Pergamon 0022-1910(95)00006-2

J. Insect Physiol. Vol. 41, No. 7, 545-553, 1995 pp. Copyright 0 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0022-1910/95 $29.00 + 0.00

Review

The Geometric Analysis of Feeding and Nutrition: a User’s Guide S. J. SIMPSON,* D. RAUBENHEIMER*

Received 12 December 1994

A practical guide is provided for the use of recently developed techniques for the analysis of feeding and post-ingestive processing in insects. We argue that a primary aim of studies in nutrition should be the experimental estimation of the positions of functional optima (the intake, nutrient and growth targets) in nutrient space. Next, rules of behavioural and physiological compromise can be investigated under nutritional and other environmental conditions which render these optima unreachable by the animal. Estimates of target positions and decision rules provide key points of reference for mechanistic, functional, developmental and evolutionary studies, carried out under laboratory or field conditions.

Feeding Nutrition Food utilization Locusts Caterpillars

Recently we have developed a geometric framework for the study of insect nutrition [expounded in Raubenheimer and Simpson (1993, 1994); Simpson and Raubenheimer (1993a) and Simpson et al. (1995) the last of these providing the most general account]. In essence, the principles behind this scheme are simple, and we believe that is has considerable power to organize and integrate data and to provide specific, testable hypotheses. Behind the framework’s simplicity, however, lie issues which are more complex. Our aim here is to provide a straightfor- ward, practical guide which will help the uninitiated to apply the framework, either in laboratory or field studies. We have also highlighted what we see to be key hypoth- eses that warrant future investigation.

TARGETS

Central to the scheme is the concept of functional targets in nutrition. There are points in multidimensional “nutrient space”, where each dimension (axis) represents a different nutrient. The optima1 amount and balance of nutrients required by an animal over a stated time period is termed the nutrient target. The proportion of this which can be optimally allocated to growth (somatic, reproductive or storage) is called the growth target. Approaching the nutrient and growth targets involves two stages: feeding, and post-ingestive processing. The

*Department of Zoology and University Museum, University of Oxford, South Parks Road, Oxford OX1 3PS, U.K.

amount and balance of nutrients which must be ingested to achieve the nutrient target, at minimal cost to fitness, is termed the intake target. The intake and nutrient targets differ from each other within each dimension by the amount of nutrient that is unavoidably wasted when feeding on the optima1 diet. Such wastage reflects intrin- sic constraints on post-ingestive processing.

The nutrient, growth and intake targets are so-called “global” optima, i.e. they represent the best possible outcome for that genotype (over a specified period during its ontogeny) under the environmental conditions prevailing during its evolution (Raubenheimer and Simp- son, 1994). Even higher fitness might be attained, leading, in the extreme, to the organism tending towards “univer- sal” optimality, either through further evolution or a change in the environment (e.g. with the introduction of an alien food plant species or a change in climate). In a universally optima1 animal the intake and nutrient targets would be the same, i.e. it would waste nothing in acquiring and processing nutrients from an optima1 diet.

Foods are represented within nutrient space as lines radiating from the origin, their angle to the various axes being determined by the relative amounts of different nutrients they contain. Depending, among other things, on which foods are available to an animal, the targets may or may not be reachable. When an animal is constrained, either by properties of the food (includ- ing its nutritional, physical and allelochemical characteristics) or other ecological factors (e.g. tempera- ture or predators) from reaching a target, it would be

546 REVIEW

expected to make the best possible compromise under available conditions and achieve the “local” fitness optimum (“point of best compromise” Sense Rauben- heimer and Simpson, 1993).

The relationship between a point of best compromise and a target indicates the weighting assigned by the animal’s control mechanisms to balancing the conse- quences of under- or overshooting the target under those specific circumstances. An array of such points across nutrient space forms a pattern describing a rule of compromise (“decision rule”) (Raubenheimer and Simp- son, 1993; Simpson and Raubenheimer, 1993a).

We have argued that estimating the positions of the targets should be a primary aim of nutritional studies, since such estimates provide the necessary referents for testing physiological, functional, ecological or evolution- ary hypotheses (Simpson and Raubenheimer, 1993a; Raubenheimer and Simpson, 1994). There are, however, practical decisions which must be made before embarking upon experiments to locate the targets: (a) which nutrient axes (how many dimensions) will be investigated, across which ranges and in what detail will they be sampled?, and (b) how will variation in levels of these nutrients be achieved in the experimental foods? Jointly, these affect how representative are estimates of target points.

STAGE 1: PRELIMINARY PRACTICALITIES

(a) Restricting the number of axes to investigate

Given that growing insects require more than 30 nutrients, to explore the whole of nutrient space in an experiment would usually not be feasible. It is therefore necessary to restrict the scope of an investigation to a subset of axes, and to work within biologically realistic ranges on these axes.

A balance needs to be struck between what is exper- imentally tractable and how representative are the esti- mates obtained of the target positions. The latter will depend on how many axes are chosen and the number of levels of each nutrient tested. An experiment involving 5 levels of each of two nutrients would require 25 treatment groups. Adding a third axis would necessitate either 125 treatments, or choosing a smaller number of levels per nutrient.

One way of reducing the number of axes included in the experimental model is to combine two or more nutrients into a single, compound axis. A common example is to use one axis for nitrogenous nutrients (see Raubenheimer and Simpson, 1993; Simpson and Raubenheimer, 1993a; Abisgold et al., 1994; Simpson et al., 1994). In other cases, nutrient space has been collapsed into two dimensions by plotting one specific (“focal”) axis [minerals (Trumper and Simpson, 1993); protein (Stockhoff, 1994); water (Raubenheimer and Gade, 1994)] against an axis com- pounded of all other nutrients.

It is important to ensure that both the blend of constituent nutrients in compound axes (e.g. the type of protein), and the levels of all the excluded nutrients, are

in balanced proportions with respect to the animal’s needs. Otherwise, observed outcomes might represent points of best compromise, rather than the target co- ordinates for the focal nutrients. However, there are both good experimental and theoretical grounds for suggesting that close approximations of target positions for the focal nutrients can be obtained when levels of micronutrients (vitamins, minerals, sterols, etc.) are within broad tolerance zones (see Trumper and Simpson, 1993; Abisgold et al., 1994). Given the leverage that the macronutrients have over behavioural and physiological control systems (Simpson and Raubenheimer, 1993b; Trumper and Simpson, 1993) it is important that they be present at close to optimal proportions when they are not the focal nutrients in a study.

An efficient experimental programme for investigating a phytophagous insect would be first to study the macronutrients (e.g. protein and carbohydrate) as focal axes in a bicoordinate plot, ensuring that all other nutrients are present in the test foods at levels known to support good performance over the experimental time scale. If so desired, the resulting estimates of optimal macronutrient levels can then be used as the starting point for investigations into other dietary constituents. These further nutrients could then be tested one at a time against an axis compounded of all other nutrients. Collectively, such 2-dimensional snap-shots will build to a multidimensional picture.

Another way of ensuring that intake of non-focal nutrients is close to optimal is to allow the animal to regulate one or more of these independently of the focal nutrients. For instance, when working with locusts (Locusta migratoria), we have avoided the problem of choosing an appropriate level of water in the food by providing dry foods and allowing the insects to drink ad libitum. It is, of course, necessary to ensure that the insect has the capability to regulate in this way.

(b) Generating variation in food nutrient levels

There are three ways of generating the necessary variation in the nutrient levels of test foods: (1) use artificial foods which are to some degree chemically defined; (2) make use of naturally occurring variation in real foods, and (3) experimentally generate variation in real foods.

Because the experimenter has complete control over their content, chemically-defined artificial foods have many advantages. It is critical, however, that they can support good performance over the period under inves- tigation, otherwise pathological responses are likely. Unfortunately, suitable artificial foods are not available for many species (although geometric analyses may provide a systematic means of developing such diets).

Natural foods also have advantages, as well as practi- cal drawbacks. They offer the possibility of working under laboratory, semi-field or field conditions and, if chosen correctly, are indisputably ecologically “rel- evant”. Care must be taken, however, in quantifying the nutritional composition of such foods. There are four

REVIEW 547

aspects here which should be borne in mind: (1) analyti- cal procedures should be used which appropriately measure the nutrients in question (e.g. under some circumstances total nitrogen is not a good indicator of protein content); (2) only those food parts which are actually ingested by the insects should be measured (e.g. it is pointless to grind up a whole leaf, if only the adaxial, epidermal tissue between veins is eaten); (3) given that there may be marked temporal variation in food quality, it is important that food parts are sampled by the experimenter close to the time when associated tissue was ingested by the insects, and (4), ideally, nutrients should be measured in the biochemical (allelochemical, pH, etc.) context encountered in natural foods.

STAGE 2: ESTIMATING TARGET POSITIONS

Targets are defined functionally (i.e. in terms of fitness). Their positions (integrated over a given time period) can be estimated either (1) by measuring the performance consequences of feeding for that period on food varying in nutrient levels, or (2) by studying the features of the mechanisms involved in nutritional homeostasis. To the extent that such mechanisms (whether behavioural or physiological) are the result of natural selection, their output should tend towards achieving the functional optima.

The following discussion will consider the simple case of two nutrient axes, A and B, where all other nutrients are present in balanced proportions. Simple methods for estimating the intake, growth and nutrient targets are presented. It is not necessary that all three targets be estimated by all possible means. Very useful data can be derived when only one (say, the intake target) is measured by one of the methods.

(a) Using performance criteria

When performance over a given period is plotted against proportions of A and B in the food, then the highest point on the resulting 3-dimensional surface indicates the composition of the nutritionally optimal food. From here it is possible to locate the targets. The intake target is the amount of A and B ingested over that period by insects fed the optimal food. Similarly, the growth target is the amount of growth derived by these animals from ingested A and B, and the nutrient target is ingested A and B which is used for all the animal’s requirements, including growth, to fuel metabolism and to produce necessary secretions.

The most pressing practical issue is which perform- ance criteria to use? By definition, they must represent inclusive fitness.

In some cases a single, simply measured variable exposes the performance peak, while for other species that same variable either does not resolve a peak, or worse, leads to incorrect inferences. For instance, in larval pea aphids (Acvrthosiphon pisum) dry weight growth provides a good indicator of performance (Abis- gold et ul., 1994; Simpson et al., 1994) while in locusts

it is entirely misleading (Raubenheimer and Simpson, 1993). Also in the locust, development time and mor- tality both distinguish a broad performance plateau but are unable to resolve where on the plateau the optimal food lies (Simpson and Raubenheimer, 1993a). Accord- ingly, more performance variables are needed in the model. In an ongoing laboratory study we are measuring a suite of variables, including development time, time spent locomoting and feeding, respiratory rate, mor- tality, size and body composition at the adult moult, resistance to starvation as an adult, and female repro- ductive success. Complex trade-offs exist between such life-history characteristics and these should be included in the final fitness model (Stearns, 1992).

(b) Using evidence from mechanisms

The approach here is to measure the outcome over a stated period of the behavioural and physiological mech- anisms which produce nutritional homeostasis. This is achieved by identifying any points of intake and growth which are defended in nutrient space. In other words, it is important to show that changed nutritional circum- stances result in the animal altering its response (e.g. diet selection or volume eaten), thus conserving the outcome (point achieved in nutrient space). Defence is not demon- strated when the animal conserves its reponse, allowing the outcome to vary.

Intake target. An example of a simple yet powerful experimental design is illustrated in Fig. 1. It is based on an experiment performed on locust nymphs (Chambers et al., 1995). Here locusts were provided with one of four choices of foods varying in protein and carbohydrate content. The four treatment groups reached the same point in nutrient space, although achieving that point involved their eating quite different dry-weights of the choice foods. A similar example can be found in Trumper and Simpson (1993) for salts vs non-mineral nutrients.

The only point of caution is that foods must be chosen which, between them, encompass the intake target (otherwise the target cannot be reached), but which are not so extreme in composition as to elicit pathological responses. For instance, in a study with two nutrients, A and B, the most all-encompassing foods would be those lying on the axes themselves (i.e. A but no B vs B but no A). An insect given such a choice could, in theory, move anywhere on the nutrient plane. The risk is, however, that such foods are so unbalanced as either not to be perceived by the insect as edible, or to elicit unrepresentative responses from control systems which have evolved to operate in the simultaneous presence of both nutrients (see Raubenheimer and Gade, 1994).

Another means of identifying the intake target is to make use of utilization plots (Raubenheimer and Simp- son, 1994). This involves constructing nutrient budgets for insects fed foods varying in nutrient content (Raubenheimer and Simpson, 1995). The method is illustrated diagrammatically in Fig. 2.

Growth target. The most effective means of estimating

548 REVIEW

the growth target is to provide insects with one of several foods varying in nutrient content, then to measure total consumption over the experiment and, at the end, deter- mine the chemical composition of the carcases. Achiev- ing similar growth points despite having ingested different amounts of nutrients, indicates that growth is regulated by differential utilization of ingested nutrients (Raubenheimer and Simpson, 1993; Simpson and Raubenheimer, 1993a; Fig. 3). Presenting data in the form of utilization plots is the best way of visualizing the target coordinate for each individual nutrient (Rauben- heimer and Simpson, 1994; see Fig. 2).

Nutrient target. Utilization plots also provide a power- ful tool for locating the nutrient target (Raubenheimer and Simpson, 1994; Fig. 2). What is required is to distinguish the amount of ingested nutrient which is wasted from that which is utilized (for growth, metab- olism, etc.), and to plot one or the other against intake of that nutrient (e.g. see Zanotto et al., 1993).

(c) Compatibility of measures: performance criteria us mechanisms

Estimates of target positions based on performance criteria should agree with those gained by studying mechanisms. If they do not, then some critical dietary or

*foodA

Food palrings:

AvsC

1 B “S D

f 30- A vs D

B “S c z

.E 20- a

I .fclod B fcQdD

z 10 - l

foodc

01 0 10 20 30 40 50

% Nutrient A in food

6

defended IT

l -

Dry weights eaten to reach IT.

167mgAvs333mgC 333mgBvs167mgD

167mgAvs167mgD 333 mg 6 vs 333 mg C

0 50 100 150 200

Nutrient A eaten (mg)

FIGURE 1. A simple experimental design for estimating the position

of the intake target for two nutrients, A and B. Insects are provided with one of four pairings of food and the quantity eaten is measured over a given time period. (B) plots amounts of A and B eaten. All four

treatments have converged at the same point through insects eating

different dry weights of their foods. Such “defence” of a point on the nutrient plane is good evidence for that point representing the intake

target (IT) for A vs B. In this example the intake target lies on the

mid-rail, but this need not be the case.

A 200

160

=

E 160

P 140

j; 120

$Y 100 4 E 60

E 60 ; 40 IT(A)

20

I.J

x = x 8 NT(A) IT(A) 0

0

0 50 100 150 200 0 50 100 150 200

Nutrient A eaten (mg) Nutrient B eaten (mg)

ids’;/

z’ 0 50 100 150 200 I 0 50 100 150 200

Nutrient A eaten (mg) Nutrient B eaten (mg)

FIGURE 2. Diagrammatic illustrations of utilization plots, based on

data from the hypothetical experiment which is featured in Table 1 and

in the other figures. The plots are derived from gravimetric budgets (see

Raubenheimer and Simpson, 1994, 1995). (A) and (B) plot the amount

of nutrient (A or B) which is wasted (i.e. is not utilized for fitness-en-

hancing purposes such as growth, to fuel essential metabolism, or to

produce necessary secretions) against the amount of that nutrient

eaten. The change in slope which occurs for both nutrients at an intake

of 100 mg indicates the position of the intake target coordinate (IT)

(see Fig. 1). The nutrient target coordinate for each nutrient (NT) is

simply the intake target corrected for unavoidable wastage and can

thus be derived as follows: NT, = IT, x (1, - WJ/l”. where I and I+’

are the amount of nutrient n eaten and wasted, respectively. (C) and

(D) plot the amount of nutrient which is lost from the body (i.e.

respired, secreted, egested, etc.) against intake. Again, the point of

slope change indicates the intake target coordinate for that nutrient.

The growth target coordinate (GT) is the intake target corrected for

“non-growth”, i.e.: GT, = IT, x (1, - 0,)/1,, where D is the amount of

n not retained for growth (dissociated). The upper graphs (A) and (B)

could equally well have plotted nutrient utilized for fitness-enhancing

purposes (rather than that wasted) against intake, while in the lower

graphs (C) and (D), nutrient retained for growth could have been

substituted for nutrient dissociated from the insect (see Raubenheimer

and Simpson, 1994).

other ecological variable has been neglected and be- comes the focus for further work (Simpson and Rauben- heimer, 1993a). For instance, laboratory experiments exclude certain key ecological variables, such as preda- tion and disease, which might be expected to be a function of developmental period and behavioural time budgets. These risks can be explored mathematically by introducing time-dependent mortality into fitness models. Field studies could then test the predictions of such models. For example, it may be that incorporating a mortality risk into a laboratory-derived fitness model leads to alignment of the target estimate with that recorded from studies of mechanisms. Such a mortality level then becomes a prediction for field testing.

REVIEW 549

140

120

100

G

?z 60 m Z -$ 60

z'

40

20

0

intake array

l

Grwiih array

0 20 40 60 60 100 120 140

Nutrient A (mg)

FIGURE 3. Intake and growth arrays from data shown in Table 1.

Note the convergence of A- and B-derived growth relative to the points

of intake. This point of convergence indicates the position of the

growth target.

STAGE 3: EXAMINING DECISION RULES

Having estimated the position of a target with respect to two or more nutrients it now becomes possible to investigate the nature of any interactions occurring between the regulatory systems for those nutrients. The aim is first to establish whether or not such systems interact. If they do, then the next question to ask is whether one is weighted more heavily than the other in terms of physiological regulation, and, if so, to what extent. This could in turn be considered in terms of the fitness consequences for the animal.

These questions can be answered by providing insects with one of a range of foods (nutrient rails) which render the targets unachievable, and then interpreting the shapes of the arrays of behavioural and physiological outcomes (Raubenheimer and Simpson, 1993; Trumper and Simpson, 1993; Abisgold et al., 1994). The form of an array defines the decision rule employed by insects unable to reach the target.

For explanatory purposes we will illustrate the pro-

cedures involved by reference to the hypothetical example shown in Fig. 3 and Table 1.

(a) Generating an array

An experiment has been undertaken in which insects (standardized for developmental and nutritional history) were provided with one of 7 foods varying in the ratio of A:B. The position of the intake target is known from earlier work (Fig. 1). Survivorship, intake, growth and faecal composition were recorded over a given period. The resulting 2-dimensional arrays of intake and growth for the 5 foods which supported successful development are shown in Fig. 3.

@) Selecting treatments

When interpreting the shape of an array it is import- ant to include only those treatments in which insects have successfully demonstrated nutritional homeostasis. For instance, the behavioural and utilization strategies followed in treatments where most insects died (perhaps because the ratios of nutrients in the foods were too extreme, or because the foods were overly diluted by indigestible bulk) are not usefully compared with those where insects grew and survived normally. In our hypo- thetical example the two extreme foods did not support growth and survivorship, while insects performed equally well on the other 5 foods in terms of the performance criteria that were used [they all achieved the growth target and had similar survivorship (Figs 3 and 4)]. More complete fitness measures would almost certainly have indicated a peak rather than a plateau, but it is clear that those insects on the performance plateau have demonstrated successful homeostasis and are thus worth comparing, while insects fed the extreme foods should be excluded.

Real data might not be so clean (although Fig. 3 is closely based on what we found for locusts), and in such instances it may be necessary to choose a somewhat arbitrary threshold for “successful” performance (see, e.g. Abisgold et al., 1994).

TABLE I. An hypothetical data set showing intake by insects of 5 foods (Fl-5) varying in their ratio of two nutrients, A and B. The nutrients

differ in molecular weight (A = 200, B = 100) and energy content (A = 8 kcal/g, B = 4 kcal/g). The same data are plotted in three different units:

dry weight, mols, and energy equivalents. Columns headed B:A contain the ratio of the two nutrients in each food (i.e. the slope of the food rail); A and B are the amounts of each nutrient eaten, and k is the scaling factor which, when applied to nutrient A, moves the point of intake

reached on a given food rail so that it is at 90” to the intake target (hence, transforms the intake and intake target points to be consistent with

closest distance optimization) (see equation 3). Food 3 (in bold) is optimal and its rail bears the intake target

Units

dry weight eaten (mg) mol x 10m4 eaten kcal eaten _

Fl.

F2. F3.

F4. F5.

B:A A B k B:A A B k B:A A B k

0.25 117.6 29.4 1 0.5 5.88 2.94 2 0.125 0.94 0.12 0.5

0.5 120.0 60.0 1 1.0 6.00 6.00 2 0.25 0.96 0.24 0.5

1.0 100.0 100.0 - 2.0 5.00 10.00 - 0.50 0.80 0.40 -

2.0 60.0 120.0 1 4.0 3.00 12.00 2 1.00 0.48 0.48 0.5

4.0 29.4 117.6 I 8.0 1.47 11.76 2 2.00 0.24 0.47 0.5

REVIEW 550

100

80

ISI

.E 2 2 60

z Q I 5 40

9 0

20

0 I

4

I Performance plateau \

I I I I / ,

( RatioB:A 1’oO:O SE:20 67333 50450 33?67 2Cf13o o:ioo ) Rail

FIGURE 4. A plot of performance against food from the hypothetical

experiment. Note how foods lacking A or B failed to support

survivorship, while all other treatments lie on a performance plateau

which will also contain the optimal food. More detailed performance

criteria would be likely to resolve the foods on the plateau,

(c) Inadmissible regions in the array

There are regions in nutrient space where it is expected that no points should lie. Animals should neither simultaneously undershoot or overshoot the target in all dimensions (see Fig. 5). If treatment means do lie in these regions for a particular study, and the insects nevertheless perform sufficiently well to be included in the analysis (see above), then it suggests that additional nutrient dimensions need to be included in the model when interpreting the form of the array. For instance, erecting a third nutrient axis

200

0

lndereaten A, overeaten B Overeaten A and B

Undereaten A and B Overeaten A, undereaten B

c 0 50 100 150 200

Nutrient A eaten (mg)

FIGURE 5. A graph indicating the basic regions on a nutrient plane

with respect to the intake target (IT). Insects which end up in the bottom left quadrant have failed to reach the target level for either nutrient, while those in the top right quadrant have exceeded the target

for both nutrients. The two most likely explanations for treatment means falling in these two regions are: (1) that the insects have

exhibited pathological responses, and (2) that the model has omitted key nutrients and requires one or more further axes to be included.

by splitting a compound axis into two components may turn a 2-dimensional array with inadmissible data points into a continuous 3-dimensional form.

(d) Do the nutrients interact?

Certain shapes of array indicate that there is no interaction between the mechanisms regulating intake (or utilization) of different nutrients. In the case of 2-dimensional intake arrays, the behavioural rules de- scribing those arrays are: (1) eat until the intake level of nutrient A (or B) is reached, irrespective of how much B (or A) is consumed; (2) eat until at least the intake target levels for A and B are reached, even if this means eating more than the target level for one to reach the target for the other; and (3) eat until the intake target level for A or B is reached then stop eating, even if the target for the other nutrient is not reached [see Fig. l(bd) in Simpson and Raubenheimer, 1993a]. An example which is consistent with the first of these can be seen for salt vs non-mineral intake in locusts (Trumper and Simpson, 1993). Here, the mechanisms regulating intake of non-mineral nutrients (mainly protein and carbohydrate) are so strong relative to those for mineral intake as to overwhelm the latter, although salt intake is regulated if the insect is able to do so independently of non-mineral nutrients.

(e) Is the interaction symmetrical?

Other forms of array [e.g. Fig. l(e-f) in Simpson and Raubenheimer, 1993a] indicate that regulatory mechan- isms for the various nutrients interact. Such interactions may be completely symmetrical, such that the mechan- isms weight over- and under-eating equally, both within and between nutrient axes. Alternatively, they may be asymmetrical, either with one nutrient being weighted more strongly than the other, or else under- and over- eating being regulated with unequal strength within a dimension.

A completely symmetrical interaction leads to re- sponse arrays which describe the rule “closest distance optimization” (CDO; see Raubenheimer and Simpson, 1993; Simpson and Raubenheimer, 1993a; Abisgold et al., 1994; Simpson et al., 1994; Stockhoff, 1994). Here the animal moves as close as possible to the target in the available nutrient space.

Defined mathematically for 2 dimensions, the CD0 array passes through the target and forms an arc which is part of a circle with its centre halfway between the target and the origin. (In 3 dimensions it is a sphere rather than a circle, and in N dimensions an N-dimen- sional hypersphere.) Accordingly, the formula for locat- ing the closest point on a food rail to the target on a nutrient plane is:

x-coordinate = (a + b.s)/( 1 + s’) (1)

y-coordinate = s. (a + b.s)/( 1 + s’) (2)

where (a, b) are the coordinates of the target and s is the slope of the rail.

REVIEW 551

UNITS: MOLECULES

DRY WEIGHT

ENERGY EQUIVALENTS

L- -

0 2 4 6 0 10

Slope of rail (i.e. ratio of B:A in food)

FIGURE 6. Values of the scaling factor (k) plotted against the slope

of the rail (i.e. the ratio of B: A in the food) for different units of intake

(see Table 1 and equation 3). The entire intake array is transformed

into closest-distance optimization (CDO) by multiplying with a con-

stant, the value of which depends on the units used to plot intake. The

most biologically appropriate units are likely to be molecules eaten (see

text). in which case the transformation indicates that total intake of A

is regulated twice as strongly as that of B (i.e. k = 2) (see also Fig. 7).

(f) Quantifying asymmetry

A simple way to measure asymmetry in the weighting of regulatory mechanisms for two nutrients is to trans- form the observed array into CDO. The function achiev- ing the transformation quantifies the difference between the observed array and one in which mechanisms inter- act symmetrically.

The transformation of a 2-dimensional array [with no treatment means lying in inadmissible regions (Fig. 5)] into CD0 can be achieved by applying the following formula independently to the x-coordinate (x) of each treatment mean in the array (excluding, of course, the target point itself):

k = (s(s.x - b)/a - x) (3)

where k is a scaling factor, (a, b) are the coordinates of the target, and s is the slope of the rail (see Table 1).

The values for k are next plotted against the corre- sponding slopes of rails (Fig. 6). A zero slope for the relationship shows that a scaling constant transforms the entire array into CDO, while other shapes indicate that the relative weighting of regulatory mechanisms varies throughout the array. A k-value of 1 indicates that the array is already consistent with CD0 without further transformation.

(g) Interpreting intake arrays

The array which will be considered first in most experimental investigations is that for nutrient intake. The shape of a given intake array of total nutrient intake, and hence the transformation function quantify- ing the asymmetry in the strengths of regulatory mech- anisms, is influenced by two factors. First, the shape will vary with the units in which the axes are plotted

(Simpson and Raubenheimer, 1993a). Second, the array will be influenced by the presence of non-utilizable substances included in a nutrient axis, either as a result of that axis having been misdefined by the experimenter (e.g. through providing an unsuitable source of dietary protein), or else due to intrinsic constraints on the animal’s post-ingestive efficiency.

Units. Various units might be used for plotting intake of nutrients, including mass, mols and energy equivalents, and the shape of an array will vary accordingly (Table 1). However, since the biochemically, and thus physiologi- cally, fundamental entities are the basic nutrient mol- ecules, axes should ideally be plotted in these units. For instance, amino acids and monosaccharides are the basic components into which ingested proteins and sugars are digested and then absorbed from the gut for use by the tissues. Hence, intake of protein and carbohydrate are probably best plotted in terms of mols of constituent amino acids and monosaccharides ingested.

Arrays of physiologically usable intake. It is to be expected that the mechanisms regulating intake will have evolved to provide behavioural compensation for intrinsi- cally constrained wastage. In other words. the animal should exceed the nutrient target level of a nutrient by the amount which is inevitably wasted in post-ingestive pro- cessing (Raubenheimer and Simpson, 1994). Where post- ingestive efficiencies differ between nutrient axes, this should be reflected in the relative weighting of regulatory mechanisms controlling intake of those nutrients.

Intake arrays can be corrected for non-utilizable nutri- ents by multiplying each point in an intake array, in each dimension, by the ratio NT/IT. This transforms the array so that, instead of intercepting the intake target, it now passes through the nutrient target. The corrected array, which now represents physiologically-useful intake (for respiration, growth, secretions, etc.), can then be reinter- preted by proceeding as described in (f) above (see Fig. 7).

(h) Protein versus carbohydrate intake in chewing phyto- phages

Published data for protein versus digestible carbo- hydrate intake provide a useful case history, illustrating some of the principles outlined above and also highlight- ing the effectiveness of comparative studies.

Arrays of total nutrient intake for 5 species of chewing insect (one species each of acridid, beetle, and caterpillar, and two species of cockroach) are all consistent with CD0 when plotted as dry weight of protein and carbo- hydrate eaten, even though the species vary markedly in the position of the intake target (Simpson and Rauben- heimer, 1993a). If intake is plotted as mols of mono- saccharides and amino acids ingested (as suggested above), then the various patterns from the different species are both inconsistent with CD0 and irreconcil- able with each other. This is because the average molecu- lar weight of amino acids in the proteins (mainly casein) used is less than that for the sugars (130 vs 180 mg).

However, data indicate that dietary protein is utilized less efficiently than digestible carbohydrate (Zanotto

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et al., 1993; Waldbauer, 1968). Correcting total intake arrays for non-utilizable nutrients would reunite the patterns under CDO, provided that the percentage of protein which is physiologically useful is ca 70% that of digestible carbohydrate. Results of Zanotto et al. (1993) indicate that this is the case for the locust. If it transpires also to be true for the other species, then the fact that 5 disparate data sets can all be united under CD0 provides strong circumstantial evidence that CDO, with respect to the nutrient target, is actually the behavioural rule employed by the various species. This inference would be strengthened if further studies, either on different species or other life-stages of the same species, also prove to be consistent under the same units.

(i) Translating physiology into fitness

The units which ultimately describe the interaction between regulatory systems for two or more nutrients are those of fitness. An exciting aim would be to discover how behavioural and physiological decision rules trans- late into fitness. This would require estimating the fitness consequence associated with each point in an intake array. If, for example, fitness were to decline linearly with distance from the nutrient target (as measured in units of physiologically usable nutrient molecules in- gested) then usable nutrient molecules will have been shown to provide a scaling which is commensurate with

14 / I

Nutrient A eaten (mol x 1 O-4)

FIGURE 7. Graph showing the intake array from Fig. 3 plotted as

total number of molecules of A and B ingested (the more appropriate

units). Also shown is the intake array corrected for non-utilizable nutrients (nutrients which the insect cannot use because of intrinsic

constraints on post-ingestive efficiencies; see Fig. 2). The corrected

array passes through the nutrient target (NT). Applying equation 3 to the corrected array yields a k-value of 1, indicating the regulatory mechanisms weight under- and overeating of physiologically usable

molecules symmetrically, both within and between the two nutrient axes (i.e. CD0 applies with respect to the NT). As can be seen in Fig. 3, CD0 also appeared when total (rather than physiologically usable)

intake was plotted in units of dry weight of nutrient eaten. This results from the fact that the molecular weight of nutrient A is twice that of

B (Table I), yet B is utilized only half as efficiently as A (see Fig. 2).

Similar effects have been described in experiments

carbohydrate (see text). with protein and

fitness. If not, then a plot of fitness versus distance to the

target will define that scaling relationship.

STAGE 4: USING ESTIMATES OF TARGETS AND

DECISION RULES AS THE BASIS FOR FURTHER

STUDIES

Achieving estimates of the positions of targets and a description of decision rules enhances the effectiveness of a range of different experimental approaches. The fol- lowing is a brief selection of such possibilities.

(a) Mechanisms

Quantifying the outcome of behavioural and physio- logical mechanisms not only provides a means of locat- ing targets and identifying decision rules, it also opens up an obvious avenue of research, namely what are those mechanisms? This is presently a vigorous field, details of which may be found in two recent reviews (Simpson and Raubenheimer, 1993b; Simpson et al., 1995).

(b) The adaptive signljicance of mechanisms

A key interface in the geometric framework is that between proximate and ultimate causes. This provides a particularly fertile area of research and links directly with ecological investigations. One example is the means by which targets are achieved. So far we have considered targets as points in nutrient space, whose position is integrated over a particular time interval. In real time, however, they are perhaps better considered as optimal trajectories through nutrient space (Simpson and Raubenheimer, 1993a). The insect can be seen (in func- tional terms) as tracking these trajectories. An important issue is the relationship between the trajectory and the track. An animal may reach a certain point of nutrient intake by switching very infrequently between two complementary foods, or else by changing after each meal (or even within a meal). These tracks are non-arbi- trary and subject to selection pressure. The balance which is struck between switching too often and too rarely is a function of the past and present ecological circumstances. Experiments in which the form of trajec- tories is described can yield insights into such selective pressures (e.g. Chambers et al., 1995).

(c) Comparative studies

Once data for target positions and decision rules exist for a number of species (or genotypes within species), numerous possibilities for comparative analyses become available. Insects differing in particular life-history characteristics or ecological circumstances can be compared with respect to target positions, the be- havioural/post-ingestive decision rules they employ, their efficiency in processing ingested nutrients, the tolerance limits of homeostatic mechanisms (i.e. the region in nutrient space within which homeostasis is achieved), and so on. Some examples are given in Simpson and Raubenheimer (1993a).

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(d) Ontogeny

The geometric framework provides an ideal environ- ment for considering ontogenetic changes in feeding and nutrition (Barton Browne, 1995; Simpson and Rauben- heimer, 1993a).

(e) The interaction between nutrients and non-nutritional variables

A fruitful avenue of research would be to consider how arrays of intake and growth, achieved under cir- cumscribed nutritional parameters, become distorted by altering non-nutritional variables such as temperature regime or dietary allelochemicals.

CONCLUSIONS

There are several important advantages of considering nutrition in terms of targets in nutrient space:

(1) On the one hand, targets are functionally defined, while on the other, the geometric framework emphasizes the behavioural and physiological mechanisms employed by an animal to approach them. Hence, both adaptive and mechanistic explanations are integrated. Knowledge of one provides a solid basis for investigating the other.

(2) By focusing on the intake, nutrient and growth targets, the framework unifies studies of feeding be- haviour with those on post-ingestive processing (Raubenheimer and Simpson, 1994, 1995b; Simpson et al., 1994), hence breaking the cause-effect circle which has previously bedeviled studies of this relationship (Simpson and Simpson, 1990).

(3) The targets represent absolute reference points. Inherent in studies on nutritional compensation and dietary selection is the question “relative to what?” (Raubenheimer, 1992). Targets provide the necessary referents for interpreting such data, and provide a means of investigating the behavioural and physiological de- cision rules employed by animals when the targets cannot be reached (Simpson and Raubenheimer, 1993a).

(4) By knowing target positions (and hence the ideal requirements of the animal), potentially ambiguous con- cepts such as the “efficiency” at which insects process ingested food become clearer, as do the dangers of using traditional methodologies (Raubenheimer, 1995; Raubenheimer and Simpson, 1994).

(5) The framework emphasizes the dynamic inter- actions occurring between nutrient groups, leading to insights which are lost when nutrients are not discrimi- nated (Simpson and Raubenheimer, 1993a, b; Rauben- heimer and Simpson, 1994, 1995).

(6) Knowledge of target positions provides dependent variables for a range of subsequent investigations, e.g. studies of changes with ontogeny (Barton Browne, 1995), and analyses where genotypes differing in life- history characteristics are compared (Simpson and Raubenheimer, 1993a).

We believe that the geometric framework provides many new opportunities for the study of nutritional physiology and ecology. What is required now is that data are collected from a diverse range of species, representing a broad spectrum of life-history strategies. Hopefully, this user’s guide will stimulate and facilitate that task.

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