MATHEMATICAL MODELING OF ALLELOPATHY: BIOLOGICAL RESPONSE TO ALLELOCHEMICALS AND ITS INTERPRETATION

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1 Journal of Chemical Ecology, Vol. 19, No. 10, 1993 MATHEMATICAL MODELING OF ALLELOPATHY: BIOLOGICAL RESPONSE TO ALLELOCHEMICALS AND ITS INTERPRETATION M. AN, 1"* I.R. JOHNSON, 2 and J.V. LOVETT 2 Centre for Conservation Farming Charles Sturt University Wagga Wagga, NSW 2650, Australia 2Department of Agronomy and Soil Science University of New England Armidale, NSW 2351, Australia (Received December 4, 1992; accepted May 24, 1992) Abstract--Allelochemicals are assumed to possess specific biological properties and responses of an organism are external expressions of such properties. Based on this assumption, a mathematical model has been constructed to interpret the characteristic responses of an organism to allelochemicals. Several sets of experimental data have been compared with the model predictions; good agreement between the model and data is observed. Key Words--Allelopathy, allelochemicals, mathematical model, plant defense. INTRODUCTION Allelopathy refers to both harmful and beneficial biochemical interactions between all types of plants, including microorganisms. Up to now, probably one of the most consistent features in studies of allelopathy has been the recognition of the characteristic responses of an organism to an allelochemical, i.e., stimulation or attraction at low concentrations of allelochemicals and inhibition or repellence as the concentration increases (Lovett, 1979, 1989; Rice, 1984). These phenomena have been widely recognized in other growth-regulating chemicals, including herbicides (e.g., Devlin and Witham, 1983; Fedtke, 1982). In the present work, a model for allelopathic interactions is presented and *To whom correspondence should be addressed /93/ / Plenum Publishing Corporation

2 2380 AN ET AL. compared with experimental data from the literature. The model also gives some insight into the underlying processes involved. DESCRIPTION OF THE MODEL It is hypothesized that the characteristic response to allelochemicals is a result of the character of the allelochemicals themselves. An allelochemical is assumed to have two complementary attributes: stimulation and inhibition. These attributes need not necessarily have the same sites of action. As concentration changes, the relative dominance of stimulation and inhibition by the allelochemical alters, so determining the overall property of the allelochemical. This can only be shown through the biological responses when an allelochemical acts on an organism, and is referred to as the biological property of the allelochemical (as opposed to physical or chemical). It is assumed that both stimulation and inhibition have a sigmoidal response to allelochemical concentration. The equation used to define this response is Vm[X] q v - (1) K q -t- [X] q where [X] is the allelochemical concentration, v is the response, Vm is the response at saturating concentration, K is the concentration at which v = Vm/2, and q is a constant that controls the shape of the curve. Equation 1 can be derived from enzyme kinetics to represent the reaction speed of an enzyme substrate reaction with q active sites per enzyme molecule for the substrate (e.g., see Thornley and Johnson, 1990, Chapter 2). It is used here on the basis that it is a versatile curve that has the appropriate response and has some basis in biology. Equation 1 is illustrated in Figure 1. Let the biological response to the allelochemical be P% of the control (no allelochemical present), and let S and I be biological responses to the stimulatory and inhibitory attributes of the allelochemical respectively, so that, applying equation 1, and SmX q S (2a) Kqs + X q Im X q I (2b) K q + X q Note that the same parameter q is used for both S and I; this could be relaxed if necessary. In the study of allelopathy, biological responses to allelochemical are fre-

3 MATHEMATICAL MODELING OF ALLELOPATHY 2381 q=8 q=l g. I I I I Substrate concentration X FIG. 1. The sigmoidal response curve for the reaction response v as a function of substrate concentration, [X], as defined by equation 1. q = 1 and q = 8 are indicated, and the intermediate curves are for q = 2 and 4 as indicated. There is an asymptote at v = Vm; and v = Vm]2 when [X] = K (from Thornley and Johnson, 1990). quently expressed as percent of control. With the control set at 100% it is, therefore, hypothesized that the biological response to allelochemical, P% of control, is given by P = S-I (3a) where S and I are given by equations 2a,b. It is further assumed that P --, 0 as X ~ r which implies I m = S m (3b) Note that although this relationship is derived in the limit P ~ 0 as X --* co the equality applies since Im and Sm are not themselves taken as limiting values. Equation 3a with 2a,b and 3b define the model. Note that there is a maximum at xq = icq - ~Smgqs_ ~-mg q (3C) although this will not exist for all parameter values. Equation 3c is derived by

4 2382 AN ET AL. differentiating equation 3a with 2a,b with respect to X, equating to zero, and solving the resulting equation for X. While we could pursue the range of parameter values for which this maximum will exist further, there is little point since we anticipate that the data to be analyzed will generally display a maximum. The parameters Sin, Ks, Im, K1, and q should be regarded as parameters to be derived by comparing the model to experimental data. Before doing so, it is useful to look at the general behavior of the model, as shown in Figure 2a-c for a range of parameter values. It can be seen from Figure 2 that the model has the general characteristics of the biological response to an allelochemical. At low concentrations, stimulation is observed. As the concentration increases, the pattern changes and inhibition dominates, eventually to the extent that P --, 0. The biological property of an allelochemical will vary both with the allelochemical and species under consideration. APPLICATIONS OF THE MODEL We now turn our attention to the behavior of the model in comparison with experimental data. In the following illustrations, all direct measurements of biological responses have been converted to percent control values. All fitted curves are derived by minimizing the residual sum of squares for the difference between the actual data values and the corresponding value of the model, equations 2a,b and 3a,b. Figures 3-6 show the model behavior compared with experiments testing morphological biological responses to single allelochemicals which are from, or contained in, living plants. Note that the data shown in Figure 6 do not show stimulation, although it is possible that stimulation may have been observed had a lower allelochemical concentration been included in the series. Figure 7 shows the emergence of wild oats (Avena ludoviciana) in response to wheat straw leachate, which is known to contain allelochemicals (e.g., Lynch, 1977). Figures 8 and 9 show the response to alletochemicals at the biochemical level, which is similar to that observed in gross morphological studies (Roshchina et al., 1986; Lovett et al., 1989). It can be seen from the illustrations presented here that the model behavior is in good agreement with a wide range of experimental data taken from the literature. CONCLUSIONS The characteristic responses of an organism to an allelochemical, namely, stimulation or attraction at low allelochemical concentrations and inhibition or repellence as the concentration increases, are almost universally observed. The

5 MATHEMATICAL MODELING OF ALLELOPATHY T " (a) I \"" Allelochemical concentration, X, arbitrary units 200 Co) 8 g 150 \ i00 -g. 50 c~ 0 0 I I I I I AUelochemical concentration, X, arbitrary units 200 (c) 8 g 150 o= i00 "~ 50 0 i ~ i..... r..... I Allelochemicad concentration, X, arbitrary units FIG. 2. Biological response, P, to an auelochemical, X, as given by equation 3a with equations 2a,b and 3b. P is measured as a percentage of control and X has arbitrary units. Default parameter values are S,n = 100, Im = 200, KS = 5, Kx = 30, q = 2, corresponding to the solid lines. The other parameter values are: (a) Sm= 50 (---), 150 (----); (b) Ks = 1 (---), 10 (----); (c) K 1 = 15 (---), 100 (----).

6 2384 AN ET AL '~ v 50" -o doo 800 ' I000 ' Benzylamine concentration (#g/ml) F~G. 3. Response of radicle length of linseed to benzylamine. S,~ = 30, Im , Ks = 70,/<1 = 400, q = 2.5 (data from Lovett et al., 1989). 130 o 115 b 5 1o0 u }- as 70~', I00 Gramine concentration (~I/ml) FIG. 4. Response of radicle length of white mustard to gramine. Sm= 30, I m = 130, K s = 3, K~ = 190, q = 0.8 (data from LoveU et al., 1989).

7 MATHEMATICAL MODELING OF ALLELOPATHY " 4-* llff (9 88- ~j ~S 7& Q 65 ~o 4~ 66 Wo i~o Hordenine concentration (/=I/ml) FIG. 5. Response of radicle length of white mustard to hordenine. S m = 40, lm= 140, Ks = 0.6, /<1 = 200, q = 0.4 (data from Lovett et al., 1989). 150" o 100 w 5 a) ~ 50- O 2bo q 1000 Castanospermine concentration (ppb) FIG. 6. Effect of castanospermine on root growth of lettuce. Sm= 10, I m = 110, K s = 10, KI = 300, q = 2 (data from Stevens and Molyneux, 1988).

8 2386 AN ET AL g loo g 5o I I i I I Wheat leachate concentration (g straw/litre) FIO. 7. Emergence of wild oats (,4. ludoviciana) in response to applied wheat straw leachate. Sm = 45, Im = 145, Ks = 3, KI = 45, q = 2.5 (data from Purvis and Jessop, 1985) ~ 125" w "~ IOO c~ I I I I t Scopolamine concentration (%) FiG. 8. Response of co-amylase activity to scopolamine. S m = 25, Im= 125, KS = O. 1, Kz = 10, q = 1.5 (data from Lovett et al., 1989).

9 MATHEMATICAL MODELING OF ALLELOPATHY u) 1257 o: 100" 4~ q ~ ~ ~,, '----I a-terpineo] concentration (~g) FIG. 9. Olfactory response of weevil larvae to c~-terpineol. S m = 40, Im = 140, Ks = 0.5, Kt = 400, q = 0.5 (data from Selander et al., 1974). model presented here simulates well such responses for a wide range of experimental conditions. The model is simple in structure and is based on the hypothesis that the response to allelochemicals is simultaneously stimulatory and inhibitory in nature: the balance between stimulation and inhibition alters as allelochemical concentration changes. We feel that the model provides a useful means for analyzing experimental data, and may be of value in attempts to predict allelopathic effects in practice. Acknowledgments--We thank the Rural Industries Research and Development Corporation for financial support. REFERENCES DEVLIN, R.M., and WITHAM, F.H Plant Physiology, 4th ed., PWS Publishers, California, 577 pp. FEDTKE, C Biolochemistry and Physiology of Herbicide Action. Springer-Verlag, Berlin, 202 pp. LOVETT, J.V The ecological significance of odor in weeds. Proceedings 5th Conference Asian-Pacific Weed Science Society, Sydney, pp LOVETT, J.V Allelopathy research in Australia: An update, pp , in C.H. Chou and G.R. Waller (eds.). Phytochemical Ecology: Allelochemicals, Mycotoxins, Pheromones and Allomones. Institute of Botany, Academia Sinica Monograph Series No. 9. Taipei, Taiwan.

10 2388 AN ET AL. LOVETT, J.W., RYUNTYU, M.Y., and LIU, D.L Allelopathy, chemical communication, and plant defense. J. Chem. Ecol. 15: LvyCR, J.M Phytotoxicity of acetic acid produced in the anaerobic decomposition of wheat straw. J. Appl. Bacteriol. 42: PURVIS, C.E., and JEssoP, R.S Biochemical regulation of wild oat germination and growth by wheat and wheat crop residues. Proceedings, 1985 British Crop Protection Conference-- Weeds, pp RICE, E.L Allelopathy, 2rid ed. Academic Press, New York, 422 pp. ROSHCH1NA, V.V., RUZIENA, R.K., and MUKttIN, E.N Capsaicine from the fruits of red pepper Capsicum annuum L. as a regulator of photosynthetic electron transport. J. Biochem. Microbiol. 22: SELANDER, J., KALO, P., KANGAS, E., and PERTUNNEN, V Olfactory behaviour of Hylobium abietis L. (Coleoptera, Curculionidae). I. Response to several terpenoid fractions isolated from Scots pine phloem. Ann. Entomol. Fenn. 40: STEVENS, K.L., and MOLYNEUX, R.J Castanospermine--a plant growth regulator. J. Chem. Ecol. 14: THORNLEY, J.H.M., and JOHNSON, I.R Plant and Crop Modelling: A Mathematical Approach to Plant and Crop Physiology. Clarendon Press, Oxford, 669 pp.