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General proof of the validity of a new tensor equation of plant growth
ARTICLE IN PRESS Journal of Theoretical Biology 256 (2009) 584–585
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
General proof of the validity of a new tensor equation of plant growth M. Pietruszka ´ ska 28, PL-40032 Katowice, Poland Department of Plant Physiology, Faculty of Biology and Environmental Protection, University of Silesia, ul. Jagiellon
a r t i c l e in f o
a b s t r a c t
Article history: Received 30 May 2008 Received in revised form 28 October 2008 Accepted 1 November 2008 Available online 11 November 2008
Plant cell/organ growth may be partly described by a local tensor equation. We provide a mathematical proof that the Lockhart (global) equation is the diagonal component of this tensor equation. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Anisotropy Local equation Plant growth
1. Introduction Lockhart (1965) proposed a simple-first order time differential equation 1 dV ¼ FðP Y Þ V dt
(1)
which may be summarized as describing the relative growth rate dV/V (where V denotes the cell volume) linear dependence on hydrostatic pressure P in excess of the yielding threshold Y. The proportionality coefficient F denotes the plastic extensibility and in the most studies is treated as a constant (e.g., Cosgrove, 1986; Proseus et al., 2000). The Lockhart equation describes the elongation of plant cells during growth in a satisfactory fashion, providing that one takes into account only water and osmotic relations and ignores any preferential growth directionality. In elongation growth, as described by the Lockhart equation, the expanding volume V (which is a global quantity) is coordinateindependent, and in reality fails to accurately describe growth correctly because of course such growth must be treated as coordinate-dependent (i.e. local). Any kind of tropism (like photoor gravi-tropism), as direction-dependent phenomenon, requires tensor description, and the Lockhart equation must be regarded as too simplistic. Many papers have been written aiming to describe the problem of plant growth based on the solution of a differential equation (e.g., Lockhart, 1965; Moltz and Boyer, 1978; Ortega, 1985; Boyer and Silk, 2004). None of these, however, was able to adequately describe cases where elongation growth, as perturbed Corresponding author. Tel.: +48 32 2009453.
E-mail address: [email protected] 0022-5193/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2008.11.001
by physical or chemical agents, could account for the existing anisotropies. Such growth anisotropies, involve asymmetric growth of stem or root, and are permanent, directed growth reactions generated by plants in response to external stimuli, such as light or gravity. To resolve the problem of coordinate system dependent anisotropies, a tensor approach to plant growth has been developed (e.g., Hejnowicz and Sievers, 1995; Hejnowicz, 1997; Silk and Erickson, 1978, 1979). A local equation for anisotropic growth, based on (a) theory of visco-elasticity (see e.g., Atkin and Fox, 2005), and (b) the Lockhart equation, has been proposed recently by Pietruszka and Lewicka (2007). Here, we show that this approach is mathematically appropriate.
2. Theory Let us consider the time dependent spatial coordinate system (local) equation, capable of describing anisotropic plant cell/organ growth in the form d qxi ¼ FGij dt qxj
(2)
where x is the displacement vector field describing cell deformation, x denotes the Cartesian coordinates fx1 ; x2 ; x3 g and the stress field G accounts for the internal pressure. This generalized equation, referring to many physiological phenomena, subsumes the global (Lockhart) equation as a special limiting case. In fact, it turns out that the Lockhart equation Eq. (1) is only the diagonal part of Eq. (2) providing that Gij describes the stress tensor. To better demonstrate Eq. (2), we utilize (a) the definition of Lie derivative along the displacement vector field x acting on the
ARTICLE IN PRESS M. Pietruszka / Journal of Theoretical Biology 256 (2009) 584–585
manifold M. (The volume form o changes as follows Lxo ¼ (div x)o where Lx is the Lie derivative along the displacement vector field). We also make use of (b) Pascal’s principle, that states that pressure applied to a confined fluid at any point is transmitted undiminished through the fluid, in order to fulfill the isotropy and homogeneity condition. Here we also assume that F, like in the original Lockhart equation does not depend on Cartesian coordinates. Hence, calculating explicitly the trace of Eq. (2), X dX qx dij i ¼ F dij Gij dt i qxj i this yields d qxi ¼ FG dt qxi By virtue of (b) we can put G ¼ tr(Gij), and utilize the properties of the Kronecker delta function dij ¼ {1 if i ¼ j and 0 if iaj}. Since the divergence of the vector field x is given by the derivative divðxÞ ¼ qi xi then we may thus write d div x ¼ FG dt Because the divergence of the vector field x is related to the change of the volume by the formula dV/V ¼ div(x), we eventually obtain d ln V ¼ FG dt However, as we see, if we substitute PY for G, this is exactly the Lockhart (global as opposed to local) equation, we wanted to derive from Eq. (2). If G is homogeneous in all space coordinates, then this equation has a general solution in the form Z t Fðt0 ÞGðt0 Þ dt0 V ðtÞ ¼ V 0 exp 0
where V0 can be interpreted as a volume for the initial time V0 ¼ V(t ¼ t0). 3. Conclusions Many experiments in plant physiology are subjected to the specific and directional external perturbations. At the very core of the Lockhart equation, as coordinate-independent model, there is the problem that this simplistic approach is unable to report on growth effects caused by a unidirectional stimulus, like the action of (collimated) light, gravitational field or external pressure
585
(force). In practice, there is a need for a growth equation, like the Eq. (2), that includes a local coordinate system. The latter has been recently successfully applied to describe spatial redistribution of auxin during phototropic response and the connection between light perception and auxin protein carriers (PIN’s) relocation (Pietruszka and Lewicka, 2007). Its applicability to the problems connected with the gravitropic response has also been shown (Lewicka and Pietruszka, 2007). Here we provide the fundamental reasoning, that such an approach to the problems of anisotropic plant (cell/organ) growth is plausible. We show explicitly that the tensor equation Eq. (2) reduces to the wellknown Lockhart equation by fulfilling the assumptions of isotropy and homogeneity. Based on our recent experience, we hope that further use of Eq. (2) will give more insight into biological processes where the problem of unidirectional stimulus is considered. We also believe that the derivation of an anisotropic form of the Lockhart equation, that may be tied to a visco-elastic model of plant cell wall extensibility whose relevance to living plant cell behavior has recently been called into question, may have some usefulness for those interested in modeling cell volume growth.
References Atkin, R.J., Fox, N., 2005. An Introduction to the theory of elasticity. Dover, New York, USA. Boyer, J.S., Silk, W.K., 2004. Hydraulics of plant growth. Funct. Plant Biol. 31, 761–773. Cosgrove, D.J., 1986. Biophysical control of plant cell growth. Annu. Rev. Plant. Physiol. 37, 377–405. Hejnowicz, Z., 1997. Graviresponses in herbs and trees: a major role for the redistribution of tissue and growth stresses. Planta 203, 136–146. Hejnowicz, Z., Sievers, A., 1995. Tissue stresses in organs of herbaceous plants i. Poisson ratios of tissues and their role in determination of the stresses. J. Exp. Bot. 46, 1035–1043. Lewicka, S., Pietruszka, M., 2007. Anisotropic plant cell elongation due to orthogravitropism. J. Math. Biol. 54, 91–100. Lockhart, A., 1965. An analysis of irreversible plant cell elongation. J. Theor. Biol. 8, 264–275. Moltz, F.J., Boyer, J.S., 1978. Growth-induced water potentials in plant cells and tissues. Plant Physiol. 62, 423–429. Ortega, J.K.E., 1985. Augmented growth equation for cell wall expansion. Plant. Physiol. 79, 318–320. Pietruszka, M., Lewicka, S., 2007. Anisotropic plant growth due to phototropism. J. Math. Biol. 54, 45–55. Proseus, T.E., Zhu, G.-L., Boyer, J.S., 2000. Turgor, temperature and the growth of plant cells: using Chara corallina as a model system. J. Exp. Bot. 51, 1481–1494. Silk, W.K., Erickson, R.O., 1978. Kinematics of hypocotyl curvature. Am. J. Bot. 65, 310–319. Silk, W.K., Erickson, R.O., 1979. Kinematics of plant growth. J. Theor. Biol. 76, 481–501.
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
General proof of the validity of a new tensor equation of plant growth M. Pietruszka ´ ska 28, PL-40032 Katowice, Poland Department of Plant Physiology, Faculty of Biology and Environmental Protection, University of Silesia, ul. Jagiellon
a r t i c l e in f o
a b s t r a c t
Article history: Received 30 May 2008 Received in revised form 28 October 2008 Accepted 1 November 2008 Available online 11 November 2008
Plant cell/organ growth may be partly described by a local tensor equation. We provide a mathematical proof that the Lockhart (global) equation is the diagonal component of this tensor equation. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Anisotropy Local equation Plant growth
1. Introduction Lockhart (1965) proposed a simple-first order time differential equation 1 dV ¼ FðP Y Þ V dt
(1)
which may be summarized as describing the relative growth rate dV/V (where V denotes the cell volume) linear dependence on hydrostatic pressure P in excess of the yielding threshold Y. The proportionality coefficient F denotes the plastic extensibility and in the most studies is treated as a constant (e.g., Cosgrove, 1986; Proseus et al., 2000). The Lockhart equation describes the elongation of plant cells during growth in a satisfactory fashion, providing that one takes into account only water and osmotic relations and ignores any preferential growth directionality. In elongation growth, as described by the Lockhart equation, the expanding volume V (which is a global quantity) is coordinateindependent, and in reality fails to accurately describe growth correctly because of course such growth must be treated as coordinate-dependent (i.e. local). Any kind of tropism (like photoor gravi-tropism), as direction-dependent phenomenon, requires tensor description, and the Lockhart equation must be regarded as too simplistic. Many papers have been written aiming to describe the problem of plant growth based on the solution of a differential equation (e.g., Lockhart, 1965; Moltz and Boyer, 1978; Ortega, 1985; Boyer and Silk, 2004). None of these, however, was able to adequately describe cases where elongation growth, as perturbed Corresponding author. Tel.: +48 32 2009453.
E-mail address: [email protected] 0022-5193/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2008.11.001
by physical or chemical agents, could account for the existing anisotropies. Such growth anisotropies, involve asymmetric growth of stem or root, and are permanent, directed growth reactions generated by plants in response to external stimuli, such as light or gravity. To resolve the problem of coordinate system dependent anisotropies, a tensor approach to plant growth has been developed (e.g., Hejnowicz and Sievers, 1995; Hejnowicz, 1997; Silk and Erickson, 1978, 1979). A local equation for anisotropic growth, based on (a) theory of visco-elasticity (see e.g., Atkin and Fox, 2005), and (b) the Lockhart equation, has been proposed recently by Pietruszka and Lewicka (2007). Here, we show that this approach is mathematically appropriate.
2. Theory Let us consider the time dependent spatial coordinate system (local) equation, capable of describing anisotropic plant cell/organ growth in the form d qxi ¼ FGij dt qxj
(2)
where x is the displacement vector field describing cell deformation, x denotes the Cartesian coordinates fx1 ; x2 ; x3 g and the stress field G accounts for the internal pressure. This generalized equation, referring to many physiological phenomena, subsumes the global (Lockhart) equation as a special limiting case. In fact, it turns out that the Lockhart equation Eq. (1) is only the diagonal part of Eq. (2) providing that Gij describes the stress tensor. To better demonstrate Eq. (2), we utilize (a) the definition of Lie derivative along the displacement vector field x acting on the
ARTICLE IN PRESS M. Pietruszka / Journal of Theoretical Biology 256 (2009) 584–585
manifold M. (The volume form o changes as follows Lxo ¼ (div x)o where Lx is the Lie derivative along the displacement vector field). We also make use of (b) Pascal’s principle, that states that pressure applied to a confined fluid at any point is transmitted undiminished through the fluid, in order to fulfill the isotropy and homogeneity condition. Here we also assume that F, like in the original Lockhart equation does not depend on Cartesian coordinates. Hence, calculating explicitly the trace of Eq. (2), X dX qx dij i ¼ F dij Gij dt i qxj i this yields d qxi ¼ FG dt qxi By virtue of (b) we can put G ¼ tr(Gij), and utilize the properties of the Kronecker delta function dij ¼ {1 if i ¼ j and 0 if iaj}. Since the divergence of the vector field x is given by the derivative divðxÞ ¼ qi xi then we may thus write d div x ¼ FG dt Because the divergence of the vector field x is related to the change of the volume by the formula dV/V ¼ div(x), we eventually obtain d ln V ¼ FG dt However, as we see, if we substitute PY for G, this is exactly the Lockhart (global as opposed to local) equation, we wanted to derive from Eq. (2). If G is homogeneous in all space coordinates, then this equation has a general solution in the form Z t Fðt0 ÞGðt0 Þ dt0 V ðtÞ ¼ V 0 exp 0
where V0 can be interpreted as a volume for the initial time V0 ¼ V(t ¼ t0). 3. Conclusions Many experiments in plant physiology are subjected to the specific and directional external perturbations. At the very core of the Lockhart equation, as coordinate-independent model, there is the problem that this simplistic approach is unable to report on growth effects caused by a unidirectional stimulus, like the action of (collimated) light, gravitational field or external pressure
585
(force). In practice, there is a need for a growth equation, like the Eq. (2), that includes a local coordinate system. The latter has been recently successfully applied to describe spatial redistribution of auxin during phototropic response and the connection between light perception and auxin protein carriers (PIN’s) relocation (Pietruszka and Lewicka, 2007). Its applicability to the problems connected with the gravitropic response has also been shown (Lewicka and Pietruszka, 2007). Here we provide the fundamental reasoning, that such an approach to the problems of anisotropic plant (cell/organ) growth is plausible. We show explicitly that the tensor equation Eq. (2) reduces to the wellknown Lockhart equation by fulfilling the assumptions of isotropy and homogeneity. Based on our recent experience, we hope that further use of Eq. (2) will give more insight into biological processes where the problem of unidirectional stimulus is considered. We also believe that the derivation of an anisotropic form of the Lockhart equation, that may be tied to a visco-elastic model of plant cell wall extensibility whose relevance to living plant cell behavior has recently been called into question, may have some usefulness for those interested in modeling cell volume growth.
References Atkin, R.J., Fox, N., 2005. An Introduction to the theory of elasticity. Dover, New York, USA. Boyer, J.S., Silk, W.K., 2004. Hydraulics of plant growth. Funct. Plant Biol. 31, 761–773. Cosgrove, D.J., 1986. Biophysical control of plant cell growth. Annu. Rev. Plant. Physiol. 37, 377–405. Hejnowicz, Z., 1997. Graviresponses in herbs and trees: a major role for the redistribution of tissue and growth stresses. Planta 203, 136–146. Hejnowicz, Z., Sievers, A., 1995. Tissue stresses in organs of herbaceous plants i. Poisson ratios of tissues and their role in determination of the stresses. J. Exp. Bot. 46, 1035–1043. Lewicka, S., Pietruszka, M., 2007. Anisotropic plant cell elongation due to orthogravitropism. J. Math. Biol. 54, 91–100. Lockhart, A., 1965. An analysis of irreversible plant cell elongation. J. Theor. Biol. 8, 264–275. Moltz, F.J., Boyer, J.S., 1978. Growth-induced water potentials in plant cells and tissues. Plant Physiol. 62, 423–429. Ortega, J.K.E., 1985. Augmented growth equation for cell wall expansion. Plant. Physiol. 79, 318–320. Pietruszka, M., Lewicka, S., 2007. Anisotropic plant growth due to phototropism. J. Math. Biol. 54, 45–55. Proseus, T.E., Zhu, G.-L., Boyer, J.S., 2000. Turgor, temperature and the growth of plant cells: using Chara corallina as a model system. J. Exp. Bot. 51, 1481–1494. Silk, W.K., Erickson, R.O., 1978. Kinematics of hypocotyl curvature. Am. J. Bot. 65, 310–319. Silk, W.K., Erickson, R.O., 1979. Kinematics of plant growth. J. Theor. Biol. 76, 481–501.