- Email: [email protected]
Dimensionless Numbers for Plant Biology
since 2010 and the simultaneous reduction to synonymy of a similar number of genera, based on new insights into phylogenetic history. Species number increased from 3709 to 3973, which corresponds to a net increase of 4% of accepted species names caused by new species description as well as recognition of synonymized names as valid species names. Many synonyms could only be detected by inspecting the original vouchers. However, inspecting vouchers often revealed that these indeed represented well-separated taxa, which had been overlooked and are therefore not easily accessible. This agrees with the estimation that herbaria might already be housing 50% of the approximately 70 000 species of flowering plants yet to be described [6]. The latest release of BrassiBase (August 1, 2017, version 1.2a) reports taxonomic changes since 2006 (mean: 3.86 changes/corrections per species) affecting any evolutionary lineage or taxonomic level, and includes model systems like Arabidopsis, Brassica, and Eutrema/ Thellungiella, with 1.2, 0.9, and 0.3 changes per species, respectively. This is an important point, when generally focusing on ‘model systems’: Taxonomy matters (also see Taxonomics) and is often still not solved in model plants [7]. In the case of Arabidopsis, critical taxonomic evaluation led to a genuine and comprehensive taxon sampling providing the first whole nuclear genomebased phylogeny of an entire plant genus [8], which probably will result in additional taxonomic changes in the near future. Often, taxonomists are criticized for not presenting a final framework of names, in particular for model species groups, which should be, of course, hierarchically structured (family, tribe, genus, species, subspecies). The request for a final taxonomic system is understandable, but further insight will also have large effects on taxonomy in the future and may result in
6
Trends in Plant Science, January 2018, Vol. 23, No. 1
new taxonomic groupings with either new Acknowledgments names or the reuse of available ‘old’ We thank Doug Soltis and Christiane Kiefer for helpful names. With BrassiBase we aim at releas- discussions while drafting the manuscript, and we are ing checklist versions with a unique ‘ver- very grateful to Ishan Al-Shehbaz for his continuous collaboration, support, and all of his contributions on sion code’, while continuously working on Brassicaceae taxonomy and dedicate this contribuupdates. Synonymous taxon names must tion to him. M.A.K was supported by the German be kept and carefully documented, Research Foundation (DFG) grant KO2302/13. because they are often the only link to 1 important previous knowledge (e.g., pro- Centre for Organismal Studies (COS) Heidelberg, Biodiversity and Plant Systematics/Botanical Garden and vided by respective herbarium vouchers). Herbarium (HEID), Heidelberg University, Heidelberg, Germany
Plant family-wide databases that are not focusing on crop plants and that are well curated are rare. Successful platforms such as eMONOCOT or GrassBase (KEW Gardens, London/UK) are driven by experts in taxonomy and systematics and are maintained by respective institutions. However, these databases do not provide data that can be explored for developing research hypotheses or for directly performing research. Tools such as GBIF (Global Biodiversity Information Facility) allow exploring worldwide distribution data sets, but information retrieved through those is neither curated nor critically assessed. In summary, there is an urgent need for linking species-specific information to accepted names. Several global and major activities try to name, catalog, and group plant biodiversity [e.g., iDigBio, https://www.idigbio.org/; International Plant Names Index (IPNI), http://www. ipni.org/] and to link this information with knowledge focusing on distribution information, morphological descriptions, and bibliographies, but also to integrate spatial environmental and phylogenetic data [9]. Critical to all of these databases are the respective species checklists, which are often either missing or not consistent among regions of the world. As a consequence, approaches have been developed to automatically select and propagate ‘accepted names’ (The Plant List, http://www.theplantlist.org/) and to use those for the backbone of respective plant biodiversity databases, which is often not the best choice as shown here for the Brassicaceae.
*Correspondence: [email protected] (M.A. Koch). https://doi.org/10.1016/j.tplants.2017.10.005 References 1. Patterson, D.J. et al. (2010) Names are key to the big new biology. Trends Ecol. Evol. 25, 686–691 2. Hinchliff, C.E. et al. (2015) Synthesis of phylogeny and taxonomy into a comprehensive tree of life. Proc. Natl. Acad. Sci. U. S. A. 112, 12764–12769 3. Al-Shehbaz, I.A. et al. (2006) Systematics and phylogeny of the Brassicaceae (Cruciferae): an overview. Plant Syst. Evol. 259, 89–120 4. Koch, M.A. et al. (2012) BrassiBase: tools and biological resources to study characters and traits in the Brassicaceae – version 1.1. Taxon 61, 1001–1009 5. Warwick, S.I. et al. (2006) Brassicaceae: species checklist and database on CD-Rom. Plant Syst. Evol. 259, 249–258 6. Bebber, D.P. et al. (2010) Herbaria are a major frontier for species discovery. Proc. Natl. Acad. Sci. U. S. A. 107, 22169–22171 7. Koch, M.A. and German, D.A. (2013) Taxonomy and systematics are key to biological information: Arabidopsis, Eutrema (Thellungiella), Noccaea and Schrenkiella (Brassicaceae) as examples. Front. Plant Sci. 4, 267 8. Novikova, P. et al. (2016) Sequencing of the genus Arabidopsis reveals a complex history of nonbifurcating speciation and abundant trans-specific polymorphism. Nat. Genet. 48, 1077–1082 9. Soltis, E.S. and Soltis, P.S. (2016) Mobilizing and integrating big data in studies of spatial and phylogenetic patterns of biodiversity. Plant Diversity 38, 264–270
Forum
Dimensionless Numbers for Plant Biology Joseph K.E. Ortega1,* Dimensionless numbers are ubiquitous in the physical sciences because they provide insight into physical processes, organize large
quantities of data, facilitate ‘scale L PC relative volumetric water uptake rate analysis’ and establish ‘similarity’. ðaÞ Pwv ¼ ¼ relative volumetric growth rate vs Here I explore the use of dimensionless numbers in plant biology, focusing on the expansive growth vsT relative volumetric transpiration rate ¼ ðbÞ PTv ¼ rate of plant, fungal, and algal cells. relative volumetric growth rate vs Dimensionless numbers are ubiquitous in engineering and the physical sciences. Generally, dimensionless numbers can be obtained using the Buckingham Pi Theorem (see Glossary) and/or from governing equations using dimensional analysis [1]. Experiments are conducted to obtain relevant data. Then, the data are used to calculate the magnitude of the dimensionless number (P) (Figure 1). P can be used to: (i) provide insight into processes of interest; (ii) establish simple relationships between relevant variables; (iii) reduce the number of graphs needed to correlate data; (iv) conduct ‘scale analyses’; and (v) establish ‘similarity’ between processes. Here I explore how dimensionless numbers can benefit research in plant biology, focusing on the expansive growth rate of walled cells. Expansive growth rates of plant, fungal, and algal cells depend on two interrelated biophysical processes: the net wateruptake rate (osmotic water uptake rate minus transpiration rate) and the deformation rates of the cell walls [i.e., the irreversible (plastic) deformation rate plus the reversible (elastic) deformation rate]. Governing equations for these biophysical processes have been derived and established [i.e., the Augmented Growth Equations (AGEs)] [2–4] (Box 1). Dimensional analyses conducted on the AGEs identified seven dimensionless numbers (a–g) [2]. Dimensional analysis provides a physical interpretation for the dimensionless P parameters as the ratio of two biophysical processes. The first subscript of the Ps refers to the numerator and the second subscript refers to the denominator. So, for Pwv (a), the subscript ‘w’ refers to the relative volumetric water uptake rate and the subscript ‘v’ refers to the relative volumetric growth rate.
f PC relative volumetric plastic deformation rate of the wall ¼ vs relative volumetric growth rate
ðcÞ Ppv ¼
ðdÞ Pev ¼
PC e
ðeÞ Pwe ¼ ðfÞ PTe ¼
ðgÞ Ppe ¼
eL vs
¼
e vsT PC vs
relative volumetric elastic deformation rate of the wall relative volumetric growth rate
¼
relative volumetric water uptake rate relative volumetric elastic deformation rate of the wall
¼
relative volumetric transpiration rate relative volumetric elastic deformation rate of the wall
ef relative volumetric plastic deformation rate of the wall ¼ vs relative volumetric elastic deformation rate of the wall
The expansive growth rates of plant, fungal, and algal cells are directly related to the stress relaxation rates of their respective cell walls [4,5]. The dimensionless governing equation for wall stress relaxation can be obtained from the dimensionless AGEs [2,6]. It is shown that the dimensionless stress relaxation rate of the wall is controlled by the magnitude of the dimensionless number Ppe (g) [2,6].
elastic deformation rate of the walls during quasi-steady expansive growth for all three cell species. Second, the magnitudes of Ppe for these different cell species are very different from each other. This indicates that wall deformation rate behavior can be significantly different for walled cells from different species [6]. Third, it was found that the respective magnitudes of Ppe for pea stem cells of P. satinis L. (Ppe = 32) and sporangiophores of Phycomyces blakesleeanus (Ppe = 1524) Ppe Provides Insight into Wall remain constant even when f, e, and vs Stress Relaxation Rate and are altered during developmental stages Expansive Growth Rate The magnitude of Ppe has been deter- or due to changes in growth conditions or mined for plant cells in growing pea stems the addition of growth hormones [6]. of Pisum satinis L. (Ppe = 32), fungal sporangiophores of Phycomyces blakesleea- Ppe Establishes Simple nus (Ppe = 1524), and algal internode Relationships between Relevant cells of Chara corallina (Ppe = 564) [6]. Variables Three things are learned from these Ppe establishes a simple relationship dimensionless numbers. First, the magni- between f, e, and vs. Furthermore, tudes of Ppe for these different cell spe- because Ppe is constant for pea stem cies are very large – much greater than cells of P. satinis L. (Ppe = 32) and sporunity. This demonstrates that the plastic angiophores of P. blakesleeanus deformation rate is much greater than the (Ppe = 1524), the AGEs do not have to
Trends in Plant Science, January 2018, Vol. 23, No. 1
7
Insight into processes
Conduct experiments to obtain relevant biophysical variables
∏
Glossary
Establish simple relaonship between variables Reduce number of graphs needed to correlate data Conduct ‘scale analyses’
Establish ‘similarity’
Figure 1. The Dimensionless Number P (Red Box) Is Calculated from a Combination of Relevant Biophysical Variables Obtained from Experiments (Blue Box). Once P is calculated it can be used for multiple tasks (green boxes): to obtain insight into the process of interest; to establish relationships between variables; to reduce the number of graphs needed to correlate data; to conduct ‘scale analyses’; and to establish ‘similarity’.
be solved to determine the relative expansive growth rate, vs, when f or e changes. Instead, vs may be obtained from the following relationships: vs = fe/32 for pea stem cells of P. satinis L.; and vs = fe/1524 for sporangiophores of P. blakesleeanus [6].
data presented for growing pea stem cells of P. satinis L. [7] and sporangiophores of P. blakesleeanus [8,9] can be presented and summarized in simple bar graphs using Ppe [6].
Dimensionless Numbers Can Assist in Scale Analyses
Ppe Reduces the Number of Inherent in the dimensionless numbers Graphs Needed to Correlate Data (a–g) is a comparison of the relative magBecause Ppe establishes a simple rela- nitudes of different biophysical protionship between f, e, and vs, most of the cesses. For example, Ppe = 32 indicates Box 1. Dimensional and Dimensionless AGEs AGEs (Equations I, II, and III) [3,4] Equation I describes, in relative terms, the rate of change in water volume in the cell, vw, as the difference between the volumetric rate of water uptake, L (Dp P) and transpiration, vT. vw ¼ LðDpPÞ vT
½I
Equation II describes, in relative terms, the rate of change in volume of the cell wall chamber, vcw, as the sum of the volumetric irreversible (plastic) deformation rate f(P PC) and the volumetric reversible (elastic) deformation rate, (1/e) dP/dt, of the cell wall. 1 dP ½II vcw ¼ fðPPC Þ þ e dt Equation III describes the rate of change of turgor pressure, P. 1 dP ¼ LðDpPÞ vT fðPPC Þ e dt
½III
Dimensionless AGEs (Equations IV, V, and VI) [2,6] Dimensionless variables are designated with an asterisk (*). Equations IV, V, and VI are the dimensionless forms of Equations I, II, and III, respectively. The coefficients (P parameters) at the beginning of each term in Equations IV, V, and VI correspond to the P parameters in the text (a–g). vw ¼ Pwv ðDp P Þ PTv vT
½IV
dP vcw ¼ Ppv ðP 1Þ þ Pev dt
½V
dP ¼ Pwe ðDp P Þ PTe vT Ppe ðP 1Þ dt
½VI
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Trends in Plant Science, January 2018, Vol. 23, No. 1
Buckingham Pi Theorem: a formalized procedure for the deduction of dimensionless groups A: area of the plasma membrane LP: hydraulic conductivity of the plasma membrane L A L: Vp ¼ relative hydraulic conductance of the plasma membrane P: turgor pressure (gage pressure) relative to the atmosphere PC: critical turgor pressure (to be exceeded before plastic extension begins) t: time V: volume Vcw: volume of the cell wall chamber Vw: volume of water in the cell VT:volume of water lost through transpiration v: VdVdt ¼ relative rate of change in volume of the cell cw vcw: VdV ¼ relative rate of change cw dt in volume of the cell wall chamber vs: VdVdt ¼ steady or quasi steady relative rateof change in volume of the cell T ¼ relative rate of change vT: VdV T dt in water lost via transpiration volume w ¼ vw: VdV w dt relative rate of change in water volume in the cell e: volumetric elastic modulus of the cell wall f: relative irreversible extensibility of the cell wall P: dimensionless number Dp: osmotic pressure difference across the plasma membrane L(Dp P): relative volumetric rate of water uptake f(P PC): relative volumetric plastic deformation rate of the cell wall dP e dt : relative volumetric elastic deformation rate of the cell wall
that the ratio of the relative volumetric plastic and elastic deformation rates of the cell wall is 32. Importantly, other ratios may be obtained from the dimensionless numbers. For example, if you want a comparison of the magnitudes of the relative volumetric water uptake rate and the relative volumetric plastic deformation rate of the cell wall, it may be determined by the ratio, Pwv/Ppv = Pwp = L/f. Comparisons of the sums and differences of processes are easily determined with dimensionless numbers. For example, if a comparison of the relative magnitudes of the net water uptake rate and the wall deformation rate is wanted, it may be obtained as follows: (Pwv PTv)/ (Ppv + Pev) = Pwd [6]. Pwd was determined for growing pea stem cells of P.
satinis L. (Pwd = 7.8) and stage IV spor- that the magnitude of Ppe for the creep- Acknowledgment angiophores of P. blakesleeanus ing frozen–thawed walls in a low-pH Much of the research with dimensional analyses and (Pwd = 11.7) [6]. buffer is an order of magnitude smaller dimensionless numbers was supported by NSF grant than that of naturally growing stage IV MCB-0948921 to J.K.E.O. sporangiophores [6]. Thus, lowering the 1Department of Mechanical Engineering, University of Dimensionless Numbers Can Be pH alone cannot by itself produce the Colorado Denver, 1200 Larimer Street, PO Box 173364, Used to Determine Similarity Similarity between processes can be wall deformation similarity that produces Denver, CO 80217-3364, USA achieved by ensuring that the magnitudes the wall stress relaxation observed dur- *Correspondence: [email protected] (J.K.E. Ortega). of the relevant dimensionless numbers ing expansive growth. https://doi.org/10.1016/j.tplants.2017.09.020 are identical [1]. For example, ‘wall deformation similarity’ for growing walled cells Future Research References is achieved when the magnitude of Ppe is In the future dimensionless numbers can 1. Fox, R.W. et al., eds (2003) Introduction to Fluid Mechanics (6th edn), John Wiley and Sons the same in two different walls because be used to explore other processes in 2. Ortega, J.K.E. (2016) Dimensional analysis of expansive wall stress relaxation is central to expan- plant biology. For example, the dimengrowth of cells with walls. Res. Rev. J. Bot. Sci. 5, 17–24 sive growth [5]. Constant-tension exten- sionless numbers Pwv, PTv, Pwe, and 3. Ortega, J.K.E. (1990) Governing equations for plant cell growth. Physiol. Plant. 79, 116–121 sion experiments have been used to PTe could be used for research concern4. Geitmann, A. and Ortega, J.K.E. (2009) Mechanics and study how frozen–thawed cell walls are ing the effects of drought, climate modeling of plant cell growth. Trends Plant Sci. 14, 467–478 loosened by various pH buffers and pro- change, and water stress on plant cell 5. Cosgrove, D.J. (1993) How do plant cell walls extend? Plant Physiol. 102, 1–6 teins to produce creep [10]. Constant- growth rate. In addition, new dimension6. Ortega, J.K.E. (2017) Dimensionless number is central to tension extension experiments con- less numbers can be obtained and used stress relaxation and expansive growth of the cell wall. Sci. Rep. 7, 3016 ducted on frozen–thawed cell walls of to explore changes in active solute constage IV sporangiophores of P. blake- centrations and osmotic pressure and 7. Cosgrove, D.J. (1985) Cell wall yield properties of growing tissue: evaluation by in vivo stress relaxation. Plant Physiol. sleeanus demonstrate that lowering the the mass transport of carbohydrates 78, 347–356 pH of the bathing solution produces (and other materials) to the cell wall. Scale 8. Ortega, J.K.E. et al. (1989) In vivo creep and stress relaxation experiments to determine the wall extensibility and creep rates that are within the range of analysis has been shown to have great yield threshold for the sporangiophores of Phycomyces. Biophys. J. 56, 465–475 the elongation growth rates observed in utility in plant sciences [12]. As shown vivo [11]. So, is lowering the pH in the here and previously [6], dimensionless 9. Ortega, J.K.E. (2004) A quantitative biophysical perspective of expansive growth for cells with walls. In Recent wall the mechanism by which the spor- numbers can facilitate scale analyses of Research Developments in Biophysics (Vol. 3) (Pandalai, S. G., ed.), In pp. 297–324, Transworld Research Network angiophores loosen their walls to pro- biophysical processes relevant to the 10. Cosgrove, D.J. (1989) Characterization of long-term duce wall deformation during expansive growth rate of walled cells. extension of isolated cell wall from growing cucumber expansive growth? When the protocol The use of dimensionless numbers in hypocotyls. Planta 177, 121–130 for the constant-tension extension scale analysis of plant processes 11. Ortega, J.K.E. et al. (2015) Cell wall loosening in the fungus, Phycomyces blakesleeanus. Plants 4, 63–84 experiments was modified so that Ppe deserves more consideration in future 12. Niklas, K.J. (1994) Plant Allometry: The Scaling of Form could be determined, the results show research. and Process, University of Chicago Press
Trends in Plant Science, January 2018, Vol. 23, No. 1
9
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Trends in Plant Science, January 2018, Vol. 23, No. 1
new taxonomic groupings with either new Acknowledgments names or the reuse of available ‘old’ We thank Doug Soltis and Christiane Kiefer for helpful names. With BrassiBase we aim at releas- discussions while drafting the manuscript, and we are ing checklist versions with a unique ‘ver- very grateful to Ishan Al-Shehbaz for his continuous collaboration, support, and all of his contributions on sion code’, while continuously working on Brassicaceae taxonomy and dedicate this contribuupdates. Synonymous taxon names must tion to him. M.A.K was supported by the German be kept and carefully documented, Research Foundation (DFG) grant KO2302/13. because they are often the only link to 1 important previous knowledge (e.g., pro- Centre for Organismal Studies (COS) Heidelberg, Biodiversity and Plant Systematics/Botanical Garden and vided by respective herbarium vouchers). Herbarium (HEID), Heidelberg University, Heidelberg, Germany
Plant family-wide databases that are not focusing on crop plants and that are well curated are rare. Successful platforms such as eMONOCOT or GrassBase (KEW Gardens, London/UK) are driven by experts in taxonomy and systematics and are maintained by respective institutions. However, these databases do not provide data that can be explored for developing research hypotheses or for directly performing research. Tools such as GBIF (Global Biodiversity Information Facility) allow exploring worldwide distribution data sets, but information retrieved through those is neither curated nor critically assessed. In summary, there is an urgent need for linking species-specific information to accepted names. Several global and major activities try to name, catalog, and group plant biodiversity [e.g., iDigBio, https://www.idigbio.org/; International Plant Names Index (IPNI), http://www. ipni.org/] and to link this information with knowledge focusing on distribution information, morphological descriptions, and bibliographies, but also to integrate spatial environmental and phylogenetic data [9]. Critical to all of these databases are the respective species checklists, which are often either missing or not consistent among regions of the world. As a consequence, approaches have been developed to automatically select and propagate ‘accepted names’ (The Plant List, http://www.theplantlist.org/) and to use those for the backbone of respective plant biodiversity databases, which is often not the best choice as shown here for the Brassicaceae.
*Correspondence: [email protected] (M.A. Koch). https://doi.org/10.1016/j.tplants.2017.10.005 References 1. Patterson, D.J. et al. (2010) Names are key to the big new biology. Trends Ecol. Evol. 25, 686–691 2. Hinchliff, C.E. et al. (2015) Synthesis of phylogeny and taxonomy into a comprehensive tree of life. Proc. Natl. Acad. Sci. U. S. A. 112, 12764–12769 3. Al-Shehbaz, I.A. et al. (2006) Systematics and phylogeny of the Brassicaceae (Cruciferae): an overview. Plant Syst. Evol. 259, 89–120 4. Koch, M.A. et al. (2012) BrassiBase: tools and biological resources to study characters and traits in the Brassicaceae – version 1.1. Taxon 61, 1001–1009 5. Warwick, S.I. et al. (2006) Brassicaceae: species checklist and database on CD-Rom. Plant Syst. Evol. 259, 249–258 6. Bebber, D.P. et al. (2010) Herbaria are a major frontier for species discovery. Proc. Natl. Acad. Sci. U. S. A. 107, 22169–22171 7. Koch, M.A. and German, D.A. (2013) Taxonomy and systematics are key to biological information: Arabidopsis, Eutrema (Thellungiella), Noccaea and Schrenkiella (Brassicaceae) as examples. Front. Plant Sci. 4, 267 8. Novikova, P. et al. (2016) Sequencing of the genus Arabidopsis reveals a complex history of nonbifurcating speciation and abundant trans-specific polymorphism. Nat. Genet. 48, 1077–1082 9. Soltis, E.S. and Soltis, P.S. (2016) Mobilizing and integrating big data in studies of spatial and phylogenetic patterns of biodiversity. Plant Diversity 38, 264–270
Forum
Dimensionless Numbers for Plant Biology Joseph K.E. Ortega1,* Dimensionless numbers are ubiquitous in the physical sciences because they provide insight into physical processes, organize large
quantities of data, facilitate ‘scale L PC relative volumetric water uptake rate analysis’ and establish ‘similarity’. ðaÞ Pwv ¼ ¼ relative volumetric growth rate vs Here I explore the use of dimensionless numbers in plant biology, focusing on the expansive growth vsT relative volumetric transpiration rate ¼ ðbÞ PTv ¼ rate of plant, fungal, and algal cells. relative volumetric growth rate vs Dimensionless numbers are ubiquitous in engineering and the physical sciences. Generally, dimensionless numbers can be obtained using the Buckingham Pi Theorem (see Glossary) and/or from governing equations using dimensional analysis [1]. Experiments are conducted to obtain relevant data. Then, the data are used to calculate the magnitude of the dimensionless number (P) (Figure 1). P can be used to: (i) provide insight into processes of interest; (ii) establish simple relationships between relevant variables; (iii) reduce the number of graphs needed to correlate data; (iv) conduct ‘scale analyses’; and (v) establish ‘similarity’ between processes. Here I explore how dimensionless numbers can benefit research in plant biology, focusing on the expansive growth rate of walled cells. Expansive growth rates of plant, fungal, and algal cells depend on two interrelated biophysical processes: the net wateruptake rate (osmotic water uptake rate minus transpiration rate) and the deformation rates of the cell walls [i.e., the irreversible (plastic) deformation rate plus the reversible (elastic) deformation rate]. Governing equations for these biophysical processes have been derived and established [i.e., the Augmented Growth Equations (AGEs)] [2–4] (Box 1). Dimensional analyses conducted on the AGEs identified seven dimensionless numbers (a–g) [2]. Dimensional analysis provides a physical interpretation for the dimensionless P parameters as the ratio of two biophysical processes. The first subscript of the Ps refers to the numerator and the second subscript refers to the denominator. So, for Pwv (a), the subscript ‘w’ refers to the relative volumetric water uptake rate and the subscript ‘v’ refers to the relative volumetric growth rate.
f PC relative volumetric plastic deformation rate of the wall ¼ vs relative volumetric growth rate
ðcÞ Ppv ¼
ðdÞ Pev ¼
PC e
ðeÞ Pwe ¼ ðfÞ PTe ¼
ðgÞ Ppe ¼
eL vs
¼
e vsT PC vs
relative volumetric elastic deformation rate of the wall relative volumetric growth rate
¼
relative volumetric water uptake rate relative volumetric elastic deformation rate of the wall
¼
relative volumetric transpiration rate relative volumetric elastic deformation rate of the wall
ef relative volumetric plastic deformation rate of the wall ¼ vs relative volumetric elastic deformation rate of the wall
The expansive growth rates of plant, fungal, and algal cells are directly related to the stress relaxation rates of their respective cell walls [4,5]. The dimensionless governing equation for wall stress relaxation can be obtained from the dimensionless AGEs [2,6]. It is shown that the dimensionless stress relaxation rate of the wall is controlled by the magnitude of the dimensionless number Ppe (g) [2,6].
elastic deformation rate of the walls during quasi-steady expansive growth for all three cell species. Second, the magnitudes of Ppe for these different cell species are very different from each other. This indicates that wall deformation rate behavior can be significantly different for walled cells from different species [6]. Third, it was found that the respective magnitudes of Ppe for pea stem cells of P. satinis L. (Ppe = 32) and sporangiophores of Phycomyces blakesleeanus (Ppe = 1524) Ppe Provides Insight into Wall remain constant even when f, e, and vs Stress Relaxation Rate and are altered during developmental stages Expansive Growth Rate The magnitude of Ppe has been deter- or due to changes in growth conditions or mined for plant cells in growing pea stems the addition of growth hormones [6]. of Pisum satinis L. (Ppe = 32), fungal sporangiophores of Phycomyces blakesleea- Ppe Establishes Simple nus (Ppe = 1524), and algal internode Relationships between Relevant cells of Chara corallina (Ppe = 564) [6]. Variables Three things are learned from these Ppe establishes a simple relationship dimensionless numbers. First, the magni- between f, e, and vs. Furthermore, tudes of Ppe for these different cell spe- because Ppe is constant for pea stem cies are very large – much greater than cells of P. satinis L. (Ppe = 32) and sporunity. This demonstrates that the plastic angiophores of P. blakesleeanus deformation rate is much greater than the (Ppe = 1524), the AGEs do not have to
Trends in Plant Science, January 2018, Vol. 23, No. 1
7
Insight into processes
Conduct experiments to obtain relevant biophysical variables
∏
Glossary
Establish simple relaonship between variables Reduce number of graphs needed to correlate data Conduct ‘scale analyses’
Establish ‘similarity’
Figure 1. The Dimensionless Number P (Red Box) Is Calculated from a Combination of Relevant Biophysical Variables Obtained from Experiments (Blue Box). Once P is calculated it can be used for multiple tasks (green boxes): to obtain insight into the process of interest; to establish relationships between variables; to reduce the number of graphs needed to correlate data; to conduct ‘scale analyses’; and to establish ‘similarity’.
be solved to determine the relative expansive growth rate, vs, when f or e changes. Instead, vs may be obtained from the following relationships: vs = fe/32 for pea stem cells of P. satinis L.; and vs = fe/1524 for sporangiophores of P. blakesleeanus [6].
data presented for growing pea stem cells of P. satinis L. [7] and sporangiophores of P. blakesleeanus [8,9] can be presented and summarized in simple bar graphs using Ppe [6].
Dimensionless Numbers Can Assist in Scale Analyses
Ppe Reduces the Number of Inherent in the dimensionless numbers Graphs Needed to Correlate Data (a–g) is a comparison of the relative magBecause Ppe establishes a simple rela- nitudes of different biophysical protionship between f, e, and vs, most of the cesses. For example, Ppe = 32 indicates Box 1. Dimensional and Dimensionless AGEs AGEs (Equations I, II, and III) [3,4] Equation I describes, in relative terms, the rate of change in water volume in the cell, vw, as the difference between the volumetric rate of water uptake, L (Dp P) and transpiration, vT. vw ¼ LðDpPÞ vT
½I
Equation II describes, in relative terms, the rate of change in volume of the cell wall chamber, vcw, as the sum of the volumetric irreversible (plastic) deformation rate f(P PC) and the volumetric reversible (elastic) deformation rate, (1/e) dP/dt, of the cell wall. 1 dP ½II vcw ¼ fðPPC Þ þ e dt Equation III describes the rate of change of turgor pressure, P. 1 dP ¼ LðDpPÞ vT fðPPC Þ e dt
½III
Dimensionless AGEs (Equations IV, V, and VI) [2,6] Dimensionless variables are designated with an asterisk (*). Equations IV, V, and VI are the dimensionless forms of Equations I, II, and III, respectively. The coefficients (P parameters) at the beginning of each term in Equations IV, V, and VI correspond to the P parameters in the text (a–g). vw ¼ Pwv ðDp P Þ PTv vT
½IV
dP vcw ¼ Ppv ðP 1Þ þ Pev dt
½V
dP ¼ Pwe ðDp P Þ PTe vT Ppe ðP 1Þ dt
½VI
8
Trends in Plant Science, January 2018, Vol. 23, No. 1
Buckingham Pi Theorem: a formalized procedure for the deduction of dimensionless groups A: area of the plasma membrane LP: hydraulic conductivity of the plasma membrane L A L: Vp ¼ relative hydraulic conductance of the plasma membrane P: turgor pressure (gage pressure) relative to the atmosphere PC: critical turgor pressure (to be exceeded before plastic extension begins) t: time V: volume Vcw: volume of the cell wall chamber Vw: volume of water in the cell VT:volume of water lost through transpiration v: VdVdt ¼ relative rate of change in volume of the cell cw vcw: VdV ¼ relative rate of change cw dt in volume of the cell wall chamber vs: VdVdt ¼ steady or quasi steady relative rateof change in volume of the cell T ¼ relative rate of change vT: VdV T dt in water lost via transpiration volume w ¼ vw: VdV w dt relative rate of change in water volume in the cell e: volumetric elastic modulus of the cell wall f: relative irreversible extensibility of the cell wall P: dimensionless number Dp: osmotic pressure difference across the plasma membrane L(Dp P): relative volumetric rate of water uptake f(P PC): relative volumetric plastic deformation rate of the cell wall dP e dt : relative volumetric elastic deformation rate of the cell wall
that the ratio of the relative volumetric plastic and elastic deformation rates of the cell wall is 32. Importantly, other ratios may be obtained from the dimensionless numbers. For example, if you want a comparison of the magnitudes of the relative volumetric water uptake rate and the relative volumetric plastic deformation rate of the cell wall, it may be determined by the ratio, Pwv/Ppv = Pwp = L/f. Comparisons of the sums and differences of processes are easily determined with dimensionless numbers. For example, if a comparison of the relative magnitudes of the net water uptake rate and the wall deformation rate is wanted, it may be obtained as follows: (Pwv PTv)/ (Ppv + Pev) = Pwd [6]. Pwd was determined for growing pea stem cells of P.
satinis L. (Pwd = 7.8) and stage IV spor- that the magnitude of Ppe for the creep- Acknowledgment angiophores of P. blakesleeanus ing frozen–thawed walls in a low-pH Much of the research with dimensional analyses and (Pwd = 11.7) [6]. buffer is an order of magnitude smaller dimensionless numbers was supported by NSF grant than that of naturally growing stage IV MCB-0948921 to J.K.E.O. sporangiophores [6]. Thus, lowering the 1Department of Mechanical Engineering, University of Dimensionless Numbers Can Be pH alone cannot by itself produce the Colorado Denver, 1200 Larimer Street, PO Box 173364, Used to Determine Similarity Similarity between processes can be wall deformation similarity that produces Denver, CO 80217-3364, USA achieved by ensuring that the magnitudes the wall stress relaxation observed dur- *Correspondence: [email protected] (J.K.E. Ortega). of the relevant dimensionless numbers ing expansive growth. https://doi.org/10.1016/j.tplants.2017.09.020 are identical [1]. For example, ‘wall deformation similarity’ for growing walled cells Future Research References is achieved when the magnitude of Ppe is In the future dimensionless numbers can 1. Fox, R.W. et al., eds (2003) Introduction to Fluid Mechanics (6th edn), John Wiley and Sons the same in two different walls because be used to explore other processes in 2. Ortega, J.K.E. (2016) Dimensional analysis of expansive wall stress relaxation is central to expan- plant biology. For example, the dimengrowth of cells with walls. Res. Rev. J. Bot. Sci. 5, 17–24 sive growth [5]. Constant-tension exten- sionless numbers Pwv, PTv, Pwe, and 3. Ortega, J.K.E. (1990) Governing equations for plant cell growth. Physiol. Plant. 79, 116–121 sion experiments have been used to PTe could be used for research concern4. Geitmann, A. and Ortega, J.K.E. (2009) Mechanics and study how frozen–thawed cell walls are ing the effects of drought, climate modeling of plant cell growth. Trends Plant Sci. 14, 467–478 loosened by various pH buffers and pro- change, and water stress on plant cell 5. Cosgrove, D.J. (1993) How do plant cell walls extend? Plant Physiol. 102, 1–6 teins to produce creep [10]. Constant- growth rate. In addition, new dimension6. Ortega, J.K.E. (2017) Dimensionless number is central to tension extension experiments con- less numbers can be obtained and used stress relaxation and expansive growth of the cell wall. Sci. Rep. 7, 3016 ducted on frozen–thawed cell walls of to explore changes in active solute constage IV sporangiophores of P. blake- centrations and osmotic pressure and 7. Cosgrove, D.J. (1985) Cell wall yield properties of growing tissue: evaluation by in vivo stress relaxation. Plant Physiol. sleeanus demonstrate that lowering the the mass transport of carbohydrates 78, 347–356 pH of the bathing solution produces (and other materials) to the cell wall. Scale 8. Ortega, J.K.E. et al. (1989) In vivo creep and stress relaxation experiments to determine the wall extensibility and creep rates that are within the range of analysis has been shown to have great yield threshold for the sporangiophores of Phycomyces. Biophys. J. 56, 465–475 the elongation growth rates observed in utility in plant sciences [12]. As shown vivo [11]. So, is lowering the pH in the here and previously [6], dimensionless 9. Ortega, J.K.E. (2004) A quantitative biophysical perspective of expansive growth for cells with walls. In Recent wall the mechanism by which the spor- numbers can facilitate scale analyses of Research Developments in Biophysics (Vol. 3) (Pandalai, S. G., ed.), In pp. 297–324, Transworld Research Network angiophores loosen their walls to pro- biophysical processes relevant to the 10. Cosgrove, D.J. (1989) Characterization of long-term duce wall deformation during expansive growth rate of walled cells. extension of isolated cell wall from growing cucumber expansive growth? When the protocol The use of dimensionless numbers in hypocotyls. Planta 177, 121–130 for the constant-tension extension scale analysis of plant processes 11. Ortega, J.K.E. et al. (2015) Cell wall loosening in the fungus, Phycomyces blakesleeanus. Plants 4, 63–84 experiments was modified so that Ppe deserves more consideration in future 12. Niklas, K.J. (1994) Plant Allometry: The Scaling of Form could be determined, the results show research. and Process, University of Chicago Press
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