On the Relationship Between the Feedback Control of Digestion and
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On the Relationship Between the Feedback Control of Digestion and
Reference: Biol. Bull. 191: 85—9 1. (August, 1996) On the Relationship Between the Feedback Control of Digestion and the Temporal Pattern of Food Intake D. W. STEPHENS Nebraska Behavioral Biology Group, SchoolofBiologicalSciences, University ofNebraska, Lincoln, Lincoln, Nebraska 1983).Indeed, the widespreadand important phenome non of learning is best viewed (Stephens, 1991) as a scheme that capitalizes on environmental regularities that occur on a timescale that is certainly longer than seconds. In short, animal feedingbehavior seemsto be sensitive to a tangle ofdiffering timescales:in someinstancesonly the very short-term consequencesof behavior seemim The Time Scales of Foraging Behavior One ofthe simplest experiments in animal psychology is a straightforward evaluation of the effectsof time and amount on animal feeding preferences.At intervals an animal is offered a choice betweentwo options: one leads to a small amount of food quickly, while the other leads to a much larger amount of food after a longer delay. In these situations, vertebrates abhor delay (I don't know portant; in others the longer term consequences are of any studies with invertebrates). One can think of this clearly important. pattern of preference as a situation in which increasing I had this problem in mind when, coincidentally, I was time devalues amount, and by fitting classic “¿decay― invited to participate in this symposium in honor of functions to this kind ofdata we can estimate the power Vince Dethier. I thought of Dethier's work on the regu ofthis effect. It is astonishing. For feedingbluejays (stud lation of ingestion and digestion via feedback mecha ied in my laboratory, Stephens et al., 1995) the “¿pernisms (Dethier, 1976)and beganto wonder whether we ceived value― ofamount decaysby roughly 10%per sec might be able to understand some of these “¿timescale― ond (seeKagel et al., 1986 for cogent discussion of the effectsby looking inside the animal. Indeed, the mathe quantitative behavior ofthis decayphenomenon). matical technique of singular perturbation, which is of This temporal “¿discounting― of food, together with ten used to study feedback systems(Murray, 1989; Lo severalsimilar phenomena studied in the operant labo gan, 1987),representsa tantalizing possibility: in singu ratory, paints a picture in which consequencesnow are lar perturbation one finds a separatecharacterization of the primary determinants ofanimal feeding preferences. the short- and long-timescale behavior of a system, and It is asifanimals careonly about the very short term (the typically these characterizations are quite different. next few seconds).Ofcourse, a moment's reflection tells Could this be a clue as to why animals seem to have us that this can't be the whole story. Many features of different feeding preferences at short and long time animal feeding behavior seemto be organized (from an scales?In the next few pagesI present a preliminary at evolutionary point ofview) for longer term goals.Clark's tempt to answerthis question. nutcrackers (Nucifraga columbiana) diligently harvest Behavioral ecologistsare surprised when they seeani and cache piñonseedsin the fall that they consume in mals preferring a small amount ofimmediately delivered the spring (Kamil and Balda, 1991). Migratory birds in food over larger amounts that are delayed(Stephensand creasetheir food intake and put on weight severalweeks Krebs, 1986). One way to think about this problem is aheadofactually beginning to migrate (Carpenter et al., shown in Figure 1, in which rate ofingestion is plotted as a function oftime. To keep things simple, I think of the time courseofeating aslong gapswhere nothing is eaten Received 30 November 1995; accepted 15 April 1996. punctuated byjumps in ingestion rates.Notice that even This paper was originally presentedat a symposium titled Finding if we hold the averagerate of intake constant, there are Food: NeuroethologicalAspectsofForaging. The symposiumwasheld many possible patterns ranging from small, frequent at the University ofMassachusetts, Amherst, from 6 to 8 October 1995. 85 This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 86 D. W. STEPHENS A LowPulsePeriodInputStream a(t) dV1 1@(t)—¿ g( V0 —¿ V2)V1 V, < Vmax —¿@- 1@-g( V0 - V2)V1 @ V@ Vmax V2)V1-r __I I [1 I We also need two score-keeping equations to keep track of nutrient extraction (the benefits derived from the in put stream). First, we consider the absolute amount of nutrient in the gut at time t, say A(t). If C1is the concen tration of nutrient in the material flowing from the gut, then the rate at which nutrient enters the gut is C1g(V0—¿ A HighPulsePeriodInputStream a(t) V2)V3 . At any instant Time Figure 1: Two input streams give similar overall intake rates but differ in their temporal organization. blips in intake rate to infrequent, large ones. For a con stant average rate, these possibilities differ in the period icity of bouts of ingestion. In the following model I will ask whether a simple feedback-regulated digestive sys tem does better or worse (i.e., extracts nutrient more or less efficiently) as I vary the “¿pulse period―but hold the average intake rate constant. Figure 2 shows the simplified feedback system that I will analyze. To keep the terminology concrete, I will call the first compartment the “¿crop― and the second the “¿gut,― and I will suppose that the crop is simply a storage organ in which no digestion takes place, and that all di gestion occurs in the gut. I appreciate, ofcourse, that this is a vast oversimplification ofany real biological system, even of Dethier's Phormia after which it is modeled. I suppose that inputs to the crop are governed by the square wave type ofprocess mentioned above, and I will denote this input process as a(t). The volume of matter in the crop at any instant in time is V@ , and the volume the concentration of nutrient in the gut is A(t)/ V2(t), and so nutrient is lost with the effluent at rate r A(t)/ V2(t). The other process that re moves nutrient from the gut is nutrient extraction. The process ofnutricnt extraction may be quite complex, and it may be qualitatively different for different nutrients. I model extraction as a “¿Fick's Law―style absorption pro cess: that is, nutrient is “¿captured― by diffusing across the gut wall while some homeostatic process maintains the concentration of nutrient on the outside of the gut wall (i.e., changes in the concentration gradient are due only to changes in the concentration within the gut). This means that nutrient leaves the gut via extraction at a rate proportional to the wetted surface area ofthe gut (5) and the concentration the nutrient in the gut, aSA/V2 . Now, if we view the gut as a rigid container, a cylinder with fixed radius, then the wetted surface area will be propor tional to V2, say S = b V2. This relationship can be com plicated by elasticity in the gut wall and by allowing more complex gut geometries. This linear relationship is the simplest one that captured the simple idea that wetted surface area increases with gut volume. Putting this all together, I write an equation for the change in amount of nutrient in the gut: dA A A ut ‘¿2 ‘¿2 -:i-= C@g(V0- V2)V1-@-r-abV2@- @= C1g(V0- V2)V1 -‘Ø-r_icii where in the second equation I have cancelled V2and combined the two constants a and b into a single con controlled by a feedback law; specifically, matter is re stant k = ab. I remark that ifthe gut is not well-modeled leased at a rate proportional to the product of (a) the as a rigid container (say that it is more elastic than rigid) difference between the actual gut volume and a “¿set then this volume cancelation will not be valid. To keep point―V0and (b) the actual volume of the crop. I sup track of the benefit (i.e., nutrient) extracted over time, pose that there is an upper limit on the amount of matter say B(t), we would integrate the simple differential equa the crop can hold Vmax(so if the crop volume reaches tion dB/dt = kA Vmax ingestion cannot continue). Finally, I suppose that As a first attempt to understand this system, I analyzed effluent leaves the gut at constant rate r(therc arc a num it numerically; that is, I choose parameter values more or-less arbitrarily and simply integrated the differential ber of reasons to view this assumption with suspicion, equations using well-understood numerical techniques but we withhold judgment for the moment). This leads (I used a commercial package and 4th order Runge to two differential equations for the crop-gut system: in the gut is V2. Releases from the crop to the gut are This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). RELATIONSHIP BETWEEN DIGESTION AND 87 FOOD INTAKE a't) III 1______i1@ a(t) t Crop Volume= g(V@-V)V1 @.SetPoint-V0 Gut Volume= V2 Figure 2: A familiar mathematical caricature of our crop/gut system in which both the crop and gut are modeledasleaky bucketswith a feedbackcontrol law governingthe output from the crop to the gut. Kutta integration). I tried a rangeofinput functions a(t), all ofwhich were “¿square waves― that provided the same averageinput rate but varied in “¿pulse period.― The re lationship that my analysisrevealedbetweennutrient as similation rate and pulse period is plotted in Figure 3. This is the relationship we would expect ifdigestive con straints play a part in animal preferencesfor immediacy: for example, if more even flows of food yield a higher assimilation rate than uneven flows with the sameaver age intake rate, then it follows that it might sometimes make economic senseto prefer an even flow with a low rate of intake over an uneven flow with higher rate of intake. Ofcourse, thesenumerical resultsdon't establish this effect asa universal result, they simply suggestthat it can happen in some instances.To understand its gener ality we needto attack the model analytically. (i.e., they are dimensionless, or “¿pure,― numbers). This is important for two reasons.First, it is always easierto work with a smaller number of terms. Second,we need dimensionless numbers to meaningfully evaluate ap proximate solutions, becausewewant our estimateof the magnitudes oferrors to be independent ofthe systemof measurementusedby the experimenters.One way to ap proach dimensional analysis to pick “¿characteristic quantities―against which to measure different types of units in the model. In the present model we have vol umes (meters cubed), time (seconds), and amounts of nutrient (moles). Ifwe measurevolumes in units of Vmax and time in units of l/gVmax(a measureof the time re quired to drain a full crop), and nutrient quantities by CiVmax (the amount of nutrient in a full crop), can rewrite our systemas di2@J—(Vo—V@)Vi V1l Analysis df @ A first step in analyzing this system is dimensional analysis (Stephens and Dunbar, 1993), a technique in which one considersthe units ofeach ofthe model's vari ables and parameters and rearrangesthem into a new, smaller setofparameters and variablesthat haveno units ta(t) —¿ (Vo—¿ V2)V1 V < 1 %=(@o- V@1P V@)V,-@-A-kA This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). then we 88 D. W. STEPHENS O.O2@ 0.0249 0.0248 @ 0.0247 i 0.0246 0.0245 0.0244 4 6 8 10 12 14 16 18 Pulse Period Figure 3: Results of a preliminary numerical analysis of the crop/gut system showing that nutrient assimilation rate declines with increasing pulse period, at least for some parameter values. Recall that the average rate ofintake is held constant as the pulse period is increased. @=Ak where “¿hat― symbols denote that fact that the terms have been rescaled. In the remainder of this paper, I shall as sume that all terms have been appropriately resealed, and so the hat symbols will not be used. The phase plane on the null dine (we know that dV2/dt = 0 here, so any movement on this curve must be parallel to the crop vol ume V1axis). To decide whether the system is moving up or down in crop volume, we need to consider the crop volume equation, dV,/dt. Since the input stream is a square wave, the crop vol umc equation takes two forms: (1) a “¿crop filling―form @=h—(V0— V2)V3 As mentioned above, the two volume equations com pletely determine the system's dynamical behavior, whereas the two remaining equations are simply mecha nisms for score keeping. I focus, therefore, on trying to understand the volume equations. To begin, I consider the collection of crop ( V1)and gut ( V2)volumes where dV2/dt equals zero (these points represents the so-called V2null dine along which gut volume is not changing). If we plot this collection of points V2 = V0 in V1 —¿ V2space, we have a hyperbolic curve that ap proaches V2= —¿ oo as V, approaches zero, and asymptot ically approaches V0 as V@becomes large. Null dines help one understand dynamic systems because we know that the system can only be moving in one dimension where h is the height of the square wave, or (2) a “¿crop emptying―form I@= —¿(V0— V2)V1 Ofcourse, along the V2null dine, the term —¿(V0 —¿ V2)V1 =—¿r, sothat during emptying phases the crop volumes must be going down dV1/t = —¿r along the null dine; and, assuming h > r, crop volume must be increasing dV1/t = h —¿ r during filling phases. Next considering the null dines for crop volume V1, we see that for the “¿filling― equation there is a null-dine at V2 = V0 —¿ h/V, This is exactly same form as the gut volume null dine, This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 89 RELATIONSHIP BETWEEN DIGESTION AND FOOD INTAKE A. Crop Emptying B. Crop Filling V2 V2 cr;j@ Null-Cline V. Gut Null-dune Crop Null-dime Crop Null-dine V, il V, C. Overall Behavior V2 V. -r il Figure 4: (A) Null-dine V1 analysis ofthe system when the crop is emptying. The overall is a decrease in both crop and gut volumes. (B) Analysis when the crop is filling. The trajectories are confined between the two null dines, while crop and gut volumes move jointly upward toward the maximum crop volume (which we have taken to be one). (C) Because our system requires that the average intake rate exceed the outflow rate r, a limit cycle will eventually be reached in which an emptying system stays slightly above the gut-volume null dine, whereas a filling system changes direction and climbs back to the maximum volume while staying slight below the gut-volume null dine. except that h takes the place of r. Since h > r by assump tion, this null dine @5 crudely parallel and below the gut volume null dine. Moreover, since the outflow term (V0 —¿ V2)V1 = h along this null dine, gut volume must be increasing along this null dine, (i.e., dV2/dt = h —¿ r > 0). For the emptying equation there are two null dines: V2= V0 J―l= 0 “¿filling― panels ofFigure 4. We see that when the system is emptying, thejoint (J―@, V2)trajectory will be attracted to the region between the null dines and then will skirt the gut-volume null dine from above as the crop and gut volumes jointly decline. Similarly, when the system is filling, the trajectory will skirt the gut-volume null dine from below as the volumes jointly increase. I note that our system is a plausible model only when the overall rate ofintake exceeds the fixed outflow rate r(simply be cause an empty gut cannot release material In both cases, ofcourse, the outflow term ( V0—¿ V2)V, = 0, so the gut volume can only be decreasing dV2/dt = —¿r. These facts are put together in the “¿emptying― and at constant rate r). With this assumption, it follows that at some point, after many cycles offilling and emptying, the crop volume will be at its maximum value. Once this happens This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 90 D. W. STEPHENS we would expectthe systemto follow a limit-cycle quali tatively like that shown in Figure 4C: declining from the maximum crop volume while staying slightly above the gut-volume null dine, and then increasing to the maxi mum crop volume while staying slightly below the gut volume null dine. Pseudo-equilibria The qualitative observation that the trajectory skirts the gut-volume null dine suggestsa major simplification of the model; the assumption that dV2/dt = 0. Biologi cally, this is the claim that the gut volume equilibrates quickly to relatively slowly occuring changesin the crop volume and input stream. Notice that to claim that the gut volume is at equilibrium is not the sameasclaiming that the gut volume is constant; we are simply supposing that the system'strajectories slide up and down the gut volume null dine (like a bead on a wire) rather than dir culating around the gut-volume null dine as we argued above. The idea of pseudo-equilibrium may be familiar to studentsofenzyme kinetics; it is part ofthe traditional derivation of the Michealis-Menten rate law (Murray, 1974).(In addition, one can often build upon a pseudo equilibrium analysis to construct the “¿two timescale so lutions― that typify the singular perturbation technique.) Assuming that dV2/dt = 0 changesour systemto @j@f-r di V1l 1,a(t)—r V1<l V2= V0-@- dt @=r— r r A-kA = Ak Now at any instant, the rate ofchange ofcrop volume is either —¿r in an emptying phaseor h —¿ r in a filling phase, so we expect that the volume ofcrop is either (a) empty ing linearly at rate r, (b) full and unchanging, or (c) filling linearly at rate h —¿ r. Our earlier numerical analysescon firm that the crop-volume function is approximately piecewise linear. Now consider two square wave input functions that give the same overall input rate R, one with a long period (F1) and a large peak input rate (h1), and the other with a short period (P2) and small peak input rate (h2)such that A*= r k+ r r V0- —¿ V3 Using calculus, it is straightforward to show that this equilibrium amount of nutrient increases with increas ing crop volume (the first derivative is alwayspositive) at a decreasingrate (the second derivative is negative for all biologically plausible parameter combinations). This meansthat increasingthe crop volume by one unit hasa big effecton the extraction rate ifthe crop volume is low, but a one-unit increasein crop volume has only a small effect if it is already high. This downward curvature means that varying crop volumes are a bad thing; this stems from a basic mathematical result called Jensen's inequality (seeStephensand Krebs, 1986,chapter 7, for an elementary discussion). This conclusion suggeststhat even intake streamspro vide two economic advantagesover uneven ones. First, they allow the systemto maintain a higher mean extrac tion rate because of the truncation effect discussed above. Second,lower variance in crop volume translates into higher extraction rates, becausethe amount of nu trient in the gut increaseswith crop volume at a decreas ing rate. In light ofthese kinds ofconsiderations one sees predigestion storageorgans like the crop of Phormia as devicesto attenuate oscillations in intake. Soperhapsbe havioral ecologists should not be so surprised that ani mals have strong preferencesabout the temporal organi zation offood intake. Discussion R=@-@=@ P1 ageinput rate R exceedsthe outflow rate r, as discussed above. Intuitively, it is easyto seethat the input stream with the smaller pulse period will, in the long run, pro duce both a higher mean crop volume and a smaller vari ance in crop volume. This is essentially a truncation effect: becauseR > r, the system eventually reachesa point at which the crop is full V1 = 1 (in our resealed units). Starting with a full crop, the long pulse-period sys tem will drain for (P1—¿ w) time units at rate r, reaching a minimum crop volume of 1 —¿ (P1—¿ w)r, similarly, the short pulse-period system will drain for the shorter pe riod P2 —¿ w at the same rate, reaching the larger mini mum of 1 —¿ (P2—¿ w) r. Now, imagine for the moment an experiment in which we hold the crop volume fixed. In this casethe equation for the amount of nutrient in the gut (which, in turn, determines the nutrient extraction rate) is strongly at tracted to the equilibrium amount: P2 P1> P2and h1> h2. In addition, I assumethat the aver I have considered a simple caricature of the relation ship between food intake and digestion. Indeed, I have This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). RELATIONSHIP BETWEEN DIGESTION AND FOOD INTAKE 91 only sketched an analysis of this system. Despite its de Acknowledgments fects,my analysisshowsa possiblelink betweendigestive I am grateful for the helpful comments of Peter Bed regulation and an important outstanding problem in nekof and Tom Getty, and for the continued financial feeding ecology—animal preferences for immediate support of the National Science Foundation (IBN food reward. Many complications await further analysis: 8958228and IBN-9507668). What ifmore realistic reaction kinetics are used?What if Literature Cited there is feedbackcontrol of the evacuation rate ,‘? What Barkan, C. P. L., and W. L. Withiam. 1989. Profitability,rate maxi if nutrients are stored after digestion instead of (as well mization, and reward delay: a test of the simultaneous-encounter as)before?There have been many mathematical models model of prey choice with Parus atricapillus paru. Am. Nat. 134: ofdigestion: models ofoptimal passagerate; the effect of 254—272. Carpenter, F. L., D. C. Paton, and M. A. Hixon. 1983. Weight gain gut length and morphology on digestive efficiency, etc. and adjustment of feedingterritory sizein migrant hummingbirds. (Yang, 1993; Penry and Jumars, 1986; Penry and Ju Proc.Nat. Acad.Sci. USA80: 7259—7263. mars, 1987; Sibly, 1981). Moreover, many studies have Dethier, V. G. 1976. The Hungry Fly: a Physiological Study of the quantified the feedback relations involved in “¿gastric BehaviorAssociatedwith Feeding.Harvard University Press,Cam bridge, MA. emptying―(McHugh and Moran, 1979; Hainsworth, Hainsworth, F. R. 1989. Evaluating models ofcrop emptying in hum 1989).Rather than draw explicitly from this literature, I mingbirds.Auk 106:724—726. have built an elementary model crudely patterned after Kagel, J. H., L. Green, and T. Caraco. 1986. When foragers discount the Phormia crop-foregut-midgut systemstudied by Dc the future: constraint or adaptation? Anim. Behav. 34: 27 1—283. thier(l976). Kamil, A. C., and R. P. Balda. 1991. Spatial memory in seed-caching corvids. Pp. 1-25 in Psychology ofLearning and Motivation, vol. Behavioral ecologists have typically tried to explain 36, G. Bower, ed. Academic Press, San Diego. animal preferences for immediacy by looking for edo Logan, J. D. 1987. AppliedMaihematics: a ContemporaryApproach. nomic forces outside the animal. The most frequently Wiley, New York. offered explanation is that in choosing immediate food McHugh, P. R., and T. H. Moran. 1979. Calories and gastric empty ing: a regulatory capacity with implications for feeding. Am. J. reward animals are anticipating interruptions (say by a Physiol. 236: R254—R260. predator or a competitor) that might prevent them from McNamara, J. M., and A. I. Houston. 1987. A general framework for actually collecting delayed food (Kagel et al., 1986;Bar understandingthe effectsof variability and interruptions on forag kan and Withiam, 1989; McNamara and Houston, ing behaviour.Acta Biotheor.36: 3—22. 1987). My “¿explanation― differs from this because it Murray, J. D. 1974. AsymptoticAnalysis. Clarendon Press, Oxford. Murray, J. D. 1989. MathematicalBiology. Springer-Verlag, Berlin. looks inside the animal and askswhether preferencesfor Penry, D. L., and P. A. Jumars. 1986. Chemical reactor analysis and immediacy can be viewed as an attempt to managethe optimal digestion. Bioscience36: 310—315. inputs to a regulated digestive system. Both kinds of cx Penry, D. L., and P. A. Jumars. 1987. Modeling animal guts aschem planation could operate simultaneously, of course. One ical reactors.Am. Nat. 129:69—96. Sibly, R. M. 1981. Strategies of digestion and defecation. Pp. 109— would like to compare the magnitudes of predicted 139 in Physiological Ecology: an Evolutionary Approach to Re effectsin the two types of models and, if possible,create source Use. C. R. Townsend and P. Calow, eds. Sinuaer, Sunder experimental situations in which these effects could be land, MA. assessedand compared. Stephens, D. W. 1991. Change, regularityand value in the evolution ofanimal learning.Behav.Ecol. 2: 77—89. As a behavioral ecologist interested in feeding behav Stephens, D. W., and S. R. Dunbar. 1993. Dimensional analysis in ior, it is difficult to leafthrough the pagesof The Hungry behavioral ecology. Behav. Ecol. 4: 172—183. Fly(Dethier, 1976)without a senseofshame. Behavioral Stephens, D. W., and J. R. Krebs. 1986. Foraging Theory. Mono ecologists have, by and large, simply passedup the op graphs in Behavior and Ecology. Princeton University Press, portunity to incorporate mechanistic details, like those Princeton, NJ. Stephens, D. W., K. Nishimura, and K. B. Toyer. 1995. Error and revealedin the Dethier's work, into their models. I have discounting in the iteratedprisoner'sdilemma. J. Theor.Biol. 176: tried to offer a simple example of how one might com 457—469. bine Dethier-ian mechanism with the behavioral ecology Yang, Y. 1993. The insect herbivore gut as a series ofchemical reac of feeding, but behavioral ecologists have yet to capi tors: mathematicalmodeling and empirical evaluation. Ph.D. The talize on the enormous richnessofmechanistic detail. sis,University ofNebraska, Lincoln. This content downloaded from 078.047.027.170 on October 12, 2016 17:52:43 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c).