Chemical reactor models of optimal digestion efficiency with constant foraging costs

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Abstract

We develop quantitative optimization criteria for transient digestion processes in simple animal tracts that can be modeled by a semi-batch reactor or plug flow reactor. Specifically, we determine the residence time that optimizes the average net energy intake over the total residence time. The net energy is measured by the total energy intake, less the cost of foraging and digestion. Precise values for optimal residence times are presented for different chemical kinetics of substrate breakdown and of absorption. Both first-order kinetics and Michaelis–Menten kinetics are examined and compared, and it is determined how these residence times vary with foraging costs.

Introduction

Understanding nutrient acquisition and digestion in individuals form important components in an overall program of ecological modeling. Nutrient acquisition by animals can be roughly categorized into four serial steps (Woods and Kingsolver, 1999): consumption, digestion into small units, absorption, and use of newly acquired nutrients for general metabolism (e.g. tissue construction). The digestive process integrates chemical processing, transport and compartmentalization of digesta in the gut, and ultimately absorption such that large molecules (carbohydrate, proteins, fats) are cleaved into smaller units that can be transported across the gut boundary (absorption) to then be distributed within an individual for use in general metabolism. The digestive process can be quite complex. Large numbers of enzymes and buffering systems Dahlquist, 1968, Applebaum, 1985, Christopher and Mathavan, 1985, Terra et al., 1994, Marana et al., 1995 coupled to specialized organ-level responses Chapman, 1985a, Chapman, 1985b, Terra, 1990, Perrin, 1992, Yang and Joern, 1994c and organism-level regulation of movement of substrate and products through the gut Terra, 1990, Yang and Joern, 1994a are all required. Appropriate regulation of the digestive process (Chapman, 1985b), longitudinal and countercurrent movements of digesta through the gut (Terra, 1990) and of products of digestion to sites of absorption Turunen, 1985, Wright et al., 1994 are the primary processes. Additional regulation and feedbacks at the whole organism level comprise the rest of the picture. These include dietary influences of redox and pH regulation and absorption Harrison and Kennedy, 1994, Frasier et al., 2000, or feeding activity regulated by nutrient titers in body tissue (e.g. in the hemolymph in insects; Bernays, 1985, Simpson and Raubenheimer, 1993) that contribute to a consumer’s success in obtaining sufficient nutrition.

Food varies greatly in both availability and nutritional quality, especially to herbivores, and organismal needs change depending on physiological and biochemical requirements. Moreover, organisms often actively control the movement of digesta in the gut depending on the amount and quality of available food and the current physiological state of the individual (Yang and Joern, 1994a). At issue is the degree to which individuals optimize digestion rate in such a dynamic environment, and which points in the series of digestive events limit or regulate digestion to effect this goal, if indeed consumers optimize digestion Sibly, 1981, Cochran, 1987. We develop a mathematical model of digestion using chemical reactor theory Penry and Jumars, 1986, Penry and Jumars, 1987, Dade et al., 1990, Karasov and Hume, 1997, Woods and Kingsolver, 1999, Jumars, 2000a, Jumars, 2000b to provide a conceptual framework to study possible optimal digestion strategies in a mechanistic framework. This chemical reactor model is evaluated in the context of how individuals best meet nutritional needs in an optimizing framework Sibly, 1981, Cochran, 1987. Chemical reactor theory is now used regularly to analyze relationships among diet composition, food processing, and gut morphology, yielding diverse conclusions regarding how digestion interrelates with foraging for a variety of taxa Penry and Jumars, 1986, Penry and Jumars, 1987, Martínez del Rio and Karasov, 1990, Karasov and Cork, 1994, Karasov and Hume, 1997, Woods and Kingsolver, 1999, Whelan and Schmidt, 2003.

Chemical reactor models of digestion include standard models of chemical engineering (Nauman, 1987): batch reactors (BRs), continuously stirred tank reactors (CSTRs), plug flow reactors (PFRs), and various serial combinations of these. Authors use mass balance laws and simple chemical kinetics to obtain relations involving throughput rate, volume and rate constants for reactors operating at steady state. Jumars (2000a) determined the optimal throughput time for such reactors and introduced axial variation in a tubular gut by modeling it as a sequence of discrete CSTRs connected in series (Jumars, 2000b). With chewing insect herbivores in mind, Logan et al. (2002) developed a time-dependent model for a tubular gut that has variable cross-sectional area, spatial variation in its absorptive abilities, and temperature-dependent reaction rates. For insects, this model has been extended to couple a crop-like structure (CSTR) connected in series to a tubular gut (PFR) (Wolesensky et al., 2003); nutrients are absorbed into the hemolymph (CSTR) and concentration thresholds feedback to initiate feeding. In this paper we use chemical reactor models to examine an optimal, transient digestion process in a simple organism whose gut structure is either sacular (BR) or tubular (PFR). We focus on analyzing the strategy of maximizing digestibility (fraction of nutrient absorbed), less the foraging and digestion costs, as a function of the total residence time of the digesta in the gut. Residence time is inversely proportional to the throughput speed.

Other digestion strategies have also been proposed. Jumars (2000a) maximizes the absorption rate as a function of the flow-through rate under steady-state conditions. Other work with batch reactor models generally assume steady-state operating conditions with constant inflow and outflow and no volume change, or that the contents undergo reaction before emptying (simple batch reactor). Under the more likely transient conditions, however, the absorption rate is time dependent and no simple optimization criterion for absorption rate exists based on a single flow through speed; this is one topic we develop in this paper. An optimization strategy for transient digestion processes based on a performance criteria of maximizing the net average energy intake over the residence time of the digesta in the reactor is examined for digestion of sugars in nectar-eating birds (Martínez del Rio and Karasov, 1990) and more generally by Raubenheimer and Simpson (1996). In this paper we give simple algebraic formulas to calculate the optimal strategy for a transient model and compare the results with expectations from steady-state models of Jumars (2000a). Another possible consumer strategy, especially for certain insects, is to reach a nutritional goal, rather than a rate of nutrient gain (Simpson and Raubenheimer, 1993). That is, concentrations of nutrients in the hemolymph may regulate nutrient intake, resulting in an absorption rate that meets targeted nutrient concentrations or ratios of nutrient concentrations. Moreover, some experiments challenge predictions of previous chemical reactor models Karasov and Cork, 1994, Karasov and Hume, 1997. New time-dependent chemical reactor models included herein may provide new insights into the process.

The digestion systems analyzed in this paper represent extremes, although specific animal taxa exhibit unique features depending on diet and environmental constraints. For the case of the sacular gut structure, we assume that it is initially loaded with a volume V0 of a substrate S with molar concentration s0. The contents are removed at a constant volumetric flow rate (digestion speed) until the gut is emptied. The chemical dynamics include substrate breakdown into a nutrient product that is then absorbed across the boundary of the gut. This process models animals that eat discrete meals by filling the gut quickly, and then digesting the contents as it flows from the gut. It can also model a single gut structure, like the crop in some terrestrial arthropods, that empties into another gut structure, like the midgut. The latter is illustrated by a grasshopper whose crop fills in minutes, but the emptying process of the entire crop and midgut system may be on the order of hours (Yang and Joern, 1994b). For the tubular gut, we load an initial segment of the gut with substrate and permit the digesta to flow through the gut at a constant speed until the gut empties. In both gut structures our goal is to find the throughput rate that optimizes the average, net intake and develop specific formulas that express this digestion strategy.

Section snippets

Sacular gut structure

In both types of gut structures (sacular and tubular), the complex chemical kinetics of digestion is modeled by a simple two-step reaction SPabsorption.That is, a substrate S containing a nutrient breaks down (say, by hydrolysis) into the nutrient product P, which is then absorbed across the gut epithelium into the organism’s circulatory system. For example, in hornworms, protein substrate is broken down by proteolytic enzymes into small and medium-sized peptides, followed by a breakdown into

Optimal digestion strategy

We now apply optimization methods to determine quantitatively the optimum residence time for the sacular gut model. A similar qualitative analysis is given in different contexts by Sibly, 1981, Raubenheimer and Simpson, 1996, Martínez del Rio and Karasov, 1990, but without the accompanying formulas obtained here.

There is an important connection between foraging and digestion Sibly, 1981, Stephens and Krebs, 1986, Penry and Jumars, 1986, Penry and Jumars, 1987, Martinez del Rio et al., 1994,

Michaelis–Menten uptake

In some cases substrate breakdown is limited by enzyme concentrations and the nutrient products from the lumen are passed through the epithelium by facilitated transport. In this case the chemistry is modeled by Michaelis–Menten kinetics, and we replace , by RSP=k3Sk4+S,RPabsorp=k1Pk2+P.Whereupon, using the same nondimensionalization (5), but with τ=k3t/s0 as dimensionless time, the governing dynamics are s′=−sr1+s,p′=sr1+sr3pr2+p,s(0)=1,p(0)=0.Now there are three dimensionless parameters: r

Tubular gut structure

Now we compare the optimal performance of a digestion structure that can be modeled as a tubular gut (PFR) of length L and cross-sectional area A.

The reaction–advection equations governing the unknown concentrations are Woods and Kingsolver, 1999, Logan et al., 2002St=−VSx−kS,Pt=−VPx+kS−aP,where V is the constant speed of the digesta in the gut and S=S(x,t) and P=P(x,t) are the concentrations of the substrate and product at position 0<x<L and at time t. The first term on the right side of each

Conclusions

Time-dependent chemical reactor models like the one developed here will help provide a foundational, analytical, framework to assess the entire food acquisition process. We have analyzed a simple, mathematically tractable model where quantitative predictions are fully determined and correspond to qualitative expectations based on empirical evidence and intuition. Like previous efforts Penry and Jumars, 1986, Penry and Jumars, 1987, Karasov and Hume, 1997, Woods and Kingsolver, 1999, Jumars,

Acknowledgements

We thank C. Whelan and an anonymous reviewer for very helpful comments on the manuscript. A.J. is supported by NSF Grant 0087253.

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