An easy method of estimating the stress-relaxation parameters To and b

Transcription

1 Plant & Cell Physiol. 17: (1976) An easy method of estimating the stress-relaxation parameters To and b Ryoichi Yamamoto and Yoshio Masuda Department of Biology, Faculty of Science, Osaka City University Sumiyoshi-ku, Osaka 558, Japan (Received April 27, 1976) Mechanical properties of the cell wall are represented by relaxation parameters. This paper describes an easy method of estimating the stress-relaxation parameters T o and b, in which they can be calculated using stress-relaxation data directly obtained with a simple tensile tester. The value of A is calculated from the relaxation data at time = 1, 2, 3 and 5 sec, and the value of T o is calculated by multiplying 6 by T, which is calculated from the data at time = 0, 1 and 2 sec. Stress-relaxation analysis for quantitative evaluation of mechanical properties of the cell wall of higher plants has been adopted by Yamamoto et al. (6) and Cleland and Haughton (/). Generally, the stress-relaxation properties of synthetic and natural polymers can be evaluated by simulating them to models composed of a number of Maxwell viscoelastic components. Thus, stress-relaxation properties characterizing the models show the mechanical properties of polymers. Yamamoto et al. (6, 8) reported that stress-relaxation properties of the cell wall can be simulated to a Maxwell model and represented by relaxation parameters. The stressrelaxation parameters TQ, T m and b have physically defined meanings and are referred to as the minimum and maximum relaxation times and the relaxation rate, respectively. The parameters TQ and b have been also suggested to represent elongation capacity of the cell, that is, elongation which is going to occur (4, 10). We have found a good correlation between these parameters and chemical modifications of wall polysaccharides {2, Sakurai et al., in preparation). However, the method we developed {4, 6-10) needs either a FORTRAN computer system or a mini-computer system directly attached to the tensile tester of, for example, an Instron type. These procedures are either laborious or expensive and may possibly not be suitable for ordinary plant physiology experiments. If these parameters could be easily calculated using the relaxation data which are simply obtained with an inexpensive, simple tensile tester, they could be used by many plant physiologists in many laboratories. For this purpose, we tried to develop an approximation method to calculate the parameters from the chart paper data of the tensile tester. We describe the method with a simple theory which, we hope, will be useful. 835

2 836 R. Yamamoto and Y. Masuda Approximate estimation method of the parameters The stress relaxation of the cell wall was measured with a tensile tester as follows. The specimen, methanol-killed and then rehydrated, was fixed between two clamps of the tensile tester and stretched by lowering the bottom clamp (cf. 8). The stress produced by the specimen increased approximately linearly (A in Fig. 1). After the specimen had received a certain amount of initial load, usually 10 g, the clamp was stopped by pushing the stop button of the tensile tester (B in Fig. 1). Then the decay of the load was recorded and the stress-relaxation curve was obtained (C in Fig. 1). Generally, the stress-relaxation properties of natural and synthetic polymers can be simulated to models composed of a number of Maxwell viscoelastic components. Yamamoto et al. (6") reported that a simple model composed of one or two Maxwell components is not adequate for the real stress relaxation of the cell wall and that stress-relaxation properties of the cell wall fit well a model of at least four components. Using this model, the parameter, T, which is the model's smallest relaxation time, was found to be useful for analysis of the auxin effect on the cell wall (6"). The value of T can be obtained from: =l/ln {(So- (1) where St is the stress at time=i sec (6"). The reciprocal value of T is the inclination of the initial part of the stress-relaxation curve made by plotting logarithmic stress against time (Fig. 2). Thus, in a sense, T represents the rate of stress relaxation. Auxin causes a decrease in the T value when it induces cell elongation (3-7), indicating that auxin speeds up the stress relaxation of the cell wall. Yamamoto et al. (6, 8) also found that the stress relaxation of the cell wall of higher plants can be simulated to a model composed of an infinite number of Maxwell components, represented by the following equation: S=b-ln{(t+T m )l(t+t 0 )} (2) - T itfie- "' Fig Time, sec Fig. 2. Fig. 1. Stress-relaxation process recorded on Tensilon tensile tester chart paper. A: Pre-extension process. B: Starting point of stress relaxation. C: Stress-relaxation process. Fig. 1. Stress-relaxation curve made by plotting the logarithmic stress against time. The broken line represents the inclination of the initial part of the stress relaxation.

3 An easy estimation method of the stress-relaxation parameters ' Fig. 3. An actual example of reading stress relaxation data. recorded on chart paper using a handmade ruler. Stress was read from a stress-relaxation curve where S is the stress, t is time, T m and To are the maximum and minimum relaxation times, respectively, and b is the relaxation rate. Mechanical properties of the cell wall represented by this model are characterized by the relaxation parameters T m, T o and b. To and T m represent the times when the stress starts to decrease and reaches 0, respectively. While the values of T o and b have been estimated by non-linear least square method including partial differential equations, an easier estimation method is needed. Segments, 1-cm long, excised from 4-day-old etiolated oat (Avena sativa c.v. Victory) coleoptiles were incubated in buffer solution (0.01 M K-phosphate, ph 5.5) containing various concentrations of indole-3-acetic acid (IAA) for 2 hr. After incubation, they were methanol-killed, treated with 200 /*g/ml pronase (in 0.05 M K-phosphate, ph 7.0) at 37 C for 18 hr and subjected to stress-relaxation analysis as reported previously (6, 8). A Tensilon tensile tester, Model UTM-II, Toyo Measuring Instrument Co., was used. The stress-relaxation process recorded on the chart paper of the tensile tester was read as shown in Fig. 3. We then attempted to estimate the b value by an approximate least squares method using the stress data at time=l, 2, 3 and 5 sec. In the range from 1 to 5 sec, Eq. 2 can be approximated as: S=a-b-\n(t+T 0 ) where a=b-\nt m. Since this equation is linear with respect to \n(t+t 0 ), we can obtain the b value by a linear least squares method as follows: >=E [S t n-\n(t+to)-a where a = V l \n(t+t 0 ), 0=S{m(*+ T 0 )} 2, n is the data number and S t is the stress at time = /sec. If we assume that T Q =0.03 sec, the values of a*-n-p

4 838 R. Yamamoto and Y. Masuda b/s 0 Fig. 4. Fig. 5. Fig. 4. Correlation between b/.?o values estimated by a computerized system and the new method. Abscissa: A/So estimated by a computerized system. Ordinate: b/so by the new method. Fig. 5. Correlation between r and To. Oat coleoptile, mung bean and cucumber hypocotyl, and pea epicotyl cell walls were used for the determination of r and TQ. for t=\, 2, 3 and 5 sec are approximately 0.62, 0.12, 0.18 and 0.56, respectively. Therefore, the b value can be represented by the following equation: 6=0.62.^ (3). Since the b value was represented by the percent value of So (4, 10), the value determined by Eq. 3 was divided by So (See Fig. 4, Tables 1 and 2). The b value obtained by this procedure correlated well with that obtained by an Instron- HITAC-10 system (<?) as shown in Fig. 4. Our study showed that the coefficients, 0.62, 0.12, 0.18 and 0.56 in Eq. 3 are valid for estimating the b value of at least oat coleoptile and pea epicotyl cell walls, and possibly mung bean, lettuce and cucumber hypocotyl cell walls. Thus, the computerized system may not be necessary for obtaining the b value of the cell wall. Next, we attempted to determine the To value by a simple method as in the case of the b value. First, we examined the relationship between To and r values. Fig. 5 shows that the 7b value correlated fairly well with the T value, which was relatively easily estimated as mentioned above and reported previously (6"). However, the correlation was not good in the range of 7~b from 0.04 to 0.01 sec (Fig. 5, shadowed range), indicating that the r value cannot be replaced directly with T o. The mathematical relationship between TQ and r was reported as follows (6):

5 An easy estimation method of the stress-relaxation parameters To, sec Fig. 6. Correlation between T-A/SO and To of Ike oat coleoplile cell wall. The T o value was calculated by a non-linear least squares method using relaxation data measured for at least 50-msec intervals. ' See text for T-bjS 0. If we assume t 0 and S=SQ, Eq. 2 leads to then S 0 =b-\nt m IT 0 Solb=\nT m!t o. Substitution of this equation into Eq. 4 results in T o =T-b/So Since we have easy methods of estimating b and T as mentioned above, 7~b can be simply estimated using Eq. 5. We then checked whether or not the leal 7o value estimated by non-linear least squares method can be replaced with the T o value estimated by Eq. 5. Oat coleoptile segments treated with various concentrations of IAA for 2 hr were used. The value of r-b/so correlated well with the value of To, which was determined with our computerized system, even in the T o range of 0.01 sec, as shown in Fig. 6. Thus, we may be able to replace the To value with the value ofr-h/so for analyzing auxin-induced cell wall loosening as represented by a decrease in To. (4). (5). Application of the method to analysis of auxin-induced wall loosening Oat coleoptile segments treated with 10~ 5 M IAA for 3 hr were methanolkilled, pronase-treated and subjected to stress-relaxation analysis using a Tensilon tensile tester and a computerized Instron tensile tester. The stress-relaxation process was recorded on chart paper and the stress at time=0, 1, 2, 3 and 5 sec was

6 840 R. Yamamoto and Y. Masuda Table 1 Examples of calculation ofh,r and To ( = r'b/so) Treatment S o S t S 2 S T= l/ln[(5 0 Si)/(Si S 2 )] Eq. 1 (gr. weight) T O TI>ISO Eq. 5 A/So = (0.62x x _ n.., x x8.4)/10.6 -u.ima Initial T=l/ln[( )/(9.I-8.75)] =0.687 Tb = x0.687 = A/5 0 =(0.62x x _ n,. x x8.0)/10 -v.wdl None T= l/ln[(10-8.7)/( )] =0.762 T o = x = A/S 0 =(0.62x x _ n n, tq x x7.8)/9.9 -u.imy IAA T=l/ln[( )/( )] =0.569 r o = 0.O419x = read from it (Fig. 3). Examples are listed in Table 1. First, the value of b was calculated with Eq. 1 and then it was divided by So- Under these experimental conditions, the values were around 4.3% of So- The value of T was also calculated with Eq. 3. The value of T was usually in the range of 0.4 to 0.8 sec. Next, the 7"o value, which is T times b as represented by Eq. 5, was obtained. Usually, the order of T& was 0.01 sec and in rare cases, 0.1 sec. The values obtained by the method developed in this study were compared with those from the computerized system in Table 2. The b value calculated with Eq. 1 was approximately equal to the other. Under the conditions of the present study, auxin did not cause changes in the b value obtained by both systems. However, auxin caused a significant decrease in T values, as reported previously (3-7). The T value was the same for both systems. This is the reason the T values of the computerized system were omitted from Table 2. The To value was calculated by multiplication of T by b in the case of the Table 2 Effect of auxin on stress-relaxation parameters estimated by computerized and non-computerized systems Treatment Computerized system Non-computerized system (sec) &(%) r(sec)t o (sec) Initial ± ± ± ± None ± ± ± IAA 4.37± ± ± ± Mean values with standard errors (n = 20).

7 An easy estimation method of the stress-relaxation parameters 841 usual tensile tester and a non-linear least squares method in the case of a computerized system. Although the former To value (T-6/SO) was larger than the latter, auxin caused a significant decrease in To values in both cases as shown in Table 2. Judging from these facts, a computerized system is not necessary for the analysis of stress-relaxation properties of the cell wall. A system composed of a tensile tester and a recorder is sufficient for determining the values of To and b, as well as T. By the method described in this paper, the relaxation parameters To, b and T, which are useful for analysis of auxin-induced cell wall loosening, can be estimated from the data of the stress at time 0, 1, 2, 3 and 5 sec, if a tensile tester is available. This approximation method might be useful not only for oat coleoptile segments but also for other plant tissues. References ( / ) Cleland, R. and P. M. Haughton: The effect of auxin on stress-relaxation in isolated Avena coleoptiles. Plant Physiol. 47: (1971). ( 2) Kawamura, H., S. Kamisaka and Y. Masuda: Regulation of lettuce hypocotyl elongation by gibberellic acid. Correlation between cell elongation, stress-relaxation properties of the cell wall and wall polysaccharide content. Plant & Cell Physiol. 17: (1976). (3) Masuda, Y., R. Yamamoto and E. Tanimoto: Auxin-induced changes in cell wall properties and growth of Avena coleoptiles and green pea epicotyls. In Plant Growth Substances, Edited by D.J. Carr. p Springer Verlag, ( 4 ) Masuda, Y., R. Yamamoto, H. Kawamura and Y. Yamagata: Stress-relaxation properties of the cell wall of tissue segments under different growth conditions. Plant & Cell Physiol. 15: (1974). ( 5) Tanimoto, E. and Y. Masuda: Role of the epidermis in auxin-induced elongation of lightgrown pea stem segments, ibid. 12: (1971). ( 6) Yamamoto, R., K. Shinozaki and Y. Masuda: Stress-relaxation properties of plant cell walls with special reference to auxin action, ibid. 11: (1970). (7) Yamamoto, R. and Y. Masuda: Stress-relaxation properties of the Avena coleoptile cell wall. Physiol. Plant. 25: (1971). (8) Yamamoto, R., S. Fujihara and Y. Masuda: Measurement of stress-relaxation properties of plant cell walls. In Plant Growth Substances, p , Hirokawa Publishing Co., Tokyo (9) Yamamoto, R., K. Maki and Y. Masuda: Auxin and hydrogen ion actions on light-grown pea epicotyl segments III. Effect of auxin and hydrogen ions on stress-relaxation properties. Plant & Cell Physiol. 15: (1974). (10) Yamamoto, R., H. Kawamura and Y. Masuda: Stress-relaxation properties of the cell wall of growing intact plants, ibid. 15: (1974).

8