Distinguishing between elementary orientation mechanisms by means of path analysis

Anita. Behav., 1992, 43, 371-377

Distinguishing between elementary orientation mechanisms by means of path analysis SIMON BENHAMOU & PIERRE BOVET Laboratoire de Neurosciences Fonctionnelles C.N.R.S., BP71, F-13402 Marseille Cedex 9, France (Received3 May 1990; init&lacceptance 1 July 1990; final acceptance 24 June 1991;MS. number: 3660)

Abstract. It is not always possible to tell by analysing the environmental factors and an organism's receptors whether this organism has reached a given target by chance after making an unoriented movement, or by using one of two types of elementary orientation mechanism, namely taxis and differential klinokinesis. In this paper the main characteristics of these two orientation mechanisms are described. It is then shown how a statistical analysis of the organism's path can be performed to determine which of these two mechanisms is likely to be involved. This type of analysis is illustrated by means of three examples: two simulated paths, one involving taxis and the other differential klinokinesis, and an actual foraging ant's path.

When an organism (whether one is dealing with bacteria or higher animals) reaches a particular target, such as a food item or an especially suitable area, one cannot but wonder whether this has been achieved solely by chance as the result of an unoriented movement, or whether it has actively oriented itself towards the target in the course of its path. The answer is not obvious in many cases because paths, even when they are goal-oriented, often contain a large random component. In the case of a higher animal returning frequently to a given place along a fairly straight path, orientation may have been based on both egocentric (route-based) and exocentric (locationbased) spatial memories (Benhamou et al. 1990). If one excludes this possibility, however, assuming that this is the first time the target has been reached by an individual of the species studied, or that the latter does not possess these types of spatial memory, there remain two main possible elementary orientation mechanisms, which need to be clearly defined: taxis and differential klinokinesis. Our aim in this paper is to show how a statistical analysis of the path can make it possible to determine whether a given target has been reached by an organism as the result of an unoriented movement, and if not, which elementary orientation mechanism, taxis or differential klinokinesis, was involved. This statistical analysis seems to be particularly useful when analysis of the environmental factors and the organism's receptors does not suffice to 0003-3472/92/030371 + 07 $03.00/0

determine the type of stimulus to which the organism may be sensitive in orienting itself towards the target. As in previous papers (Bovet & Benhamou 1988, 1991; Benhamou & Bovet 1989; Benhamou 1989; Benhamou et al. 1990), we assume here that any path can be suitably represented as a sequence of steps and changes of direction (for convenience, changes of direction between successive steps are referred to simply as turning angles throughout this paper). This is true both with theoretical paths that are simulated on a computer and with actual paths. In the latter case, it is important to note that an original continuous path has been first discretized, i.e. represented as a time-dependent sequence of spatial coordinates, at the recording level; to be analysed, this path has to be spatially 'rediscretized' using the procedure described by Bovet & Benhamou (1988): it will then be represented as a sequence of spatial coordinates with a constant step length. This procedure makes it possible to characterize the stochastic structure of the path independently of the step length, which is obviously not a biological parameter. As an introduction to what follows, recall that any angular distribution can be characterized by a mean vector (Mardia 1972; Batschelet 1981): its orientation defines the angular mean of the distribution, and its length, ranging between 0 (uniform distribution) and 1 (punctual distribution), defines the concentration of the distribution around the

O 1992 The Association for the Study of Animal Behaviour 371

372

Animal Behaviour, 43, 3

angular mean. The mean vector length therefore constitutes an inversely related index to the dispersion of the distribution. When the distribution is centred on 0, however, it is possible, provided that the angular values have been expressed between - n and + n rad, to use classical methods to calculate its arithmetical mean (which must be close to 0) and its standard deviation, which ranges between 0 (punctual distribution) and n/~/3 rad (uniform distribution).

CHARACTERISTICS OF TAXIS AND KINESIS To be operative, both taxis and differential klinokinesis require a radial stimulation gradient field centred on the target. This type of field is provided by a physical or chemical intensity factor to which the organism may be sensitive (the potential in mathematical terms) which varies nearly monotonically (always increasing or decreasing) as a function of the distance to the centre of the field. The mathematical shape of this function is unimportant. The particular case of a virtual (located at infinity) target, which obviously cannot be reached but towards which an organism may orient itself, can be taken into account in the form of an axial stimulation gradient field: the potential varies along a fixed axis corresponding to the gradient direction. Taxis

Taxis is an orientation mechanism based on the determination of the gradient direction, i.e. the direction in which the largest local increase in the field potential occurs (Gunn et al. 1937; Fraenkel & Gunn 1961; Sch6ne 1984; Bovet & Benhamou 1990; Doucet & Dunn 1990). If the field potential varies strictly monotonically with the distance to the target, the gradient direction actually corresponds to the target direction at any point in the environment. If the field potential is subject to random spatio-temporal fluctuations or to local anomalies, as occurs with the geomagnetic field for example, the gradient direction might not correspond to the target direction. Various types of taxis have been empirically defined by Fraenkel & Gunn (1961), depending on the way organisms are assumed to determine the gradient direction: klino-taxis, tropo-taxis and

telo-taxis (mnemo-taxis which involves a spatial memory is not taken into account because we restrict the definition of taxis to elementary orientation mechanisms). This classification has been severely criticized (Jander 1975; Van der Steen & Ter Maat 1979; Bell & Tobin 1982). In particular, a reliable classification should take into account the nature of the stimulation gradient field and the type of information the organism is actually able to process to determine the gradient direction. For example, the gradient direction in an atmospheric pressure gradient field can theoretically be determined by several means, depending on the perceived stimulus: (1) by combining the measures provided by a sensitive barometer at several (at least three) non-aligned places with respect to their relative locations (determined using rudimentary path integration); or since the direction of the air flux corresponds to the gradient direction; (2) by turning a directional anemometer until the wind strength measured reaches a maximum; (3) by balancing the two measures provided concomitantly by a pair of directional anemometers oriented in two different directions; or (4) directly by relying on the directional information provided by a weathercock. A special case of taxis is provided by straight, visually guided movements towards a conspicuous target performed by animals with acute eyesight (a conspicuous target can be taken to be the centre of a radial stimulation gradient field, where the gradient direction corresponds to the direction of the light flux). In many cases of taxis, however, the steering of movement may be subject to random variations both because of local fluctuations in the field potential as mentioned above, and because of errors in perceiving the gradient direction owing to the inaccuracy of the receptors. This problem arises consistently whatever the type of taxis involved. The path leading the animal to the target will consequently be variably tortuous. In fact, when the perceived gradient direction is subject to large random variations (because of environmental turbulence as well as perception errors), a purely tactic mechanism systematically forcing the organism to orient itself in the perceived gradient direction at each step will not be very efficient because it will lead the organism to take an excessively chaotic path. One solution which is both plausible and efficient consists of incorporating a forward persistence into the tactic mechanism, in order to prevent the organism from

Benhamou & Bovet: Elementary orientation meehan&ms performing any over-abrupt turning angles. The orientation of any step 0i+ 1in a tactic path can then be expressed as a t-weighted angular mean between the gradient direction perceived at the end of the previous step qbi, and the orientation of this previous step 0i, where t (a constant without units ranging between 0 and 1) is the tactic factor

373

1.0

9

A

0.9

0-8 0"7

Oi +1

:

arctan~ tsin(qb,) + (1 - t)sin(Oi) ~ + b~ [tcos(@i) + (1 -- t)cos(0i) )

|c

0.6

with b = 0 when the denominator is positive and b = 1 otherwise. Because of random fluctuations, o o the perceived gradient direction can be written as

,i,=n+~ where 1) is the local target direction (the target direction varies at each step depending on the location of the organism in the field) and ~ is a random variable obeying an angular distribution with a null mean (mean vector orientation) and a concentration parameter (mean vector length) r. We determined the efficiency of the tactic mechanism in an axial stimulation gradient field (the target direction f2 is constant). As the drift of a tactic path D t (measured along the gradient axis from the starting point) is expected to be proportional to the path length L (E(Dt)= E[cos(0- f~)] x L), the tactic efficiency can be computed as the ratio E(Dt)/L. It is obviously equal to 0 when the tactic factor t is null, and equal to the concentration parameter r of the distribution of random fluctuations when the tactic factor is equal to 1. The tactic efficiency corresponding to intermediate values of t was determined using computer simulations: first, 500 1000-step paths were simulated with each of the nine values of t ranging arithmetically between 0-1 and 0-9, and with each of the 10 values of r ranging arithmetically between 0-1 and 1.0. As the highest tactic efficiency was reached with a value of t equal to 0.1 with most of the values of r, additional simulations involving each of the nine values of t ranging arithmetically between 0.01 and 0.09 were carried out. In most cases, the maximum efficiency of the tactic mechanism is reached with a value of t of approximately 0.1 (Fig. 1). Differential Klinokinesis

Differential klinokinesis has recently been redefined as an orientation mechanism based on the regulation of the path sinuosity (instead of the rate

0.5 0"4 0-3

0-2 O-I 0.(3

0 ' 0 0-1 0.2 0"3 0.4- 0.5 0"6 0.7 0'8 0-9

I'0

Tactic factor

Figure 1. Tactic efficiency(E(Dt)/L) as a function of the tactic factor t and the concentration parameter r of the distribution of random fluctuations in the perceived gradient direction. of change of direction as in the former definition by Fraenkel & G u n n 1961) as a function of variations in the field potential perceived during the movement (Benhamou & Bovet 1989; Benhamou 1989). The sinuosity is a purely spatial index corresponding to the 'amount of turning' associated with a given path length; it is expressed in rad/u 1/2, where u is the unit of length used (Bovet & Benhamou 1988). In contrast to taxis, differential klinokinesis therefore involves no determination of the gradient direction, but only a measure of the variation in the field potential occurring during each step. The magnitude of this variation can be measured either from the successive values provided at the end of each step by a receptor which is sensitive to the field potential, or more directly by a receptor with a sensory adaptation capacity making it sensitive to variations in the field potential. Whatever the type of receptor involved, differential klinokinesis in a radial stimulation gradient field can be defined in terms of the following sinuosity regulating mechanism (Benhamou 1989)

Animal Behaviour, 43, 3

374 S~= Sb(1 + k ( D , - D,_ ,)/P,)

where S i is the momentary sinuosity, S b the basic sinuosity (a constant expressed in rad/u 1/2, k the klinokinetic factor (a constant without units ranging between 0 and 1), D i the distance to the target after the ith step and P, the length of this step. The ratio (D i - D~_ 1)/P~ simply corresponds to the standardization between - 1 and + 1 of the variation in stimulus intensity perceived during the ith step. The orientation of the following step can be recurrently expressed as

01+1=01+ai where the turning angle a, is a random variable obeying a normal distribution (wrapped around the circle) with a null mean and a standard deviation c i = Six/P i (Benhamou & Bovet 1989). When the animal is not too close to the target, however, the radial stimulation gradient field can be suitably approximated by an axial stimulation gradient field and the differential klinokinetic mechanism can be simply expressed as

O

Figure 2. Example of two simulated paths to a target (T) 500 unit lengths (u) away; (a) involves taxis with a tactic factor t = 0.3 and a concentration parameter of the distribution of random fluctuations r= 0.3; (b) involves differential klinokinesis with a basic sinuosity Sb=0'5 rad/u 1/2and a klinokinetic factor k = 0.6.

Si = Sb[1 - k cos(0i - r where 0i is the ith step orientation and @ the gradient direction. The efficiency of this mechanism, as given by the ratio between the expectation of the klinokinetic drift (measured on the gradient axis from the starting point) and the path length (E(Dk)/L), does not depend on the basic sinuosity, and in the absence of any random fluctuations in the field potential, is approximately equal to the klinokinetic factor k (Benhamou & Bovet 1989). Generally speaking, taxis is therefore a more efficient orientation mechanism than differential klinokinesis. When no information is available, however, about the turbulence of the stimulation gradient field or the type and sensitivity of the organism's receptors, the observed orientational performance may have been achieved equally well either by taxis or differential klinokinesis. Two simulated paths, each involving one of these two elementary orientation mechanisms, are given in Fig. 2. The patterns of these paths are clearly similar. Absolute klinokinesis and absolute orthokinesis, which involve regulating the sinuosity and the speed of movement, respectively, as a function of the local value of the field potential (and not as a function of its variations as in the case of differential klinokinesis) have been shown to be efficient

space-use mechanisms in heterogeneous environments, but they are not orientation mechanisms: they enable an organism to regulate efficiently the time it spends in the various parts of its environment, but not to move towards a given target. The last possible kinetic mechanism, differential orthokinesis, which involves regulating the speed of movement as a function of variations in the field potential perceived during the movement, seems to have no biological applications (Benhamou & Bovet 1989). Furthermore, the path of an animal reaching a target after making an unoriented movement, as described by a first-order correlated random walk movement (Bovet & Benhamou 1988), may also sometimes give the mistaken impression that it is goal-oriented, especially if the sinuosity of the path is low. This type of path can be taken to constitute a particular case of the differential klinokinetic path where the klinokinetic factor k is equal to 0.

STATISTICAL DISCRIMINATION

PROCEDURE The most satisfactory way of determining whether a tactic mechanism or a differential klinokinetic mechanism or no orientation mechanism at all is

Benhamou & Bovet: Elementary orientation mechanisms used by the organism under study in a stimulation gradient field would ideally consist of determining from the biological point of view whether this organism is able to determine directly the gradient direction, or whether it is only able to perceive the variations in field potential occurring during its movements or whether it is not sensitive at all to the gradient field. Unfortunately, the answer to this question is not always easy to establish. For example, oriented movements made by bacteria in a chemical gradient field have usually been called chemotaxis (Adler 1966, 1969; Berg & Brown 1972; Koshland 1979). It is not very clear, however, whether the orientation of bacteria relies on sinuosity regulation based on variations in the chemical concentration perceived during the movements (by means of adapting chemoreceptors), which would mean that differential klinokinesis is involved, or whether this orientation relies directly on the perception of the chemical gradient direction (via the integration of information provided by several chemoreceptors without any sensory adaptation), which would mean that taxis is involved. We propose to show that, in the absence of any reliable answers in biological terms, it is theoretically possible to distinguish between differential klinokinesis and taxis by analysing the path statistically, once it has been 'rediscretized' (i.e. expressed as a sequence of steps using a constant step length; see Bovet & Benhamou 1988). A simple statistical procedure of distinguishing between taxis and differential klinokinesis (and unoriented movements) consists of computing the distribution of 'side-dependent' turning angles: since step orientations are measured counterclockwise in accordance with the trigonometrical usage, side-dependent turning angles are measured counterclockwise (a i = 0 i + 1 - 0i) if the previous step points to the right of the target (i.e. sin(01-f~i) < 0), and clockwise (u i = 0 i - 01+ 1) if the previous step points to the left of the target (i.e. s i n ( 0 i - D / ) > 0, where 0 i is the orientation of the ith step and f~i the local target direction measured at the end of this step). In the case of differential klinokinesis, the probability that the organism will turn right or left after each step is equal, whatever its previous step orientation; whereas in the case of taxis, the probability that the organism will turn right will be higher than the probability that it will turn left when its previous step is oriented to the left of the target and vice versa. Consequently, the mean of the distribution of side-dependent turning angles

375

will not differ significantly from 0 in the case of differential klinokinesis, but will be significantly higher than 0 in the case of taxis. Furthermore, no significant auto-correlation will occur between successive turning angles in the case of differential klinokinesis, whereas a negative auto-correlation is to be expected in the case of taxis because of the successive steering corrections. This distinction is not very reliable, however, because the auto-correlation may be very small in cases where taxis involves a low tactic factor. Another means of ascertaining whether a tactic or a differential klinokinetic mechanism has been used is described below. For this purpose, let us call the 'reference direction' the f-weighted angular mean between the orientation of any step and the local target direction measured at the end of this step, wherefis a parameter ranging between 0 and 1 ~i = arctan~" [fsin(~,) + (1 ---f)cos(0,)] ] + brt ([fcos(f~i) + (1 --f)cos(Oi)]J with b = 0 when the denominator is positive and b = 1 otherwise. Each step in a tactic path is more or less directly oriented towards the target, depending on the tactic factor t, and on the concentration parameter r of the distribution of random fluctuations. Consequently, the distribution of step orientations in relation to the previous reference direction (0/+ 1 - 8 i ) is centred on 0 whatever the value off, but its dispersion is minimal only for one particular non-zero value o f f depending on t and r. Using computer simulations, this minimizing value of f was found to be equal to t l/,/r. In the case of differential klinokinesis, each step first tends to be oriented in the same direction as the previous one (by construction), and, second (as a consequence), towards the target, so that the distribution of step orientations in relation to the previous reference direction is also centred on 0 for any value off, but its dispersion is minimal when the value o f f is equal to 0 (this distribution then corresponds to that of turning angles (0 i + 1 - 0i) since in this case ~i is equal to 0/). Unfortunately, there is no way to calculate directly the value of f that minimizes the dispersion of this distribution. A useful solution consists of analysing the distribution of step orientations in relation to a set of reference directions, defined by several values off, e.g. the 11 values o f f arithmetically ranging between 0 and 1. If the lowest dispersion is obtained with a value of f different from 0, this will indicate that a tactic mechanism is likely to be involved whereas a null

Animal Behaviour, 43, 3

376

value of f will indicate that a differential klinokinetic mechanism is likely to be involved. When a differential klinokinetic mechanism has been identified, it can be of great interest to determine the values of the sinuosity-regulating parameters: the basic sinuosity S b and the klinokinetic factor k. This requires sorting the turning angles cti into two equal classes depending on how close the previous step orientation and the target direction at the beginning of this step are (measured by cos(0 i - ~ i - 0): half values of the turning angles that occur after steps with orientations closest to the target direction are put in the first class and the other half in the second class. The two distributions of turning angles obtained in this way should be centred on 0. In the case of an unoriented movement, the standard deviations (Yl (first class) and (Y2 (second class) do not differ significantly from each other. This can be tested statistically using the Ftest on the variance ratio. The sinuosity of the path is then given by S = 1.2 (y/x/R, where (yis the overall standard deviation of the distribution of turning angles and R the rediscretization step length used in the path analysis (Bovet & Benhamou 1988). In the case of differential klinokinesis, in contrast, (Yl is significantly lower than (Yz-Computer simulations showed that the klinokinetic factor can be reliably computed as k = 1.5(o-2 -- (yl)/(al -~- (Y2) and the basic sinuosity as S b = 1-2 (yl/(1 --k)x/R It is important to note that a tactic mechanism leads, especially with high values of t, to a value of (yx that is lower than that of (yz, since the organism will, on average, perform larger turning angles after steps taking it away from the target than after those bringing it closer to the target. Computing the values (Yl and (yz is therefore only meaningful when path analysis has suggested that a differential klinokinetic mechanism may be involved.

EXAMPLES OF POSSIBLE APPLICATIONS

To illustrate our procedure, let us consider the two simulated paths given in Fig. 2. Statistical analysis of the tactic path (Fig. 2a) showed that the mean of the distribution of side-dependent turning

Figure 3. (a) Path of a foraging ant to a humid area (A) 20 cm from the exit hole of colony nest (C). (b) The same path after rediscretization with step length R=0.5 cm. Path analysis showed that a tactic mechanism may be involved.

angles was significantly higher (one-tailed t-test: P<0.00001) than 0, which indicated that a tactic mechanism was likely to be involved. The lowest dispersion of the distribution of step orientations in relation to various reference directions is obtained with a value o f f equal to 0-1, which is correct since this path was simulated with r=0.3 and t=0.3. Similar results were obtained with all the tactic paths simulated with various values of t and r to establish the relationf = t l/Jr. By contrast, statistical analysis of the klinokinetic path (Fig. 2b) showed that the mean of the distribution of side-dependent turning angles did not differ significantly (one-tailed t-test: P>0.25) from 0. Furthermore, the lowest dispersion of the distribution of step orientations was obtained with a value of f equal to 0, which confirmed that a differential klinokinetic mechanism was likely to be involved. Sorting analysis of turning angles yielded values for the basic sinuosity and the klinokinetic factor that were similar to those used in the simulation: Sb=0"5 rad/u 1/a and k = 0"6. Thus, statistical analysis of this path correctly showed that a differential klinokinetic mechanisms was involved, and made it possible to determine the correct values of the sinuosity-regulatingparameters. Similar results were obtained with all the klinokinetic paths simulated with various values ofS band k to deduce these

Benhamou & Bovet: Elementary orientation mechanisms

values from the values of the standard deviations ~1 and ~2. In a previous study (Bovet et al. 1989), the paths o f Serrastruma lujae ants foraging in a homogeneous wet environment were studied in the laboratory. These small nearly blind ants usually encounter conditions o f this kind in their natural environment during the humid season. During the dry season, however, they have to forage in an environment where prey (collembolans) are aggregated in humid patches. This type of environment was reproduced in the laboratory by A. Dejean. The ant colony nest was set up at the centre of an arena, and a wet disc was randomly positioned in the arena. To illustrate the statistical procedure developed above, we simply give here a sample analysis of one ant's path from the nest to the wet disc (Fig. 3). This p a t h was rediscretized with a step length R = 0 . 5 cm. The mean of the distribution of side-dependent turning angles was significantly higher (one-tailed t-test: P < 0 . 0 0 1 ) than 0. Furthermore, the lowest dispersion of the distribution of step orientations was obtained with a value of f equal to 0-3. Consequently, it can be concluded that this ant may be able, during the dry season, to rely on the humidity gradient direction to orient itself tactically towards suitable foraging areas. It is therefore possible from a fairly simple statistical analysis of a path to determine whether a particular organism reached a target after making an unoriented m o v e m e n t or by using a differential klinokinetic or tactic orientation mechanism. This type of analysis should be very useful when it is not possible to establish which type of stimulus (field potential or gradient direction) and which type o f information processing are actually used by an organism to orient itself towards a given target.

ACKNOWLEDGMENTS We are very grateful to Alain Dejean for permitting us to use his unpublished data. M a n y thanks are due to Marylin G r a n j o n for her technical help. Jessica Blanc helped to revise the English.

377

REFERENCES Adler, J. 1966. Chemotaxis in bacteria. Science, 155, 708-716. Adler, J. 1969. Chemoreceptors in bacteria. Science, 166, 1588 1597. Batschelet, E. 1981. Circular Statistics in Biology. London: Academic Press. Bell, W. J. & Tobin, T. R. 1982. Chemo-orientation. Biol. Rev., 57, 219-260. Benhamou, S. 1989. An olfactory orientation model for mammals' movements in their home ranges. J. theor. Biol., 139, 379-388. Benhamou, S. & Bovet, P. 1989. How animals use their environment: a new look at kinesis. Anim. Behav., 38, 375-383. Benhamou, S., Sauv6, J.-P. & Bovet, P. 1990. Spatial memory in large scale movements: efficiency and limitation of the egocentric coding process. J. theor. Biol., 145, 1-12. Berg, H. C. & Brown, D. A. 1972. Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature, Lond., 239, 500-504. Bovet, P. & Benhamou, S. 1988. Spatial analysis of animals' movements using a correlated random walk model. J. theor. Biol., 131, 41%433. Bovet, P. & Benhamou, S. 1990. Adaptation and orientation in animals' movements: random walk, kinesis, taxis and path integration. In: Biological Motion (Ed. by W. Alt & G. Hoffmann), pp. 297-304. BerlinHeidelberg: Springer Verlag. Bovet, P. & Benhamou, S. 1991. Optimal sinuosity in central place foraging movements. Anim. Behav. 42, 5~62. Bovet, P., Dejean, A. & Granjon, M. 1989. Trajets d'approvisionnement ~t partir d'un nid central chez la fourmi Serrastruma lujae. Insectes Sot., 36, 51-61. Doucet, P. G. & Dunn, G. A. 1990. Distinction between kinesis and taxis in terms of system theory. In: Biological Motion (Ed. by W. Alt & G. Hoffmann), pp. 498-509. Berlin-Heidelberg: Springer-Verlag. Fraenkel, G. S. & Gunn, D. L. 1961. The Orientation of Animals: Kineses, Taxes and Compass Reactions. New York: Dover. Gunn, D. L., Kennedy, J. S. & Pielou, D. P. 1937. Classification of taxes and kineses. Nature, Lond., 140, 1064. Jander, R. 1975. Ecological aspects of spatial orientation. A. Rev. Ecol. Syst., 6, 171-188. Koshland, D. E. 1979. A model regulatory system: bacterial chemotaxis. Physiol. Rev., 59, 811-862. Mardia, K. V. 1972. Statistics of Directional Data. London: Academic Press. Sch6ne, H. 1984. Spatial Orientation; the Spatial Control of Behavior in Animals and Man. Princeton: Princeton University Press. Van der Steen, W. J. & Ter Maat, A. 1979. Theoretical studies on animal orientation. I. Methodological appraisal of classifications. J. theor. Biol., 79, 223-234.