Open in a separate window Figure 1. Long time scale (in days) for which elongation of plant cells fulfills the law of great growth. Here, for data adapted from Kutschera and Koehler (1993), the two practical representations for GR(= 0.23 0.01 and = 0.107 0.002, with = 0.58 0.06 and = 1.36 0.03, with =GR(= and = ? determines different solutions depending on its sign. For r 0, the integral equals + 12 0, the perfect solution is of Equation 2 takes on the analytical form (6) For = 0, in turn, integration is easy: = GR(= = and and = ? have the same ideals for all functions, 0 (therefore, GR1 GR2 GR3; observe Fig. 2A). Moreover, the final volume (i.e. the volume after completing the elongation process, mathematically defined as a limiting value of the volume when tends to infinity), will also satisfy the relations 0. What follows from your above calculations is that the bad results in the turgor pressure lower than the pressure for = 0, which, in turn, is lower than the pressure for the positive = 0, dashed lines 0, and dotted lines 0. Thin solid lines in B and D visualize the exponential decrease of pressure when GR = 0. If, as mentioned above, = 0, the perfect solution is = 4/is the Euler quantity, 2.78. In fact, from your experimental viewpoint, the function = ? is definitely therefore as follows: its value (equal, less than, or greater than zero) displays the coupling between the growth process and mechanical properties of the cell wall; for 0 ( or = [may also depend on numerous, both internal and external, factors affecting growth. Let us notice that Number 2A resembles (at least qualitatively) some experimental results, well known in the Cidofovir tyrosianse inhibitor literature, where the influence of phytohormones (abscisic acid [ABA]; Montague, 1997; indole-3-acetic acid [IAA]; Ross et al., 2002), or abiotic factors (Cd, Pb; Obroucheva et al., 1998) on growth was analyzed. Among other experts, Montague, in particular, studied the effect of jasmonic acid and ABA on elongation of Avena internodal cells (numbers 2 and 5 in Montague, 1997). Ross while others (number 3 in Ross et al., 2002), in turn, investigated auxin-GA relationships and their influence on plant growth, while Obroucheva and her coworkers analyzed the root growth reactions to Pb in main origins of maize ( 0) than the same organ with no added growth factors ( 0). Applying inhibitors (e.g. ABA, Cd, Pb) slows the growth down ( 0) until it finally ceases. Similarly, the final volume (precisely length of coleoptile or internode cell or main root as in most cases of these experiments) is best in the case of the stimulator and least in the case of Cidofovir tyrosianse inhibitor the inhibitor. The second class of solutions of the Ortega equation (Fig. 2, C and D) would represent growth (Fig. 2C) and pressure (Fig. 2D) switch over time for different herb species or tissues. Also, it is likely to represent the character of GR(and differ, but the cell wall yielding coefficent and the elastic modulus ? do as well. This fact may slightly switch both classes of solutions. The validity of the hypotheses should also be proved empirically. While studying the elongation of, for instance, maize coleoptiles, roots, or internode cells of Chara and Nitella, we are interested in whether any qualitative and quantitative agreement between the experimentally measured pressure and Equations 6 and 7, theoretically determined, exists. If parameters and describing the growth rate satisfy the inequalities considered in the last two paragraphs of the previous section, the pressure should behave as it has been derived there. SOX9 Actually, experiments should verify which classes of solutionsEquations 6 and 7, Physique 2, A to Dare physiologically recognized in nature and under what conditions. Experimental study leading to calculation of the parameter would also be an interesting task in this research area. Nonetheless, it is worth stressing that the time dependence of herb cell turgor pressure obtained here (offered in Fig. 2, B and D, dashed lines) stays in agreement with some data obtained from experiments (e.g. see Kutschera and Koehler, 1993, 1994; Kutschera, 2000). Unfortunately, another problem also arises. Are the data obtained from the study of the Ortega equation sufficient to get reliable results? Certainly, they are not complete because the coupling (and possible dependence) between cell wall yielding and pressure has not been considered. In fact, the relation = ( em P /em ) must take place as both the physical quantities (simultaneously and reverse) influence growth. They stay in delicate balance during plant growth so alteration of the pressure induces switch of cell wall yielding. This statement is also experimentally supported by Proseus et al. (1999, 2000). Therefore, the expression = ( em t /em ), accepted in order to find analytic integrals in Equation 5, was a simplifying assumption that has largely limited our concern. The next step is to include the dependence = [ em P /em ( em t /em ), em t /em ]. Likewise, even though possible time dependence of the elastic modulus ? has been taken into account in the general procedure of solving the Ortega equation, it has been accepted as constant in time for the case of analytic solutions. In fact, cell wall features evolve during herb development. The elastic properties of the cell wall are not the same for any juvenile herb cell or the cell in the elongation stadium (when the cell wall is thin and very elastic) or, at least, for the mature cell (when the cell wall is solid and rigid; e.g. observe Proseus et al., 1999; Cosgrove, 2000; Ortega, 2004). These properties impact the elastic modulus em ? /em . Further study should include the dependence of Young’s modulus in function of time, em ? /em ( em t /em ), which naturally can be obtained from experiments; however, the challenge would be to determine it theoretically. Notes The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Sylwia Lewicka (lp.ude.su@akciwels). www.plantphysiol.org/cgi/doi/10.1104/pp.106.086751. given from your experiment. (3) The elastic modulus = is usually constant, and the initial condition of pressure = = are constant, the above formula transforms into the answer originally obtained by Ortega (1985) and Cosgrove (1985): = + (and = constant are assumed. This simplifying assumption often appears in the literature, although it is not sufficient (observe Discussion). Open in a separate window Physique 1. Long time level (in days) for which elongation of herb cells fulfills the law of great growth. Here, for data adapted from Kutschera and Koehler (1993), the two functional representations for GR(= 0.23 0.01 and = 0.107 0.002, with = 0.58 0.06 and = 1.36 0.03, with =GR(= and = ? determines different solutions depending on its sign. For r 0, the integral equals + Cidofovir tyrosianse inhibitor 12 0, the solution of Equation 2 takes on the analytical form (6) For = 0, in turn, integration is easy: = GR(= = and and = ? have the same values for all functions, 0 (thus, GR1 GR2 GR3; observe Fig. 2A). Moreover, the final volume (i.e. the volume after completing the elongation process, mathematically defined as a limiting value of the volume when tends to infinity), will also satisfy the relations 0. What follows from your above calculations is that the unfavorable results in the turgor pressure lower than the pressure for = 0, which, in turn, is lower than the pressure for the positive = 0, dashed lines 0, and dotted lines 0. Thin solid lines in B and D visualize the exponential decrease of pressure when GR = 0. If, as mentioned above, = 0, the solution = 4/is usually the Euler number, 2.78. In fact, from your experimental viewpoint, the function = ? is usually therefore as follows: its value (equal, less than, or greater than zero) displays the coupling between the growth process and mechanical properties of the cell wall; for 0 ( or = [may also depend on numerous, both internal and external, factors affecting growth. Let us notice that Physique 2A resembles (at least qualitatively) some experimental results, well known in the literature, where the influence of phytohormones (abscisic acid [ABA]; Montague, 1997; indole-3-acetic acid [IAA]; Ross et al., 2002), or abiotic factors (Cd, Pb; Obroucheva et al., 1998) on growth was analyzed. Among other experts, Montague, in particular, studied the effect of jasmonic acid and ABA on elongation of Avena internodal tissue (figures 2 and 5 in Montague, 1997). Ross as well as others (physique 3 in Ross et al., 2002), in turn, investigated auxin-GA interactions and their influence on plant growth, while Obroucheva and her coworkers analyzed the root growth responses to Pb in main roots of maize ( 0) than the same organ with no added growth factors ( 0). Applying inhibitors (e.g. ABA, Cd, Pb) slows the growth down ( 0) until it finally ceases. Similarly, the final volume (precisely length of coleoptile or internode cell or main root as in most cases of these experiments) is best in the case of the stimulator and least in the case of the inhibitor. The second class of solutions of the Ortega equation (Fig. 2, C and D) would represent growth (Fig. 2C) and pressure (Fig. 2D) switch over time for different herb species or tissues. Also, it is likely to represent the character of GR(and differ, but the cell wall yielding coefficent and the elastic modulus ? do as well. This fact may slightly switch both classes of solutions. The validity of the hypotheses should also be proved empirically. While studying the elongation of, for instance, maize coleoptiles, roots, or internode cells of Chara and Nitella, we are interested in whether any qualitative and quantitative agreement between the experimentally measured pressure and Equations 6 and 7, theoretically decided, exists. If parameters and describing the growth rate satisfy the inequalities regarded within the last two paragraphs of the prior section, the pressure should work as it’s been produced there. Actually, tests should verify which classes of solutionsEquations 6 and 7, Body 2, A to Dare realized in character and under what Cidofovir tyrosianse inhibitor physiologically.