Data analysis in plant physiology: are we missing the reality

Blackwell Science, LtdOxford, UK PCEPlant, Cell and Environment0016-8025Blackwell Science Ltd 2001 2410October 2001 742 Data analysis in plant physiology G. N. Amzallag Original ArticleBEES SGML Plant, Cell and Environment (2001) 24, 881–890 OPINION Data analysis in plant physiology: are we missing the reality? G. N. AMZALLAG The Judea Centre for Research and Development, Carmel 90404, Israel ABSTRACT In plant physiology, data analysis is based on the comparison of mean values. In this perspective, variability around the mean value has no significance per se, but only for estimating statistical significance of the difference between two mean values. Another approach to variability is proposed here, derived from the difference between redundant and deterministic patterns of regulation in their capacity to buffer noise. From this point of view, analysis of variability enables the investigation of the level of redundancy of a regulation pattern, and even allows us to study its modifications. As an example, this method is used to investigate the effect of brassinosteroids (BSs) during vegetative growth in Sorghum bicolor. It is shown that, at physiological concentrations, BSs modulate the network of regulation without affecting the mean value. Thus, it is concluded that the physiological effect of BSs cannot be revealed by comparison of mean values. This example illustrates how a part of the reality (in this case, the most relevant one) is hidden by the classical methods of comparison between mean values. The proposed tools of analysis open new perspectives in understanding plant development and the nonlinear processes involved in its regulation. They also ask for a redefinition of fundamental concepts in physiology, such as growth regulator, optimality, stress and adaptation. Key-words: adaptation; brassinosteroid; connectance; networks; noise; redundancy; stress and optimality; variability. Abbreviations: BS, brassinosteroid; PGR, plant growth regulator; PDR, plant development regulator; CV, coefficient of variation; DUCE, deterministic unidimensional cause–effect; NELI, network-like. ‘The popularity of averaging and other statistical approaches is used (unconsciously) to imply that mechanisms are not only much simpler than they really are, but even to directly mislead as to the truer state of affairs.’ A. J. Trewavas INTRODUCTION The transformation of data to mean value and standard deviation is generally performed even before analysis. By Correspondence: G. N. Amzallag. Fax: + 972 2 9960061; e-mail: nissamz@bgumail.bgu.ac.il © 2001 Blackwell Science Ltd this mode of treatment, it is generally assumed that variation around the mean has no biological significance. However, this assumption is not always justified. Variability in leaf morphogenesis fluctuates according to the phase of development in Nicotiana (Paxman 1956; Sakai & Shimamoto 1965) and Clarkia tembloriensis (Sherry & Lord 1996), suggesting an endogenous control of variation. This is confirmed by genotype differences observed in amplitude of variability in development (Roy 1963; Thomas 1969). Exposure to suboptimal conditions is known to modify variability, but this effect varies considerably, both in direction and in amplitude, according to the stage of development (Heslop-Harrison 1959; Amzallag, Seligmann & Lerner 1995). DNA transactions (such as changes in repetitive DNA and activation of transposable elements) occur during specific phases of development (Fedoroff 1989; Bassi 1990). Influencing the genome expression, these changes include a stochastic component (Rogers & Bendich 1987; Amzallag 1999a) generating variability in the phenotype. This is considered to be one of many causes of variability in development (Conrad 1990). Biological significance of noise is also suggested in physiology. For example, measuring intra-individual variability in the concentration of glucose and insulin in human blood, Kroll (1999) concluded: ‘In the past, it was thought that the source of variation was external to the internal workings of the organism, that the environment, such as temperature, food ingestion, immobilization, veinous occlusion were responsible of short-term changes . . . [but] . . . The source of biological variation for glucose and insulin comes from within the organism itself; it is endogenous.’. An endogenous source of variability is also observed for osmoregulation of shoot of salt-treated Sorghum plants (Amzallag 1999b). All these examples suggest that variability may be a parameter of biological importance. REDUNDANCY IN BIOLOGICAL PROCESSES Inadequacy of the classical mode of analysis In the classical approach, it is assumed that modification of the mean value of A aims for an effect of the tested factor, X. The lack of significant change in the mean value of A following modification of X does not mean that X has no influence at all, but rather that its influence (if really existing) on A is covered by noise, the isotropic effect of uncontrolled factors. This mode of investigation is appropriated in the 881 882 G. N. Amzallag case of a direct influence of X on A (Fig. 1a), even if X is included in a long chain of factors (Fig. 1b). It is even true for multiple pathways of influence on A (Fig. 1c). In the latter case, the influence of X on mean value of A is observed only in the case of uniformity of the Y parameter, whereas uncontrolled fluctuations on Z influence only variation around the mean value. All these modes of regulation of A are considered as cases of deterministic unidimensional cause–effect linkage (abbreviated as DUCE). However, these cases are not the exclusive modes of influence on a variable. For example, homeostatic regulations (Fig. 1d) induce a buffering capacity face to variations affecting the pathway (Fig. 1d). However, even in this case, the process may be investigated as DUCE-type after experimentally blocking the retroaction pathway (R). The method of comparison of mean values is appropriated for investigating processes fitting one of the DUCEtype pathways (or even a combination of them), but not for the case of redundancy in regulation of the variable A. The simplest case of redundancy is that of multiple homeostatic pathways of processes leading (or regulating) the variable A, autonomy (Fig. 2a) or mutually interfering (Fig. 2b). In both cases, no modification in the mean value of A is consecutive to experimental fluctuations of one of the X or Y variables. Another case of redundancy is that of the interrelations between different pathways, generating a network of processes leading to the variable A (Fig. 3). In this case, redundancy is determined by the structure of the network itself. For example, three levels of influence on the variable A may be found from the network illustrated in Fig. 3. Fluctuations of the X1-to-Xi or Y1-to-Yj variables has no influence at all on the mean value of A. Modulation of the Xi+1-to-Xm as well as Yj+1-to-Yn variables have a moderate influence on A. Only modulations of the Z1-to-Zk variables have a direct and proportional effect on the mean value of A. In this scheme, the influence of a variable is not proportional to its involvement in the pathway of regulation/formation, but rather to the position on the network. Thus, concerning networks, the involvement of a factor cannot be investigated by its influence on the mean value of the (a) (b) X A (c) X Y A ….. X1 Xn R (d) Z X1 A ….. Xn A Figure 1. Examples of deterministic unidimensional cause–effect (DUCE) linkages in control of a character. (a) Simple determinism; (b) linear chain of regulation; (c) multiple determinism; (d) homeostatic regulation. The width of the arrows in (c) symbolizes the contribution of each pathway to the global control. Continuous line: positive regulation. Dashed line: negative regulation. (a) Rx X1 …. Ry Xn A Ym …. Y1 …. Y1 Ry (b) X1 …. Xn A Ym Rx Figure 2. Redundancy in homeostatic regulation. (a) Autonomous pathways; (b) interfering pathways. measured parameter. These considerations lead to a paradox: comparison of mean values enables the study of only the DUCE-type systems or linear parts of redundant regulatory processes, but not networks and multiple homeostasis pathways (both termed network-like systems, or NELI). In the case of significant influence on mean value, the process may always be related to the DUCE system. However, the situation is confused when a lack of significant difference is observed in the mean value of A. Indeed, it remains impossible to decide whether the variable X is or is not involved in the process of regulation in the absence of appropriate tools distinguishing between DUCE and NELI pathways. Thus, it is not surprising that the DUCE type of regulation is so frequently invoked in physiology. Such a situation does not reflect its importance but rather the fact that it is the single mode of regulation analysable by classical methods. The reality of redundancy In spite of direct measurements, the importance of redundancy in regulations has been observed at all levels of biological organization. A network of regulation of gene expression is observed in Escherichia coli (Thieffry et al. 1998), and a similar mode of regulation is probably inherent to gene expression in the eucaryote cell (Thieffry & Romero 1999). The emergence of the phenotype is understood as the result of a long series of transduction networks, in which simple cause–effect relationships are not the rule (Green 1996). Metabolic pathways are far from being regulated linearly; this characteristic appears to be fundamental for the stability of the whole system (Fell 1997). The network struc- X1 … Xi … Xm Z1 Y1 … Yj … … Zk A Yn Figure 3. Schematic representation of network pattern of modulation. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Data analysis in plant physiology 883 Cholesterol Pregnenolone A1 B1 C1 A1S1 A2 AB2 C2 A2S1 A3 AB3 C3 A4 AB4 C4 AB5 C5 A1S2 AB6 ABC6 relationship between tissues and differentiated organs (Chauvet 1993; Amzallag 1999c). Stability in flower development has been related to the level of correlation between flower characters (Berg 1959, 1960). This measurement of interconnectedness may be considered globally as an estimation of the level of redundancy. A similar correspondence between networkness and stability was also observed, at the ecological level, for food webs (Law & Blackford 1992). Such stability-characterizing regulatory networks may explain the relative autonomy of development from genotypic variability (Cock 1966; Alberch 1980; Barton & Turelli 1989; Wagner & Schwenk 2000) but also its adaptive plasticity (Sultan 1992, 1995). All these considerations lead to a paradoxical situation: the NELI structure is a central property of biological systems, at all levels of organization, but it cannot be investigated before it is transformed into a DUCE structure. The current methods enable us to investigate only part of the biological reality. Worse, there are no means to estimate what is missed by such an approach. BIOLOGICAL SIGNIFICANCE OF VARIABILITY One-dimension analysis ABC7 Digoxygenine Figure 4. Example of networkness in metabolism: Biosynthetic pathways of digoxigenin from cholesterol in Digitalis. (a), (b) and (c) are three parallel biosynthetic pathways, starting with the formation of progesterone (A1), pregnen-3b,21-diol-20-one (B1), 23-nor-4,20(22)E-choladienic acid-3-one (C1), respectively (redrawn from Gershenzon & Kreis 1999). ture of biosynthesis is especially complex for many secondary metabolites. The synthesis of digoxigenin, a cardiac glycoside from Digitalis, provides an illustration of such complexity (Fig. 4). Moreover, simultaneous expression of isozymes (itself due to redundancy in genetic information) confers network properties to metabolic pathways – even those identified as linear (Igamberdiev 1999). Redundancy also exists at the subcellular level of organization. Modulation of the cytoskeleton appears as network-regulated (Pfaffmann & Conrad 2000). In plant cells, the secondary signal transduction pathways from hormone receptors also show a large redundancy (Trewavas & Malho 1997). Even cellular perception of the hormonal signal displays dual modes in plant cells, involving both dose– response and change in sensitivity (Guern 1987; Trewavas 1991). Weyers et al. (1995) propose that ‘it should always be assumed, in the absence of contrary evidence, that combined control might exist.’ This provides clear evidence towards functional redundancy in the perception of hormonal information. Redundancy also characterizes the © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 As revealed by the increase in stability of network structures, parasitic noise is buffered by redundancy. As a consequence, the level of variability in a measured parameter A may serve as an estimation of the level of redundancy in its regulation. Through this perspective, the observed variability becomes a transformation of the noise inherent to every experimental system. By its fluctuations, it is able to reveal changes in the level of redundancy in the regulation of the parameter studied. This effect may be analysed independently of changes in mean value when variability is normalized as a coefficient of variation (CV): CV(Y) = 100 ¥ SD(Y)/avg(Y). The comparison of CV values aims for changes in the network, but it cannot provide any detail about the nature of these changes. For this reason, it may be considered as a one-dimension mode of analysis of variability. Two-dimension analysis Covariance is calculated frequently in order to test the significance of the relationship between two variables, X and Y. The variation of one variable may be represented by a simple function of the second for significant correlation (P < 0·05), but nothing may be concluded about the link between X and Y for non-significant ones. As for the statistical comparison of mean values, this dichotomic method of analysis was developed in order to determine the relevant variable(s) of a DUCE-type model of regulation. However, the absolute value of a coefficient of correlation (quantifying the strength of the correlation) should not be considered only for testing significance of a correlation, but 884 G. N. Amzallag also as a measurement of the strength of the relationship between two variables. Beyond the question of the significance of the correlation, this parameter may provide information about the individual pathways of the network. Correlation coefficients (r-values) are not distributed normally. Thus, calculations cannot be performed before transforming them in z-values (normally distributed). This z-value is defined as connectance, and calculated according to Sokal & Rohlf (1981): z = 0·5.Lm[(1 + |r|)/(1 - |r|)]. Connectance may be considered for a pair of variables, but also for a parameter A in its relationship with the (X1, . . . ,Xm) other measured variables. In this case, connectance is the mean of the z-values for the relationships: C(A) = 1/m.[(z(A,X1) + . . . + z(A,Xm)]. Connectance may be also calculated for the biological system as a whole, as a mean of the z-values for all the possible relationships between the measured parameters. Although resulting from mathematical transformations, connectance reflects biological phenomena. Changes in connectance were observed during specific phases of development in Sorghum, and they were related to the adaptive phenotypic plasticity of the plant (Amzallag & Seligmann 1998; Amzallag 1999d). Connectance is also affected by hormonal treatments (CK and GA) even before any significant effect on growth (Amzallag 1999c, 2001a). The star-like pattern Connectance does not obligatorily reveal a direct linkage between two variables. For example, connectance between characters A and B may be due to their control in parallel by a third variable, C, generating a star-like mode of regulation (Fig. 5). In this case, a combination of analysis of connectance and CV may help to distinguish between networklike and star-like modes of regulation. All the variables depend on fluctuations of a single factor in a star-like structure. Consequently, all the characters are modified in parallel and proportionally to fluctuations in the regulation of C. In contrast, variability is not modified in parallel for all the characters linked through a network-like pattern. Through analysis of n populations (n replicates of the same treatment, for example), a series of n coefficients of varia- A C B D Figure 5. Schematic representation of a star-like pattern of modulation tion [CV(X1), . . . ,CV(Xn)] is determined for each character X. Thus, the linkage in variation of CV of the studied characters enables us to distinguish between star-like and NELI structures. Beyond the determination of the structure (star-like or NELI), this analysis provides some other information: in star-like systems, the highest correlated variable for comparison between CV values may be considered as the closest to the centre of the star-like structure. VARIABILITY IN THE RESPONSE TO BRASSINOSTEROIDS Brassinosteroids (BSs) are found at very low concentrations in the vegetative tissues of plants. Analysis of BS-deficient mutants in Arabidopsis reveals their essential role in plant development (Clouse, Langford & McMorris 1996; Kauschmann et al. 1996). However, being produced by all tissues and modulating a very large range of processes, BSs differ from all other identified plant growth regulators (PGRs) (Clouse & Sasse 1998). Sasse (1991) even concluded that ‘ . . . brassinolide cannot be classified as belonging to any of the known groups of plant hormones . . . it could be considered to belong to all of them!’. This obscure situation is confirmed by the paradoxical effect of BSs. For example, root elongation is inhibited by an exogenous supply of BS in Arabidopsis thaliana (Clouse et al. 1996), mungbean (Guan & Roddick 1988a), tomato (Guan & Roddick 1988b), maize and wheat (Roddick & Ikekawa 1992). In cuttings of Phaseolus vulgaris, steroids also inhibit the emergence of adventitious roots (Hewitt & Hillman 1979). In all these cases, no response is observed at low concentrations, followed by an inhibiting effect after exposure to high concentrations. From these observations, it may be concluded that BS inhibits root formation and elongation. However, root elongation in Raphanus sativus is reduced in seedlings treated with inhibitors of BS biosynthesis (Bach 1985). Consequently, BS should not be considered as a simple inhibitor of root elongation. Thus, it is even quite surprising that a positive effect on root elongation was not reported following the addition of low concentrations of BS. This contradiction may be solved by assuming that BS is always present at the optimum concentration in tissues, so that an exogenous supply may have only neutral or detrimental effects. However, the ‘informative power’ of such a mode of regulation completely disappears. Rather, it may be suggested that the physiological effect of BS is not detected by the comparison of mean values. This point is tested through analysis of variability performed on an extremely simple experimental system: the response of 8-d-old seedlings of Sorghum bicolor (genotype MP610) to the addition of BS to the root medium (halfstrength Hoagland solution, see Amzallag 1999c for details about the experimental procedures). The plants were harvested on day 18, after 10 d of treatment with BS. Shoot, adventitious and seminal roots were weighed [fresh weight (FW)] separately for each of the 12 individuals exposed to the same treatment. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Data analysis in plant physiology 885 clear influence of BS is observed, even for treatments as low as 0·1 nM (Table 2). In contrast, a significant effect on the mean value of developmental ratios is observed only for treatment with 10 nM BS (Table 2). These very simple observations reveal that the threshold of sensitivity to BS is about 100 times lower for CV than for mean values. This is especially interesting when one remembers that 10 nM is not a physiological concentration whereas 0·1 nM is compatible with the range of endogenous concentrations of BS in vegetative tissues (between 0·01 and 0·3 nM; see Adam & Marquardt 1986; Takatsuto 1994). The one-dimensional analysis of variability suggests a BS-induced change in the structure of the network. This point may be verified by quantification of the global connectance between the parameters measured. A specific drop in connectance is observed after treatment with 0·1 nM BS, whereas exposure to higher concentrations increased connectance (Table 3). The seminal root is especially affected in its relationships with shoot and adventitious roots for plants treated with 0·1 nM BS (Table 3), confirming the specific increase in CV for developmental ratios including the seminal root (Table 2). A very high connectance is calculated for plants exposed to 10 nM BS (Table 3). The linkage is so strong that anatomically unlinked characters (such as seminal and adventitious roots) became highly connected. Thus, it seems that this high connectance does not reflect an increase in redundancy but rather a transition towards a star-like pathway of regulation. This transition is confirmed by further analyses, revealing discontinuity between evolution of the regulation pattern for BS concentration between treatments with 1 and 10 nM BS (see below). Interestingly, the inhibition of growth does not occur at the stage of partial dislocation of the network (0·1 nM BS), but rather after its transformation towards a star-like system (10 nM BS). Table 1. Effect of brassinosteroids on mean (g) and coefficient of variation of vegetative organs in S. bicolor. Plants (grown hydroponically in optimal conditions: natural light intensity and photoperiod in July, aerated half-strenght Hoagland solution from day 6 following imbibition, root medium solution replaced on day 13, see Amzallag 1999c for details) were harvested 18 d after imbibition. Brassinosteroid (BS; 24-epibrassinolide purchased from Sigma Chemical Co., St Louis, USA) was added to the root solution between days 8 and 18. Twelve plants were measured for each treatment. Mean values of BS-treated plants are compared with those of control plants by a two-tailed t-test Control (no BS) 0·1 nM BS 1·0 nM BS 10 nM BS Mean CV Mean CV Mean CV Mean CV Sh AR SR Total plant 7·77 22·46 6·75NS 14·48 6·37* 16·60 4·38*** 33·56 1·83 35·43 1·54NS 27·63 1·55NS 34·42 0·786*** 30·22 2·59 39·46 2·56NS 26·31 2·23NS 24·48 1·12*** 32·64 12·19 24·80 10·86NS 13·57 10·27NS 19·23 6·29*** 32·54 Sh, shoot FW; AR, adventitious root FW; SR, seminal root FW; NS, not significant (P > 0·05); *, significant difference at P < 0·05, ***, significant difference at P < 0·005. Shoot–root relationship The comparison of mean values of shoot, adventitious and seminal root weight by two-tailed t-tests do not reveal any significant effect for plants treated with 0·1 and 1 nM (except for shoot weight at 1 nM BS). A significant difference in mean value is observed only at 10 nM BS (Table 1). Furthermore, no clear effect of BS is observed on variability (Table 1). The coefficient of variation (CV) is a normalized value, so it may be compared for different parameters measured on the same population. Accordingly, even minor variations in CV may reveal something about the system. In the case analysed here, it is interesting to observe that at 0·1 nM BS, the CV for whole-plant weight is reduced in comparison with that of separated organs (Table 1). The above-calculated CVs are strongly dependent on variability in the rate of growth. Comparing the CV of developmental ratios may eliminate this influence. Thus, a Control (no BS) 0·1 nM BS 1 nM BS 10 nM BS Mean CV Mean CV Mean CV Mean CV Control of leaf elongation The fifth leaf was the last completely unfolded one at the harvest. This is an opportunity to study the effect of BS on leaf elongation and its relation with growth of the whole plant. Mean values of sheath and blade length are modified significantly only following exposure to 10 nM BS (Table 4). However, the effect of 0·1 nM BS on the sheath : blade ratio Sh : (AR + SR) ratio AR : Sh ratio SR : Sh ratio AR : SR ratio 1·85 21·8 1·66NS 18·59 1·71NS 11·7 2·27** 5·6 0·23 19·2 0·22NS 17·74 0·24NS 24·56 0·18** 9·6 0·33 30·0 0·39NS 38·58 0·34NS 15·8 0·25* 11·3 0·77 43·0 0·69NS 60·45 0·73NS 28·52 0·72NS 21·0 Sh, shoot FW; AR, adventitious root FW; SR, seminal root FW; NS, not significant (P > 0·05); *, significant difference at P < 0·05, **, significant difference at P < 0·01. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Table 2. Influence of brassinosteroid treatments on mean and CV value of developmental parameters. Same plants and treatments as in Table 1. 886 G. N. Amzallag r coefficient for relationship between X axis Y axis Control 0·1 nM BS 1 nM BS 10 nM BS Sh (AR + SR) Sh SR Sh AR AR SR Connectance 0·825 0·321 0·909 0·989 0·584 -0·244 0·789 0·948 0·883 0·823 0·774 0·956 0·412 -0·517 0·474 0·845 0·917 0·580 1·034 1·886 Table 3. Global connectance between shoot, adventitious and seminal roots in S. bicolor exposed to brassinosteroid (BS) treatments. Same plants as in Table 1. The r coefficients are also presented for the four relationships used in the calculation of connectance [a significant correlation (P < 0·05) is observed for absolute values of r higher than 0·632] Sh, shoot FW; AR, adventitious root FW; SR, seminal root FW. (Table 4) suggests that the effect of BS focuses on development, as previously indicated for the analysis of shoot–root relationships. The mean value of organ weight and sheath length is not modified by treatments lower than 10 nM (Tables 1 & 4). However, the connectance of sheath length with growth parameters is modified by BS treatments as low as 0·1 nM. A progressive decrease in connectance is observed at 0·1 and 1 nM BS, whereas a considerable increase occurs at 10 nM. This suggests a discontinuity in the pattern of regulation towards 10 nM BS (Table 5). During leaf development in Sorghum, the sheath elongated after the blade. A high r-value for correlation between sheath and blade is observed in the absence of BS treatment (Table 5). Blade length should be considered as the first-ranked factor conditioning sheath length. Obviously, blade length is a complex factor, but it should be considered as representative of a series of regulatory processes controlling length during the unfolding of the blade. Thus, the residual value of the sheath from its correlation with the blade may be compared with other characters in order to determine the second-ranked factor of this network. In the absence of BS treatment, the seminal root (especially its ratio with shoot) appears as the second factor related to sheath length (Table 6). After treatment with 0·1 nM BS, the second determining factor is not the seminal but the adventitious root (especially its ratio with shoot) (Table 6). The lack of relationship observed for treatment with 1 nM BS may be because of the existence of another, unmeasured, secondary factor, or the fact that adventitious roots became the main factor. The latter assumption is justified Table 4. Effect of brassinosteroid (BS) treatment on mean and CV value for length (cm) of the sheath and blade of the fifth leaf. Same plants and treatments as in Table 1 Control 0·1 nM BS 1 nM BS 10 nM BS Mean CV Mean CV Mean CV Mean CV Sheath Blade Sheath : blade ratio 12·24 4·96 12·57NS 4·12 12·40NS 5·44 8·63* 9·42 26·94 5·73 26·90NS 4·10 25·82NS 4·95 19·03* 13·97 0·454 2·25 0·467* 3·29 0·480* 5·42 0·457NS 6·32 NS, not significant (P > 0·05); *, significant difference at P < 0·05. by the observation of a higher correlation between sheath and adventitious roots than between sheath and blade length (Table 5). This correlation is even strengthened (r = - 0·652) when sheath length is correlated with the adventitious root : shoot ratio. Therefore, it seems that the linkage with adventitious roots becomes the main factor controlling sheath elongation for plants treated with 1 nM BS. This point is verified by the significant correlation observed (r = 0·635) between residual value of sheath length (calculated from the correlation with the adventitious root : shoot ratio) and blade length. Therefore, in the presence of 1 nM BS, blade length becomes the second factor in control of sheath length after adventitious roots. Again, all these changes occurred before any significant modification in the mean values (Table 4). THE NEED FOR NEW CONCEPTS From the example of S. bicolor, the effect of BS on the regulation pathways without any consequence on mean value is a confirmation of redundancy in regulation. Analysis of variability reveals the involvement of BSs in the process of replacement of the seminal by adventitious root during vegetative development in Sorghum. However, all these changes remain completely cryptic in the classical comparative analysis of mean values. Two conditions are required for modification of the mean value in redundant pathways of regulation: the first is the breaking of redundancy (inducing a hierarchy in the different pathways of regulation, see Fig. 1c), and the second is the modification of the pathways in their influence on the measured variable. Thus, a modification of mean value implies a modification of all the pathways of regulation by the variable. In other words, modification of the mean value implies a transformation of the redundant structure towards a star-like pattern. This proposition indicates that the comparison of mean values enables an analysis of the biological system only after their transformation towards star-like systems. It confirms that the comparative analysis of mean values does not enable us to conclude whether the star-like system is native or an experimentally induced modification. At least two questions emerge from these considerations: 1. What does a change in mean value observed at high concentration signify? © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Data analysis in plant physiology 887 Control 0·1 nM BS 1 nM BS 10 nM BS Blade length Shoot AR SR Connectance 0·921 0·674 0·425 0·971 -0·011 -0·009 -0·074 0·723 -0·017 -0·335 -0·513 0·873 0·011 0·290 0·069 0·861 0·408 0·368 0·290 1·410 AR, adventitious root FW; SR, seminal root FW. 2. What is the biological importance of a change in the regulatory network if it does not provoke any significant effect on growth? Towards a new definition of optimality Definitions are not only conventions enabling the communication of ideas. They represent the framework from which our scientific questions emerge. On the other hand, our definitions are also conditioned by the mode of investigation. This point is clearly illustrated by the definition of stress in plant physiology. Stress is considered as a condition inducing a decrease in growth. As a corollary, optimality is defined as the environmental conditions enabling maximal rate of growth. However, these definitions are problematic, because meta-optimal conditions (such as high concentration of CO2) may induce an increase in growth accompanied by a physiological perturbation. Moreover, the response to environmental changes frequently includes developmental modifications. In the latter case, it remains impossible to determine whether changes in growth are directly caused by the stressing factor, or if they are consecutive to a change in rate of growth because of ‘developmental plasticity’. Furthermore, this definition of stress does not enable us to distinguish between tolerance (resistance to deformation), accommodation (plasticity in the modes of regulation ensuring stability of the end product), and physiological adaptation (resetting of the regulations according to the environmental modification). In many cases, the physiological adaptation to environmental modifications is revealed by an increase in tolerance rather than Table 6. The search for a secondary factor of control of sheath length and how it is influenced by brassinosteroid (BS). The residual value of the correlation between blade length (X axis) and sheath length (Y axis) is correlated with growth (adventitious and seminal root) or developmental parameters (ratio between adventitious or seminal root and shoot). Same populations of plants as in Table 5 r value for the correlation with Control 0·1 nM BS 1 nM BS 10 nM BS SR SR : Sh ratio AR AR : Sh ratio -0·651 0·447 0·129 0·173 -0·715 0·400 -0·059 0·257 -0·390 -0·597 -0·173 -0·106 -0·489 -0·674 -0·299 -0·713 Sh, shoot FW; AR, adventitious root FW; SR, seminal root FW. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Table 5. Effect of brassinosteroid (BS) on the relationship between sheath length of the fifth leaf and shoot, root weight or length of the corresponding blade. The r coefficient for each relationship with sheath length is determined, and connectance is calculated on the basis of these four r coefficients. Same plants as in Table 1 a recovery of the initial rate of growth (Amzallag & Lerner 1995; Amzallag 1999a,1999b). In S. bicolor exposed to salinity, it was observed that optimality (measured by developmental or physiological parameters) is even modified towards the new saline conditions as a consequence of physiological adaptation (Amzallag, Seligmann & Lerner 1997; Amzallag 2000). These examples show that the estimation of stress from measurements of growth is not only imprecise, but also frequently incorrect. Stress is consequent to the emergence of a limiting factor in growth and/or development. This constraining effect may be reformulated as the transformation of a redundant pattern towards a star-like pattern of regulation. Thus, optimality for the expression of a developmental character may be redefined as the conditions of redundancy in its regulation. For a specific environmental condition, redundancy may be estimated easily by the comparison of mean values, CV and connectance following variation of a secondary factor or a hormonal treatment. In these conditions, redundancy exists if a change in the patterns of regulation (revealed by changes in CV or connectance) may be observed in the absence of any significant modifications in the mean values (before the secondary factor or hormonal concentration becomes too disturbing for inducing by itself a reorganization of the regulation towards a star-like pattern). This definition of stress remains specific to each character. However, a measurement of whole-plant characters (such as rate of growth) enables a global estimation of optimality. Network regulations and adaptive plasticity Networks are not fixed structures. Developmental events (senescence or emergence of a new organ, change from vegetative to reproductive development) involves temporary drops in connectance during specific phases of development, also termed critical periods. For example, a reduced connectance has been observed during the replacement of the seminal root by the adventitious root during the early vegetative development of Sorghum (Amzallag 1999d). This drop in global connectance is the expression of a temporary decrease in influence of the whole-plant regulations on growth of an organ. This may be interpreted as a change in the sensitivity of the cells to informative molecules (Amzallag 2001a). By redefining sensitivity of the cell itself, this process enables (during critical periods) the re-emergence of redundancy in regulation. Such a process does not significantly affect the rate of growth but it holds a new significance in the context of the proposed definition of stress: the critical period enables 888 G. N. Amzallag the recovery of optimality according to the direct effect on each cell (for which sensitivity may be redefined) of environmental and/or developmental changes. This process was clearly observed during normal development (Amzallag 1999d, 2001b). As shown in S. bicolor, physiological adaptation to salinity (expressed exclusively during a developmental window, see Amzallag, Seligmann & Lerner 1993) is also clearly related to the decrease in organ connectance during the early vegetative critical period (Amzallag 1999d). It is likely that the adaptive properties inherent to developmental plasticity (Sultan 1995) are an expression of this phenomenon. Thus, although ignored in optimal conditions, it seems that changes in the regulation networks are especially important for expression of developmental plasticity and especially its adaptive dimension. The dual mode of action of informative molecules From the above considerations, it appears that the influence of informative molecules on the structure of the network remains cryptic in the case of optimal conditions because of the inherent redundancy characterizing optimality. However, this influence becomes fundamental in suboptimal conditions, by conditioning the opportunity to recover a functional, network-like regulatory system. According to the above analysis, the main physiological effect of BS on vegetative organs appears as a modulation of the network of relationships between various growing organs. For this reason, BS should not be considered as a regulator of growth, but rather as a factor of readjustment of cellular sensitivity to PGRs connecting various meristems and/or differentiating tissues. This assumption is compatible with the capacity of cells to adjust their sensitivity to a hormone (Trewavas 1991; Csaba 1994, 2000; Amzallag 2001a). Accordingly, two distinct functions may coexist for informative molecules released by the cells: (i) a role as modulators of the regulatory network; because of the importance of the network structure to development, these informative molecules should be termed plant developmental regulators (PDRs); (ii) a role as PGRs that is measured by an influence on mean value. The distinction between PGRs and PDGs is not very simple because both types of activity coexist eventually at different ranges of concentrations. In the example detailed here, BS acts as a PDR at low concentrations (0·1 and 1 nM), whereas it is a PGR when applied at high concentration (10 nM). This PGR effect is probably not always artifactual because very high concentrations of BS are found in specific organs, such as pollen (Adam & Marquardt 1986). This PGR activity of brassinosteroids is not surprising when it is considered that in unicellular organisms, steroid hormones are involved in the regulation of the cell cycle (Dahl, Biemann & Dahl 1987; Argawal 1993). A transformation from PGR to PDR activity may be observed during development without change in the range of concentrations. In S. bicolor, for example, gibberellins act as PGRs during stable periods of growth (defined as phenophases), but they show a PDR activity during critical periods of modification of the network (Amzallag 2001a). It is likely that other well-known PGRs also display a PDR activity at low concentrations or during specific phases of development. Moreover, some compounds from the socalled ‘secondary metabolism’ probably show a PDR-like activity, as suggested by their effect on sensitivity to wellknown growth regulators (Green & Corcoran 1975; Ray & Laloraya 1983; Yoshikawa et al. 1986). CONCLUSION Before being interpreted, data are organized, treated by mathematical functions and analysed by statistical methods. Through these transformations, we make many choices. Some of them are conditioned by our own hypothesis or by the tools we are using, but others issue from the general paradigms or definitions. The latter factors are completely ignored because they are accepted so universally. One of them is the postulate of separation between real effects and noise that defines not only the field of investigation, the questions that are asked, but also restricts considerably the type of responses potentially acceptable. For this reason, there is no room for redundancy in physiological investigation, and especially in the field of analysis of regulation processes. As detailed here, this situation is extremely problematic because: (i) redundant regulatory pathways are basic patterns in biology; (ii) the transition from network-like to star-like regulation is a fundamental event in differentiation and the response to stressing conditions, and (iii) adaptation to a disturbing environment is probably related to the recovery of a redundant pattern of regulation. Thus, it is the global mode of analysis leading to deterministic interpretations that has to be modified. In the new approach proposed, variability is not considered an undesirable experimental artifact. Rather, it is the transformation of the inherent noise by the biological system that provides information about its pattern of regulation. This reconsideration of variability in experimental data reveals a reality that cannot be investigated by another approach. ACKNOWLEDGMENTS Professor Tsvi Sachs is thanked here for his critical comments and opinions, which helped me to clarify some of the points discussed in this paper. REFERENCES Adam G. & Marquardt V. (1986) Brassinosteroids. Biochemistry 25, 1787–1799. Alberch P. (1980) Ontogenesis and morphological diversification. American Zoologist 20, 653–667. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Data analysis in plant physiology 889 Amzallag G.N. (1999a) Plant evolution: toward an adaptive theory. In Plant Response to Environmental Stresses: from Phytohormones to Genome Reorganization (ed. H.R. Lerner), pp. 171–245. M. Dekker, Inc., New York. Amzallag G.N. (1999b) Individuation in Sorghum bicolor: a selforganized process involved in physiological adaptation to salinity. Plant, Cell and Environment 22, 1389–1399. Amzallag G.N. (1999c) Regulation of growth: the meristem network approach. Plant, Cell and Environment 22, 483–493. Amzallag G.N. (1999d) Adaptive nature of the transition phases in development: the case of Sorghum bicolor. Plant, Cell and Environment 22, 1035–1041. Amzallag G.N. (2000) Maternal transmission of adaptive modifications in salt-treated Sorghum bicolor: a first stage in ecotypic differentiation? New Phytologist 146, 483–492. Amzallag G.N. (2001a) Maturation of integrated functions during development. I. Modifications of the regulatory network during transition periods in Sorghum bicolor. Plant, Cell and Environment 24, 337–345. Amzallag G.N. (2001b) The adaptive potential of plant development: evidence from the response to salinity. In Salinity: Environment, Plant, Molecules (eds A. Laüchli & U. Lüttge). Kluwer, Dordrecht, The Netherlands (in press). Amzallag G.N. & Lerner H.R. (1995) Physiological adaptation to environmental stresses. In Handbook for Plant Crop Physiology (ed. M. Pessarakli), pp. 557–576. M. Dekker, Inc., New York. Amzallag G.N. & Seligmann H.R. (1998) Perturbation in leaves of salt-treated Sorghum: elements for interpretation of the normal development as an adaptive response. Plant, Cell and Environment 21, 785–793. Amzallag G.N., Seligmann H. & Lerner H.R. (1993) A developmental window for salt-adaptation in Sorghum bicolor. Journal of Experimental Botany 44, 645–652. Amzallag G.N., Seligmann H. & Lerner H.R. (1995) Induced variability during the process of adaptation in Sorghum bicolor. Journal of Experimental Botany 45, 1017–1024. Amzallag G.N., Seligmann H. & Lerner H.R. (1997) Leaf malformation during early development in Sorghum. Evidence for an embryonic developmental window. Physiologia Plantarum 99, 470–476. Argawal M.K. (1993) Receptors for mammalian steroid hormones in microbes and plants. FEBS Letters 322, 207–210. Bach T.J. (1985) Selected natural and synthetic enzyme inhibitors of sterol biosynthesis as molecular probes for in vivo studies concerning the regulation of plant growth. Plant Science 39, 183– 187. Barton N.H. & Turelli M. (1989) Evolutionary quantitative genetics: how little do we know? Annual Review of Genetics 23, 337–370. Bassi P. (1990) Quantitative variations of nuclear DNA during plant development: a critical analysis. Biological Review 65, 185– 225. Berg R.L. (1959) A general evolutionary principle underlying the origin of developmental homeostasis. American Naturalist 93, 103–105. Berg R.L. (1960) The ecological significance of correlation pleiades. Evolution 14, 171–180. Chauvet G.A. (1993) Hierarchical functional organization of formal biological systems: a dynamical approach. I. The increase of complexity by self-association increases the domain of stability of a biological system. Philosophical Transactions of the Royal Society of London B 339, 425–444. Clouse S.D. & Sasse J.M. (1998) Brassinosteroids: essential regulators of plant growth and development. Annual Review of Plant Physiology and Plant Molecular Biology 49, 427–451. © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890 Clouse S.D., Langford M. & McMorris T.C. (1996) A brassinosteroid-insensitive mutant in Arabidopsis thaliana exhibits multiple defects in growth and development. Plant Physiology 111, 671–678. Cock A.G. (1966) Genetical aspects of metrical growth and form in animals. Quarterly Review of Biology 41, 131–191. Conrad M. (1990) The geometry of evolution. Biosystems 24, 61– 81. Csaba G. (1994) Phylogeny and ontogeny of hormone receptors: origin and development of hormone receptors. International Review of Cytology 155, 1–48. Csaba G. (2000) Hormonal imprinting: its role during the evolution and development of hormones and receptors. Cell Biology International 24, 407–414. Dahl C., Biemann H.P. & Dahl J. (1987) A protein kinase antigenically related to pp60 possibly involved in yeast cell cycle control: positive in vitro regulation by sterol. Proceedings of the National Academy of Sciences USA 84, 4012–4016. Fedoroff N.V. (1989) About maize transposable elements and development. Cell 56, 181–191. Fell D. (1997). Understanding the Control of Metabolism. Portland Press, London. Gershenzon J. & Kreis W. (1999) Biochemistry of terpenoids: monoterpenes, sesquiterpenes, diterpenes, sterols, cardiac glycosides and steroid saponins. In Annual Plant Reviews Vol. 2: Biochemistry of Plant Secondary Metabolism (ed. M. Wink), pp. 222–299. Sheffield Academic Press, Sheffield. Green F.B. & Corcoran M.R. (1975) Inhibitory action of five tannins on growth induced by several gibberellins. Plant Physiology 56, 801–806. Green P.B. (1996) Transductions to generate plant form and pattern: an essay on cause and effect. Annals of Botany 78, 269–281. Guan M. & Roddick J.C. (1988a) Comparison of the effects of epibrassinolide and steroidal oestrogens on adventitious root growth and early shoot development in mung bean cuttings. Physiologia Plantarum 73, 426–431. Guan M. & Roddick J.C. (1988b) Epibrassinolide-inhibition of development of excised, adventitious and intact roots of tomato (Lycopersicum esculentum): comparison with the effect of steroidal oestrogens. Physiologia Plantarum 74, 720–726. Guern J. (1987) Regulation from within: the hormone dilemma. Annals of Botany 60, S75–S102. Hewitt S. & Hillman J.R. (1979) Steroidal oestrogens and adventitious root formation in Phaseolus. Plant Physiology 63 (Suppl.), 83. Heslop-Harrison J. (1959) Variability and environment. Evolution 13, 145–147. Igamberdiev A.U. (1999) Foundations of metabolic organization: coherence as a basis of computational properties in metabolic networks. Biosystems 50, 1–16. Kauschmann A., Jessop A., Koncz C., Szekeres M., Willmitzer L. & Altmann T. (1996) Genetic evidence for an essential role of brassinosteroids in plant development. Plant Journal 9, 701– 713. Kroll M.H. (1999) Biological variation of glucose and insulin includes a deterministic chaotic component. Biosystems 50, 189– 201. Law R. & Blackford J.C. (1992) Self-assembling food webs: a global viewpoint of coexistence of species in lokta-volterra communities. Ecology 73, 567–578. Paxman G.J. (1956) Differentiation and stability in the development of Nicotiana rustica. Annals of Botany 20, 331–347. Pfaffmann J.O. & Conrad M. (2000) Adaptive information processing in microtubule networks. Biosystems 55, 47–58. Ray S.D. & Laloraya M.M. (1983) Interaction of gibberellic acid, abscisic acid, and phenolic coumpounds in the control of hypo- 890 G. N. Amzallag cotyl growth of Amaranthus caudatus seedlings. Canadian Journal of Botany 62, 2047–2052. Roddick J.G. & Ikekawa N. (1992) Modification of root and shoot development in monocotyledon and dicotyledon seedlings by 24-epibrassinolide. Journal of Plant Physiology 140, 70–74. Rogers S.O. & Bendich A.J. (1987) Ribosomal RNA genes in plants: variability in copy number and in the intergenic spacer. Plant Molecular Biology 9, 509–520. Roy S.B. (1963) The variation of organs of individual plants. Journal of Cell Science 58, 147–176. Sakai K.I. & Shimamoto Y. (1965) Developmental instability in leaves and flowers of Nicotiana tabacum. Genetics 51, 801–813. Sasse J.M. (1991) The case for Brassinosteroids as endogenous plant hormones. In Brassinosteroids: Chemistry, Bioactivity and Applications (eds H.G. Cutler, T. Yokota & G. Adam) ACS Symposium Series 474, pp. 158–166. American Chemical Society, Washington, DC. Sherry R.A. & Lord E.M. (1996) Developmental stability in flowers of Clarkia tembloriensis. Journal of Evolutionary Biology 9, 911–930. Sokal R.R. & Rohlf F.J. (1981). Biometry, 2nd edn. Freeman, San Francisco, CA. Sultan S.E. (1992) Phenotypic plasticity and the neo-Darwinian legacy. Evolutionary Trends in Plants 6, 61–71. Sultan S.E. (1995) Phenotypic plasticity and plant adaptation. Acta Botanica Neerlandica 44, 363–383. Takatsuto S. (1994) Brassinosteroids: distribution in plants, bioassays and microanalysis by gas chromatography-mass spectrometry. Journal of Chromatography A 658, 3–15. Thieffry D. & Romero D. (1999) The modularity of biological regulatory networks. Biosystems 50, 49–59. Thieffry D., Huerta A.M., Perez-Rueda E. & Collado-Vives J. (1998) From specific gene regulation to global regulatory networks: a characterization of Escherichia coli transcriptional network. Bioessays 20, 433–440. Thomas R.L. (1969) Inter-population variation in perennial ryegrass. 2. Population variance coefficients. Heredity 24, 85– 90. Trewavas A. (1991) How do plant growth substances work? Plant, Cell and Environment 14, 1–12. Trewavas A. (1999) The importance of individuality. In Plant Response to Environmental Stresses: from Phytohormones to Genome Reorganization (ed. H.R. Lerner), pp. 27–50. M. Dekker, Inc., New York. Trewavas A.J. & Malho R. (1997) Signal perception and transduction: the origin of the phenotype. Plant Cell 9, 1181–1195. Wagner G.P. & Schwenk K. (2000) Evolutionarily stable configurations: functional integration and the evolution of phenotypic stability. Evolutionary Biology 31, 155–217. Weyers J.D.B., Paterson N.W., A’Brook R. & Peng Z.Y. (1995) Quantitative analysis of the control of physiological phenomena by plant hormones. Physiologia Plantarum 95, 486–494. Yoshikawa M., Gemma H., Sobakima Y. & Masago H. (1986) Rooting cofactor activity of plant phytoalexins. Plant Physiology 82, 864–866. Received 6 February 2001; received in revised form 6 June 2001; accepted for publication 6 June 2001 © 2001 Blackwell Science Ltd, Plant, Cell and Environment, 24, 881–890