz and Jason T. Truong for conducting the experiments whose results are presented in Figure 4. Author Contributions J.K.E.O. conceived the ideas, conducted the analyses, conducted the calculations and estimates, and wrote the manuscript. Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 15 www.nature.com/scientificreports/ Additional Information Supplementary information accompanies this paper at doi:10.1038/s41598-017-03002-6 Competing Interests: The authors declare that they have no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Cre- ative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not per- mitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. © The Author(s) 2017 Scientific Repo R ts | 7: 3016 | DOI:10.1038/s41598-017-03002-6 16

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