Plant Physiology 142:1346-1349 (2006)
© 2006 American Society of Plant Biologists
SCIENTIFIC CORRESPONDENCE
General and Analytic Solutions of the Ortega Equation
Sylwia Lewicka*
Department of Plant Physiology, Faculty of Biology and Environmental Protection, University of Silesia, PL–40032 Katowice, Poland
Ortega (1985) proposed a simple first-order differential equation to describe plant cell extension and, unlike the Lockhart equation (Lockhart, 1965), it takes into account elastic deformation of the cell wall. Cosgrove (1985) and Ortega (1985) solved the equation for the particular case of a previously growing plant cell that is deprived of its water source; thus, GR = 0. The results (confirmed by experiments) show that P(t) decreases exponentially to the turgor threshold Y due to the cell wall-loosening process. In this article, their results are extended to the more general case, where water absorption is included; thus, GR  0. The problem of the growth rate change in the course of time is also considered here. For the large-scale period (days), growth is well described by the sigmoid curve and consists of three phases: acceleration, linear growth with maximal velocity, and cessation of cell elongation (Fogg, 1975; Schopfer and Mohr, 1995). Then, from the mathematical point of view, the growth rate is of the type of  t2 exp(–t2) (a time derivative of the sigmoid curve). This function, however, leads to nonanalytical solutions with no clear interpretations. In this study, the time-dependent growth rate GR(t) is modeled with a mathematical function that approximates the exact function very accurately. Importantly, the new approximate function leads to analytical solutions with new predictions for the behavior of P(t) and valuable interpretations of the parameters within the solutions.
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THEORY
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The proposed procedure of solving the Ortega equation for a general case is as follows. (1) Water uptake is taken into account (i.e. 1/V dV/dt 0). (2) Growth rate is given from the experiment. (3) The elastic modulus  , as well as the cell wall yielding  , may generally depend on time, thus:
For further consideration, we rearrange the original Ortega equation
 | (1) |
into the following useful form
 | (2) |
where pressure P = P(t), turgor threshold Y is constant, and the initial condition of pressure P = P0 for t0 = 0. (Because the focus of this article is theoretical analysis [through the Ortega equation] of the relationships between growth rate and change in turgor pressure during all three phases of elongation growth, the initial condition stands for the pressure in the beginning of the first phase [where the elongation process starts]. In experiments, the studied seedlings are growing initially in the same experimental conditions. Then, at t = t0, segments are cut from the elongation zone of the seedlings and constitute a sample isolated from the water supply, while the remaining seedlings continue growing in the aerated water solution. Of course, one may more generally take t0 > 0, i.e. in any elongation phase; still, because we are interested in pressure alteration during the whole elongation process, t0 = 0 can be accepted, without loss of generality.)
First, we solve the homogeneous equation (GR = 0, no water uptake)
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Assuming and are constant, the above formula transforms into the solution originally obtained by Ortega (1985) and Cosgrove (1985): P = Y + (P0 – Y) exp(–t). Now, the way of solving the inhomogeneous Equation 2 is standard and based on the constant variation method. This article presents only the final result:
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The above general solution of the Ortega equation consists of two terms; the first term describes the exponential decrease of the turgor pressure when the cell is deprived of its water supply, and the second term expresses the additional pressure resulting from the presence of the water source. Thus, the left side of Equation 5 represents the pressure of a turgid plant cell.
The general expression, Equation 5, depends on the evolution of the growth rate in time. One may distinguish two time scales of growth (Cosgrove, 2000). The large-scale period covers the whole life cycle of the plant cell: meristematic growth, vacuolization, and, finally, maturation and cessation of growth. Elongation is then well described by the sigmoid curve and fulfills the law of great growth (Fogg, 1975; Schopfer and Mohr, 1995). The short-scale period (in the range of seconds to hours) exhibits dynamic fluctuations in time (Cosgrove, 2000). The general solution, Equation 5, may be applied to both time scales; the only difference is the form of the growth rate in the function of time. In the case of the short scale, the second term in Equation 5 may be calculated only via numerical methods because of the complexity of GR(t). In the case of the large scale, the second term in Equation 5 may also be calculated numerically, in general, because the growth rate is then described by a time derivative of the sigmoid curve, t2 exp(–t2). Notwithstanding, a good idea would be to propose the function GR(t) = at4 exp(–bt), which is characterized by two fitting (physiologically explained; see "Discussion") parameters, a and b, and leads to analytic solutions of the Ortega equation. The choice is valuable because analytic solutions provide for clear and more insightful interpretations; it is also justified because both functions, namely, t2 exp(–t2) and at4 exp(–bt), are adequate to describe the growth rate in function of time in the large-scale period (e.g. see Fig. 1
). Following Cosgrove (1985) and Ortega (1985), in this article , = constant are assumed. This simplifying assumption often appears in the literature, although it is not sufficient (see "Discussion").
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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 functional representations for GR(t) are fitted and shown in the plot. The time derivative of the well-known sigmoid curve at2 exp(–bt2) is visualized by the dashed line (a = 0.23 ± 0.01 and b = 0.107 ± 0.002, with R2 = 0.99), whereas the proposed function at4 exp(–bt) is visualized by the solid line (a = 0.58 ± 0.06 and b = 1.36 ± 0.03, with R2 = 0.98).
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Therefore, the definitions follow: I = GR(t) exp(t) dt = at4 exp(rt) dt and r = – b. The parameter r determines different solutions depending on its sign. For r 0, the integral I equals a/r·[exp(rt)·W(t) – W(0)]. The polynomial W(t) = t4 – 4t3/r + 12t2/r2 – 24t/r3 + 24/r4 is of the fourth order. For the choice of GR(t) = at4 exp(–bt) and r 0, the solution of Equation 2 takes on the analytical form
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For r = 0, in turn, integration is easy: I = GR(t) exp(t) dt = at4 dt = a/5 t5 (the integration constant is equal to zero because of the initial condition). Therefore, the final equation follows:
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Now, the comparison of these expressions (Eqs. 6 and 7) gives some general correctness or, in other words, classes of the solutions and yields some interpretation of the parameter r. One may check that the sign of r, together with the magnitude of a, divides the set of solutions of the Ortega equation into two classes. It is obvious that, from the general formula 5, the evolution of the turgor pressure in time P(t) depends on the growth rate. Accepting the growth rate in the form of GR(t) = at4 exp(–bt), which is described by two independent parameters, a and b, one may conclude that the solutions should also be classified by these parameters. In fact, the calculations prove that there are two classes described by a and r = – b. (To perform them, it is sufficient to calculate and compare time derivatives of the third terms [the additional pressures] in Eqs. 6 and 7.)
Let P1(t), P2(t), and P3(t) be the solutions of the Ortega equation for which r1 < 0, r2 = 0, and r3 > 0, respectively; above is equivalent to b1 > , b2 = , and b3 < . Undoubtedly, b1 > b2 > b3. (We assume , have the same values for all functions, P1, P2, and P3.) If a1  a2 a3, then GR1/GR2 = a1/a2 exp[(b2 – b1)t] < 1 and GR2/GR3 = a2/a3 exp[(b3 – b2)t] < 1 for every t > 0 (thus, GR1 < GR2 < GR3; see 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 t tends to infinity), will also satisfy the relations V1f = Vi + 24a1/b15<Vi + 24a2/b25 = V2f<Vi + 24a3/b35 = V3f, where Vi is the initial volume and Vf is the final volume. Such character of variability of the growth rate affects, through Equations 6 and 7, the form of the pressure P(t). One can check that the pressures P1, P2, and P3 fulfill the relations P1(t) < P2(t) < P3(t) at every instant t > 0. What follows from the above calculations is that the negative r results in the turgor pressure lower than the pressure for r = 0, which, in turn, is lower than the pressure for the positive r. The functions P1, P2, and P3 for which r1 < 0, r2 = 0, r3 > 0, and a1 a2 a3 represent one class of the solutions of the Ortega equation and are shown in Figure 2B.
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Figure 2. Two possible classes of solutions of the Ortega equation when the growth rate is modeled with GR(t) = at4 exp(–bt). Plots on the left side (A and C) represent an example (modeled) time dependence of the plant cell volume for which the growth rates GR1, GR2, and GR3 satisfy the conditions a1 a2 a3 (A) and a1 > a2 or a2 > a3 (C). Plots on the right side (B and D) represent the theoretically predicted changes in pressure over time, P1(t), P2(t), P3(t), corresponding to these functions [according to Eq. 5, solutions of the Ortega equation depend on GR(t)]. Thus, B corresponds to A and D corresponds to C. In all plots, solid lines represent the volume or pressure for which r = 0, dashed lines r < 0, and dotted lines r > 0. Thin solid lines in B and D visualize the exponential decrease of pressure when GR = 0.
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If, as mentioned above, r1 < 0, r2 = 0, r3 > 0, with the parameters a1, a2, and a3 satisfying a1 > a2 or a2 > a3, then the growth rates GR1, GR2, and GR3 have a different character. Initially, GR1 > GR2, then the functions intersect and, finally, GR1 < GR2, similarly to GR2 and GR3. (In general, it may not apply to GR1 and GR3; see Fig. 2C.) The final volumes V1f, V2f, and V3f may fulfill various inequalities depending on the magnitude of a1/a2 and a2/a3. The parameters a1, a2, and a3, which satisfy the relations a1 > a2 or a2 > a3, as well as the character of GR1, GR2, and GR3 dependency on time, lead to solutions P1, P2, and P3 of the following properties. Starting from t = 0, the solution P1 is greater than P2, then the functions intersect at a certain point and P1 becomes smaller; in turn, the solution P3 is smaller than P2, but, after exceeding a certain time instance, it becomes greater. (In general, it may not apply to P1 and P3.) The pressures P1, P2, and P3, which belong to the second class of solutions as described in this paragraph, are plotted in Figure 2D.
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DISCUSSION
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The article presents a theoretical approach that consists in accepting the growth rate functional dependence on time as given from the experiment. Then the general solution takes on the form given in Equation 5.
For the case of a large-scale growth period, the growth rate versus time may be approximated by the function at4 exp(–bt) with two parameters strictly related to the quantities measured in experiments: the maximal growth rate GRm and the time tm at which the maximal growth rate takes place: b = 4/tm and  , where e is the Euler number, e  2.78. In fact, from the experimental viewpoint, the function at4 exp(–bt) could also be rewritten in the form including both the empirical quantities:
 | (8) |
The proposed function fits well to the experimental data (see Fig. 1) and leads to analytical solutions as well. Then, Equation 5 transforms to Equation 6 or 7. This approach also provides the new parameter r, which reflects the relationship between growth and irreversible/reversible properties of the cell wall via the simple mathematical expression r = – b. The irreversible/reversible properties of the cell wall are represented by the cell wall yielding coefficient (plastic deformation) and Young's elastic modulus , respectively. The parameter b, in turn, is related to the quantities GRm and tm, thereby to the width and the height of the considered curve. Because both quantities GRm and tm depend on b, one may conclude that this parameter determines the strength of growth. Interpretation of r is therefore as follows: its value (equal, less than, or greater than zero) reflects the coupling between the growth process and mechanical properties of the cell wall; for r 0 (b < or b > , which can be expressed in the form b = ± [c]), there exists an additional set of parameters [c] coupled to biological processes affecting growth. Such parameters would reflect enhancement or dissipation of the growth rate and consequently, via Equations 6 and 7, increase or loss of pressure in the plant cell. Worth emphasizing is the fact that the validity of this theoretically based conclusion is strengthened by some predominantly experimental studies (i.e. Proseus et al., 1999, 2000; Ortega, 2004).
One may suppose that the parameter r may also depend on various, both internal and external, factors affecting growth. Let us notice that Figure 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 studied. Among other researchers, 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 and others (figure 3 in Ross et al., 2002), in turn, investigated auxin-GA interactions and their influence on plant growth, while Obroucheva and her coworkers studied the root growth responses to Pb in primary roots of maize (Zea mays; figure 2 in Obroucheva et al., 1998). These figures, in addition to Figure 2A presented in this article, reflect the fact that, when applying growth stimulators (e.g. IAA), a plant organ elongates faster (r > 0) than the same organ with no added growth factors (r 0). Applying inhibitors (e.g. ABA, Cd, Pb) slows the growth down (r < 0) until it finally ceases. Similarly, the final volume (precisely length of coleoptile or internode cell or primary root as in most cases of these experiments) is greatest 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) change over time for different plant species or tissues. Also, it is likely to represent the character of GR(t) and P(t) changes in various environmental conditions (e.g. temperature, pH, water and soil factors, light, etc.).
These hypotheses must, however, be put forward in a way that leaves no doubts as to their appropriateness. This is due to the fact that when applying the mentioned growth regulators as well as for different species, tissues, or organs, not only do parameters a and b differ, but the cell wall yielding coefficent and the elastic modulus do as well. This fact may slightly change 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 a and b 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. Actually, experiments should verify which classes of solutions—Equations 6 and 7, Figure 2, A to D—are physiologically realized in nature and under what conditions. Experimental study leading to calculation of the parameter r would also be an interesting task in this research area. Nonetheless, it is worth stressing that the time dependence of plant cell turgor pressure obtained here (presented 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 = (P) must take place as both the physical quantities (simultaneously and opposite) influence growth. They stay in delicate balance during plant growth so alteration of the pressure induces change of cell wall yielding. This statement is also experimentally supported by Proseus et al. (1999, 2000). Therefore, the expression = (t), accepted in order to find analytic integrals in Equation 5, was a simplifying assumption that has largely limited our consideration. The next step is to include the dependence = [P(t),t].
Likewise, although the 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 plant development. The elastic properties of the cell wall are not the same for a juvenile plant 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 thick and rigid; e.g. see Proseus et al., 1999; Cosgrove, 2000; Ortega, 2004). These properties affect the elastic modulus . Further study should include the dependence of Young's modulus in function of time, (t), which naturally can be obtained from experiments; however, the challenge would be to determine it theoretically.
Received July 14, 2006;
accepted September 25, 2006;
published December 6, 2006.
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FOOTNOTES
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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 (slewicka{at}us.edu.pl).
www.plantphysiol.org/cgi/doi/10.1104/pp.106.086751
* E-mail slewicka{at}us.edu.pl; fax 48322009361.
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LITERATURE CITED
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Cosgrove DJ (1985) Cell wall yield properties of growing tissue. Plant Physiol 78: 347–356[Abstract/Free Full Text]Cosgrove DJ (2000) Expansive growth of plant cell walls. Plant Physiol Biochem 38: 109–124[CrossRef][Web of Science][Medline] Fogg GE (1975) The Growth of Plants. Richard Clay (The Chaucer Press), Bungay, Suffolk, UK Kutschera U (2000) Cell expansion in plant development. Rev Bras Fiziol Veg 12: 65–95 Kutschera U, Koehler K (1993) Turgor pressure and elongation growth in developing sunflower hypocotyls. J Plant Physiol 141: 757–758 Kutschera U, Koehler K (1994) Cell elongation and osmotic pressure in developing sunflower hypocotyls. J Exp Bot 274: 591–595 Lockhart JA (1965) An analysis of irreversible plant cell elongation. J Theor Biol 8: 264–275[CrossRef][Web of Science][Medline] Montague MJ (1997) Exogenous jasmonic and abscisic acids act differentially in elongating tissues from oat stem segments. J Plant Growth Regul 16: 11–19[Medline] Obroucheva NV, Bystrova EI, Ivanov VB, Antipova OV, Seregin IV (1998) Root growth responses to lead in young maize seedlings. Plant Soil 2000: 55–61[CrossRef] Ortega JKE (1985) Augmented growth equation for cell wall expansion. Plant Physiol 79: 318–320[Abstract/Free Full Text] Ortega JKE (2004) A quantitative biophysical perspective of expansive growth for cell with walls. In SP Pandalai, ed, Recent Research Developments in Biophysics, Vol 3, Part II. Transworld Research Network, Kerala, India, pp 297–324 Proseus TE, Ortega JKE, Boyer JS (1999) Separating growth from elastic deformation during cell enlargement. Plant Physiol 119: 775–784[Abstract/Free Full Text] Proseus TE, Zhu GL, Boyer JS (2000) Turgor, temperature and the growth of plant cells: using Chara corallina as a model system. J Exp Bot 51: 1481–1494[Abstract/Free Full Text] Ross JJ, O'Neill DP, Wolbang CM, Symons GM, Reid JB (2002) Auxin-gibberellin interactions and their role in plant growth. J Plant Growth Regul 20: 346–353 Schopfer P, Mohr H, editors (1995) Plant Physiology. Springer, Berlin
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TOP
THEORY
DISCUSSION
