QUANTITATIVE ASPECTS OF RUMINANTDIGESTION ANDMETABOLISM

Second Edition

QUANTITATIVE ASPECTSOF RUMINANT DIGESTIONANDMETABOLISMSecond Edition

Edited by

J. Dijkstra

Animal Nutrition GroupWageningen UniversityThe Netherlands

J.M. Forbes

Centre for Animal SciencesUniversity of LeedsUK

and

J. France

Centre for Nutrition ModellingUniversity of GuelphCanada

CABI Publishing

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Library of Congress Cataloging-in-Publication Data

Quantitative aspects of ruminant digestion and metabolism / edited by J. Dijkstra, J. M. Forbes,and J. France.- -2nd ed.

p. cm.Includes index.ISBN 0851998143 (alk. paper)1. Rumination. 2. Digestion. 3. Metabolism. 4. Ruminants. I. Dijkstra, J. (Jan), 1964 II.

Forbes, J. M. (John Michael), 1940III. France, J. IV. Title.

QP151.Q78 2005573.31963- -dc22

2004029078ISBN 0 85199 8143

Typeset by SPI Publishing Services, Pondicherry, IndiaPrinted and bound in the UK by Biddles Ltd, Kings Lynn

Contents

Contributors ix

1. Introduction 1J. Dijkstra, J.M. Forbes and J. France

DIGESTION

2. Rate and Extent of Digestion 13D.R. Mertens

3. Digesta Flow 49G.J. Faichney

4. In Vitro and In Situ Techniques for Estimating Digestibility 87S. Lopez

5. Particle Dynamics 123P.M. Kennedy

6. Volatile Fatty Acid Production 157J. France and J. Dijkstra

7. Nitrogen Transactions in Ruminants 177J.V. Nolan and R.C. Dobos

8. Rumen Microorganisms and their Interactions 207M.K. Theodorou and J. France

v

9. Microbial Energetics 229J.B. Russell and H.J. Strobel

10. Rumen Function 263A. Bannink and S. Tamminga

METABOLISM

11. Glucose and Short-chain Fatty Acid Metabolism 291R.P. Brockman

12. Metabolism of the Portal-drained Viscera and Liver 311D.B. Lindsay and C.K. Reynolds

13. Fat Metabolism and Turnover 345D.W. Pethick, G.S. Harper and F.R. Dunshea

14. Protein Metabolism and Turnover 373D. Attaix, D. Remond and I.C. Savary-Auzeloux

15. Interactions between Protein and Energy Metabolism 399T.C. Wright, J.A. Maas and L.P. Milligan

16. Calorimetry 421R.E. Agnew and T. Yan

17. Metabolic Regulation 443R.G. Vernon

18. Mineral Metabolism 469E. Kebreab and D.M.S.S. Vitti

THE WHOLE ANIMAL

19. Growth 489G.K. Murdoch, E.K. Okine, W.T. Dixon, J.D. Nkrumah,J.A. Basarab and R.J. Christopherson

20. Pregnancy and Fetal Metabolism 523A.W. Bell, C.L. Ferrell and H.C. Freetly

21. Lactation: Statistical and Genetic Aspects of SimulatingLactation Data from Individual Cows using a Dynamic,Mechanistic Model of Dairy Cow Metabolism 551H.A. Johnson, T.R. Famula and R.L. Baldwin

vi Contents

22. Mathematical Modelling of Wool Growth at the Cellularand Whole Animal Level 583B.N. Nagorcka and M. Freer

23. Voluntary Feed Intake and Diet Selection 607J.M. Forbes

24. Feed Processing: Effects on Nutrient Degradationand Digestibility 627A.F.B. Van der Poel, E. Prestlkken and J.O. Goelema

25. Animal Interactions with their Environment:Dairy Cows in Intensive Systems 663T. Mottram and N. Prescott

26. Pasture Characteristics and Animal Performance 681P. Chilibroste, M. Gibb and S. Tamminga

27. Integration of Data in Feed Evaluation Systems 707J.P. Cant

Index 727

Contents vii

Contributors

R.E. Agnew, Agricultural Research Institute of Northern Ireland, LargePark, Hillsborough BT26 6DR, UK.

D. Attaix, Institut National de la Recherche Agronomique, Unite de Nutri-tion et Metabolisme Proteique, Theix, 63122 Ceyrat, France.

R.L. Baldwin, Department of Animal Science, University of California,Davis, CA 95616-8521, USA.

A. Bannink, Division of Nutrition and Food, Animal Sciences Group,Wageningen University Research Centre, PO Box 65, 8200 AB Lelys-tad, The Netherlands.

J.A. Basarab, Western Forage/Beef Group, Lacombe Research Centre,6000 CandE Trail, Lacombe, Alberta T4L 1W1, Canada .

A.W. Bell, Department of Animal Science, Cornell University, Ithaca, NY14853, USA.

R.P. Brockman, St. Peters College, Muenster, Saskatchewan S0K 2Y0,Canada.

J.P. Cant, Department of Animal and Poultry Science, University ofGuelph, Guelph, Ontario N1G 2W1, Canada.

P. Chilibroste, Facultad de Agronoma, Estacion Experimental M. A. Cassi-noni, Ruta 3 km 363, CP 60000, Paysandu, Uruguay.

R.J. Christopherson, Department of Agricultural, Food and NutritionalScience, University of Alberta, Edmonton, Alberta T6G 2P5, Canada.

J. Dijkstra, Animal Nutrition Group, Wageningen Institute of Animal Sci-ences, Wageningen University, PO Box 338, 6700 AH Wageningen,The Netherlands.

W.T. Dixon, Department of Agricultural, Food and Nutritional Science,University of Alberta, Edmonton, Alberta T6G 2P5, Canada.

R.C. Dobos, Beef Industry Centre of Excellence, NSW Department ofPrimary Industries, Armidale, 2351 Australia.

ix

F.R. Dunshea, School of Veterinary and Biomedical Sciences, MurdochUniversity, Murdoch, WA 6150, Australia; and Department of Pri-mary Industries, Werribee, VIC 3030, Australia.

G.J. Faichney, School of Biological Sciences A08, University of Sydney,NSW 2006, Australia.

T.R. Famula, Department of Animal Science, University of California,Davis, CA 95616-8521, USA.

C.L. Ferrell, USDA ARS, Meat Animal Research Center, Clay Center, NE68933, USA.

J.M. Forbes, Centre for Animal Sciences, School of Biology, University ofLeeds, Leeds LS2 9JT, UK.

J. France, Centre for Nutrition Modelling, Department of Animal andPoultry Science, University of Guelph, Guelph, Ontario N1G 2W1,Canada.

M. Freer, CSIRO Plant Industry, GPO Box 1600, Canberra, ACT 2601,Australia.

H.C. Freetly, USDA ARS, Meat Animal Research Center, Clay Center, NE68933, USA.

M. Gibb, Institute of Grassland and Environmental Research, North WykeResearch Station, Okehampton, Devon EX20 2SB, UK.

J.O. Goelema, Pre-Mervo, PO Box 40248, 3504 AA Utrecht, The Nether-lands.

G.S. Harper, CSIRO, Division of Livestock Industries, St. Lucia, QLD4067, Australia.

H.A. Johnson, Department of Animal Science, University of California,Davis, CA 95616-8521, USA.

E. Kebreab, Centre for Nutrition Modelling, Department of Animal andPoultry Science, University of Guelph, Guelph, Ontario N1G 2W1,Canada.

P.M. Kennedy, CSIRO Livestock Industries, J.M. Rendel Laboratory, Rock-hampton, QLD 4701, Australia.

D.B. Lindsay, Division of Nutritional Sciences, School of Biosciences, Uni-versity of Nottingham, Sutton Bonington Campus, Loughborough,Leicestershire LE12 5RD, UK.

S. Lopez, Department of Animal Production, University of Leon, 24071Leon, Spain.

J.A. Maas, Centre for Integrative Biology, University of Nottingham, Sut-ton Bonnington, Leicestershire LE12 5RD, UK.

D.R. Mertens, USDA Agricultural Research Service, US Dairy ForageResearch Center, Madison, WI 53706, USA.

L.P. Milligan, Department of Animal and Poultry Science, University ofGuelph, Guelph, Ontario N1G 2W1, Canada.

T. Mottram, Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK454HS, UK.

G.K. Murdoch, Department of Agricultural, Food and Nutritional Science,University of Alberta, Edmonton, Alberta T6G 2P5, Canada.

x Contributors

B.N. Nagorcka, CSIRO Livestock Industries, GPO Box 1600, Canberra,ACT 2601, Australia.

J.D. Nkrumah, Department of Agricultural, Food and Nutritional Science,University of Alberta, Edmonton, Alberta T6G 2P5, Canada.

J.V. Nolan, School of Rural Science and Agriculture, University of NewEngland, Armidale, 2351 Australia.

E.K. Okine, Department of Agricultural, Food and Nutritional Science,University of Alberta, Edmonton, Alberta T6G 2P5, Canada.

D.W. Pethick, School of Veterinary and Biomedical Sciences, MurdochUniversity, Murdoch, WA 6150, Australia.

N. Prescott, Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK454HS, UK.

E. Prestlkken, Felleskjpet Forutvikling, Department of Animal and Aqua-cultural Sciences, Agricultural University of Norway, PO Box 5003,N-1432 As, Norway.

D. Remond, Institut National de la Recherche Agronomique, Unite deNutrition et Metabolisme Proteique, Theix, 63122 Ceyrat, France.

C.K. Reynolds, Department of Animal Sciences, The Ohio State University,OARDC, 1680 Madison Avenue, Wooster, OH 44691-4096 USA.

J.B. Russell, Agricultural Research Service, USDA and Department ofMicrobiology, Cornell University, Ithaca, NY 148531, USA.

I.C. Savary-Auzeloux, Institut National de la Recherche Agronomique,Unite de Recherches sur les Herbivores, Theix, 63122 Ceyrat, France.

H.J. Strobel, Department of Animal Sciences, University of Kentucky,Lexington, KY 40546-0215, USA.

S. Tamminga, Animal Nutrition Group, Wageningen Institute of AnimalSciences, Marijkeweg 40, 6709 PG Wageningen, The Netherlands.

M.K. Theodorou, BBSRC Institute for Grassland and Environmental Re-search, Aberystwyth, Dyfed SY23 3EB, UK.

A.F.B. Van der Poel, Wageningen University, Animal Nutrition Group,Marijkeweg 40, 6709 PG Wageningen, The Netherlands.

R.G. Vernon, Hannah Research Institute, Ayr KA6 5HL, UK.D.M.S.S. Vitti, Animal Nutrition Laboratory, Centro de Energia Nuclear na

Agricultura, Caixa Postal 96, CEP 13400-970, Piracicaba, SP, Brazil.T.C. Wright, Department of Animal and Poultry Science, University of

Guelph, Guelph, Ontario N1G 2W1, Canada.T. Yan, Agricultural Research Institute of Northern Ireland, Large Park,

Hillsborough BT26 6DR, UK.

Contributors xi

1 IntroductionJ. DIJKSTRA,1 J.M. FORBES2 and J. FRANCE3

1Animal Nutrition Group, Wageningen Institute of Animal Sciences,Wageningen University, P.O. Box 338, 6700 AH Wageningen,The Netherlands; 2Centre for Animal Sciences, School of Biology,University of Leeds, Leeds LS2 9JT, UK; 3Centre for Nutrition Modelling,Department of Animal & Poultry Science, University of Guelph, Guelph,Ontario N1G 2W1, Canada

Preamble

Ruminant animals have evolved a capacious set of stomachs that harbourmicroorganisms capable of digesting fibrous materials, such as cellulose. Thisallows ruminants to eat and partly digest plants, such as grass, which have ahigh fibre content and low nutritional value for simple-stomached animals.Thus, animals of the suborder Ruminantia, being plentiful and relatively easyto trap, became prime targets of hunters and, eventually, were domesticatedand farmed. Today, ruminants account for almost all of the milk and approxi-mately one-third of the meat production worldwide (Food and AgricultureOrganization, 2004) (Fig. 1.1). It is not surprising, then, that a great deal ofresearch has been carried out on the digestive system of ruminants, leading tostudies on the peculiarities of metabolism that cope with the unusual productsof microbial digestion. The reading list at the end of this chapter gives some ofthe books in which the biology of ruminants is reviewed.

As qualitative knowledge increased, so it became possible to developquantitative approaches to increase understanding further and to integratevarious aspects. Initially this was achieved by more complex statistical analysis,but in recent years this has been supplemented by dynamic mathematicalmodels that not only summarize existing data but also show where gaps inknowledge exist and where further research should be done. The purpose ofthis book is to bring together the quantitative approaches, concerned withelucidating mechanisms, used in the study of ruminant digestion, metabolismand related areas. In this introductory chapter, we describe briefly the specialfeatures of the ruminant and the potential for quantitative description ofruminant physiology to contribute to our understanding. We also indicate thechapters in which detailed consideration is given to each topic. This chapter isbased firmly on Chapter 1 of the previous edition of this book (Forbes andFrance, 1993). However, all the subsequent chapters in this second edition are

CAB International 2005. Quantitative Aspects of Ruminant Digestionand Metabolism, 2nd edition (eds J. Dijkstra, J.M. Forbes and J. France) 1

either major revisions of the old chapters or, in the majority of cases, com-pletely new chapters written either by old or new authors.

Special Features of the Ruminant

The gastrointestinal tract

ReticulorumenAs there is no sphincter between the rumen and the reticulum and theyfunction to a large extent as a single organ, they are usually consideredtogether. Feed, after being chewed during eating, enters the reticulorumenwhere it is subjected to microbial attack and to the mixing and propulsive forcesgenerated by coordinated contractions of the reticulorumen musculature. Thismuscular activity results in the pattern of movement of digesta that is showndiagrammatically in Fig. 1.2. It is coordinated not only to mix the digesta butalso to allow the removal of fermentation gases by eructation, the regurgitationof digesta for rumination, which is largely responsible for the physical break-down of digesta particles (see Chapter 5), and the passage of digesta out of thereticulorumen through the reticulo-omasal orifice (see Chapter 3). The rate andextent of degradation in the reticulorumen and developments in techniques toestimate the rate and extent are described in Chapters 2 and 4, respectively.

The microbial activity in the reticulorumen gives the host the ability to eatand utilize forages. Chapters 8 and 9 review the dynamics and energetics of thismicrobial population. Most of the material digested in the rumen yields short-chain fatty acids, known as volatile fatty acids (VFA), which are absorbedthrough the rumen wall. Acetic acid is produced in the greatest quantities,around 2050 moles per day in dairy cows, while propionic acid is usuallyproduced at about one-third of the rate of acetic acid. Butyric acid accounts foraround 10% of the total acid production, while valeric and isovaleric acids each

Beef and veal

BuffaloGoatMutton and lambOther ruminants

Non-ruminants

Non-ruminants

Buffalo

SheepGoat

Cow

Fig. 1.1. Relative contribution of various groups of ruminants and non-ruminants to theproduction of meat (left graph) and milk (right graph) worldwide in 2003 (Food and AgricultureOrganization, 2004).

2 J. Dijkstra et al.

form about 1% to 2%. The ratio of acetic:propionic acids is higher for foragediets than for concentrate diets (see Chapters 6 and 10).

Much of the dietary protein, as well as the urea that is recycled via thesaliva, is metabolized to ammonia. Both ammonia and amino acids or smallpeptides are available for microbial protein synthesis (see Chapters 7 and 10).

OmasumDigesta pass from the reticulum to the omasum via a sphincter, the reticulo-omasal orifice. The omasum is filled with about 100 tissue leaves (the laminae),which almost completely fill the lumen. The role of the omasum is not wellunderstood but it is known that water, ammonia, VFA and inorganic electro-lytes are absorbed in the omasum and that ammonia and, presumably, someVFA are produced there.

AbomasumFrom the omasum, digesta pass to the abomasum, the compartment equivalentto the monogastric stomach. As in monogastrics, acid and enzymes are secretedin the abomasum and are mixed with the digesta by the muscular activity of theorgan. However, whereas in monogastric animals there is a circadian rhythm inthis activity associated with the feeding pattern, abomasal motor activity exhibitsan ultradian rhythm as a consequence of the relatively continuous passage ofdigesta from the reticulorumen. Distension of the abomasum inhibits reticuloru-men emptying but is the main stimulus for emptying of the abomasum.

The small intestineThe small intestine comprises three segments: the duodenum, jejunum andileum. Digesta pass from the duodenum along the small intestine as a conse-quence of contractions that start at the gastroduodenal junction due to thegeneration of electrical activity at this junction in the form of migrating motorcomplexes (MMC). These also show an ultradian rhythm resulting in cyclicalvariations in flow over periods of 90 to 120 min. The velocity of propagationof MMC in the jejunum of normally fed sheep is 18 cm/min, which is similar tothe value of 20 cm/min for the velocity of digesta flow in the jejunum of sheep.The agreement between these measurements confirms the concept thatpropulsive activity of the small intestine is directly mediated by MMC. The

E

A

O

RRo

C

V

D

DB

VB

Fig. 1.2. Movement of digesta within thereticulorumen, omasum and abomasum:oesophagus (E), reticulum (R), reticulo-omasal orifice (Ro), cranial sac (C), dorsalrumen (D), ventral rumen (V), dorsal blindsac (DB), ventral blind sac (VB), omasum (O)and abomasum (A).

Introduction 3

increases in digesta flow that occur with increasing intake are the result ofincreases in the amount of digesta propelled per contraction rather than in thenumber of contractions. Digestion in the small intestine is similar to that insimple-stomached animals.

The large intestineThe flow of digesta to the caecum and proximal colon from the ileum is intermit-tent and can be followed byperiods of quiescence,whichmay range from30 minto 5 h.Digesta in the caecumand proximal colon are subjected to both peristalticand antiperistaltic contractions so that digesta are mixed as well as being movedtowards the distal colon. There is further VFA production and absorption in thelarge intestine but its main function is probably the absorption of water.

The flow of digesta through the distal colon differs between sheep andcattle. In sheep, bursts of spiking activity, which last less than 5 s and do notpropagate, result in the segmenting contractions that are responsible for theformation of faecal pellets as the digesta pass through the spiral colon. Bycontrast, in cattle bursts of spiking activity of long duration propagate along thespiral colon. These occur as several phases of hyperactivity per day and areassociated with the propulsion of large volumes of digesta. As a consequence,faeces are voided by cattle as an amorphous mass.

Metabolic adaptations

The intermediary metabolism of ruminants has adapted to the consequences ofthe production of VFA in the rumen in a number of ways (see Chapters 11 and12). Acetate is absorbed into the ruminal venous drainage, some of it being usedas an energy source by ruminal tissue, and used throughout the body for fatsynthesis, including milk fat, and as an energy source. Propionate, passing fromthe rumen in thehepaticportal vein, is takenupalmost completely by the liver andused, togetherwithaminoacids, forgluconeogenesis.Theglucose releasedby theliver is necessary for lactose synthesis in themammary gland, for fructose synthe-sis in the placenta and by the nervous system, although the latter can use ketonessufficiently to continue to functionwith very low blood glucose levels. Butyric acidis, to a large extent, metabolized in the rumen wall, to 3-hydroxy-butyrate.

Rumen fermentation also produces ammonia and that not utilized by themicrobes is absorbed and converted in the liver to urea. Much of this is secretedin the saliva, which is produced continuously in copious amounts, or isabsorbed through the rumen wall to be available once again for microbialprotein synthesis. Protein that escapes rumen degradation is digested and theconstituent amino acids absorbed.

Metabolic regulation is discussed in Chapter 17, while metabolic adapta-tions of ruminants are included in Chapter 13 (fat metabolism), Chapter 14(protein turnover), Chapter 15 (energyprotein interactions) and Chapter 18(mineral metabolism). Besides, since all life processes including growth, workand animal production (milk, eggs, wool) use energy, methods to study energymetabolism in relation to dietary changes are reviewed in Chapter 16.

4 J. Dijkstra et al.

Consequences of ruminant adaptations

The ability of the ruminant to utilize forages high in fibre is exploited in manyagricultural production systems. However, the slow rate of digestion means thatfeed particles remain in the rumen for long periods and rumen capacitybecomes a limiting factor to further intake; the slower and less complete thedigestion of a particular feed, the greater is the importance of physical factors,compared to metabolic factors, in the control of feed intake (see Chapter 23).The ability of ruminants to select a balanced diet from imbalanced foods offeredin choice has become better established since publication of the first edition ofthis book and modelling of intake has been extended to food choice in thischapter.

Feeding large amounts of rapidly fermented carbohydrate producessudden changes in acid and gas production that are sometimes beyond theadaptive ability of the animal. The pH of rumen fluid falls from a normal levelof 6.0 to 6.2, causing cessation of motility and reduction in feed intake.Excessive gas production causes bloat, under some circumstances, and a re-duced acetate:propionate ratio depresses milk fat synthesis. A consequenceof microbial protein synthesis in the rumen is that some of the protein inthe diet can be replaced by non-protein nitrogen, typically urea. High-qualityprotein sources can be protected against ruminal degradation to obtainmore benefit from their superior balance of amino acids or to better matchthe amount of degradable carbohydrates. Moreover, and depending on thestarch degradation characteristics, starch sources may be protected againstruminal degradation to avoid low pH levels, or starch degradation may beenhanced to promote energy supply to the microbes in the rumen. The effectof various technological treatments on nutrient digestibility is discussed inChapter 24.

These adaptations and their metabolic consequences have importanteffects on productive processes; these are discussed in Chapter 19 (growth),Chapter 20 (pregnancy), Chapter 21 (lactation) and Chapter 22 (wool).

In the developed world, cattle are often kept in automated, intensivesystems. In these intensive systems, a much better management control overthe environmental effects is achieved. It is therefore important to understandhow cattle interact with their environment, in order to optimize the design andmanagement of cattle production systems, and also in view of animal welfare.The topic of animalenvironment interaction is discussed in Chapter 25.

Since forages are generally the main part of the ruminant diet, botanical,physical and chemical characteristics of the forage are important in determin-ing the nutritive value for the ruminant. Ruminants will adapt their intakebehaviour (in terms of, for example, eating and ruminating time and biterate and bite mass characteristics) to changes in such forage characteristics.The interaction between the pasture and the animal is discussed in Chapter 26.

Finally, various systems have been developed to evaluate the feeding valueof diet ingredients and to predict the animal response to intake of a given set offeed ingredients. The various approaches to the integration of data in feedevaluation systems are discussed in Chapter 27.

Introduction 5

Quantitative Approaches to Ruminant Physiology

Traditionally, quantitative research into digestion and metabolism in ruminants,as in many other areas of biology, has been empirically based and has centredon statistical analysis of experimental data. Whilst this has provided much of theessential groundwork, more attention has been given in recent years to im-proving our understanding of the underlying mechanisms that govern theprocesses of ruminant digestion and metabolism, and this requires an increasedemphasis on theory and mathematical modelling. The primary purpose of eachof the subsequent chapters of this book, therefore, is to bring together thequantitative approaches concerned with elucidating mechanism in a particulararea of ruminant digestion and metabolism. Given the diverse scientific back-grounds of the contributors of each chapter, the imposition of a rigid format forpresenting the mathematical material has been eschewed, though basic math-ematical conventions are adhered to. Before considering each area, however, itis necessary to review the nature and implications of organizational hierarchy(levels of organization), and to review the different types of model that maybe constructed.

Organizational hierarchy

Biology, including ruminant physiology, is notable for its many organizationallevels. It is the existence of the different levels of organization that give rise tothe rich diversity of the biological world. For the animal sciences, a typicalscheme for the hierarchy of organizational levels is shown in Table 1.1. Thisscheme can be continued in both directions and, for ease of exposition, thedifferent levels are labelled . . . , i 1, i, i 1, . . . . Any level of the scheme canbe viewed as a system, composed of subsystems lying at a lower level, or as asubsystem of higher level systems. Such a hierarchical scheme has someimportant properties:

1. Each level has its own concepts and language. For example, the terms ofanimal production such as plane of nutrition and liveweight gain have littlemeaning at the cell or organelle level.

Table 1.1. Levels of organization.

Level Description of level

i 3 Collection of organisms (herd, flock)i 2 Organism (animal)i 1 Organi Tissuei 1 Celli 2 Organellei 3 Macromolecule

6 J. Dijkstra et al.

2. Each level is an integration of items from lower levels. The response of thesystem at level i can be related to the response at lower levels by a reductionistscheme. Thus, a description at level i 1 can provide a mechanism forbehaviour at level i.3. Successful operation of a given level requires lower levels to functionproperly, but not necessarily vice versa. For example, a microorganism canbe extracted from the rumen and can be grown in culture in a laboratory, sothat it is independent of the integrity of the rumen and the animal, but therumen (and hence the animal) relies on the proper functioning of its microbesto operate normally itself.

Three categories of model are briefly considered in the remainder of thischapter: teleonomic, empirical and mechanistic. In terms of this organizationalhierarchy, teleonomic models usually look upwards to higher levels, empiricalmodels examine a single level and mechanistic models look downwards, con-sidering processes at a level in relation to those at lower levels.

Teleonomic modelling

Teleonomic models (see Monod, 1975, for a discussion of teleonomy) areapplicable to apparently goal-directed behaviour, and are formulated explicitlyin terms of goals. They usually refer responses at level i to the constraintsprovided by level i 1. It is the higher level constraints which can selectcombinations of the lower level mechanisms, which may lead to apparentlygoal-directed behaviour at level i. Currently, teleonomic modelling plays only aminor role in biological modelling, though this role might expand. It has not, asyet, been applied to problems in ruminant physiology though it has found someapplication in plant and crop modelling (Thornley and Johnson, 1989).

Empirical modelling

Empirical models are models in which experimental data are used directly toquantify relationships, and are based at a single level (e.g. the whole animal) inthe organizational hierarchy discussed above. Empirical modelling is concernedwith using models to describe data by accounting for inherent variation in thedata. Thus, an empirical model sets out principally to describe, and is based onobservation and experiment and not necessarily on any preconceived biologicaltheory. The approach derives from the philosophy of empiricism and adheresto the methodology of statistics.

Empirical models are often curve-fitting exercises. As an example, considermodelling voluntary feed intake in a growing, non-lactating ruminant. Anempirical approach to this problem would be to take a data set and fit a linearregression equation, possibly:

I a0 a1W a2dW=dt a3D (1:1)

Introduction 7

where I denotes the intake, W, liveweight, D, measure of diet quality anda0, a1, a2, and a3 are parameters.

We note that level i behaviour (intake) is described in terms of level iattributes (liveweight, liveweight gain and diet quality). As this type of model isprincipally concerned with prediction, direct biological meaning cannot beascribed to the equation parameters and the model suggests little about themechanisms of voluntary feed intake. If the model fits the data well, theequation might be extremely useful though it is specific to the particularconditions under which the data were obtained, and so the range of its predict-ive ability will be limited.

Mechanistic modelling

Mechanistic models, which underlie much of thematerial presented in this book,seek to understand causation. A mechanistic model is constructed by looking atthe structure of the system under investigation, dividing it into its key compon-ents and analysing the behaviour of the whole system in terms of its individualcomponents and their interactions with one another. For example, a simplifiedmechanistic description of intake and nutrient utilization for our growing rumin-antmight contain five components, namely two body pools (protein and fat), twoblood plasma pools (amino acids and carbon metabolites) and a digestive pool(rumen fill), and include interactions such as protein and fat turnover, gluconeo-genesis from amino acids and nutrient absorption. Thus, the mechanistic mod-eller attempts to construct a description of the system at level i in terms of thecomponents and their associated processes at level i 1 (and possible lower), inorder to gain an understanding at level i in terms of these component processes.Indeed, it is the connections that interrelate the components that make a modelmechanistic. Mechanistic modelling follows the traditional philosophy andreductionist method of the physical and chemical sciences.

Mechanistic modelling gives rise to dynamic differential equations. There isa mathematically standard way of representing mechanistic models called therate:state formalism. The system under investigation is defined at time t by qcomponents or state variables: X1, X2, . . . , Xq. These variables representproperties or attributes of the system, such as visceral protein mass, quantityof substrate, etc. The model then comprises q first-order differential equations,which describe how the state variables change with time:

dXi=dt fi(X1, X2, . . . , Xq; S); i 1, 2, . . . , q (1:2)

where S denotes a set of parameters, and the function fi gives the rate ofchange of the state variable Xi.

The function fi comprises terms that represent individual processes (withdimensions of state variable per unit time), and these rates can be calculatedfrom the values of the state variables alone, with of course the values of anyparameters and constants. In this type of mathematical modelling, the differ-ential equations are formed through direct application of the laws of science

8 J. Dijkstra et al.

(e.g. the law of mass conservation, the first law of thermodynamics) or byapplication of a continuity equation derived from more fundamental scientificlaws.

If the system under investigation is in steady state, the solution to Eq. (1.2)is obtained by setting the differential terms to zero and manipulating to give anexpression for each of the components and processes of interest. Radioisotopicdata, for example, are usually resolved in this way, and indeed, most of thetime-independent formulae presented in this book are derived likewise. How-ever, in order to generate the dynamic behaviour of any model, the rate:stateequations must be integrated.

For simple cases, analytical solutions are usually obtained. Such models arewidely applied in ruminant digestion studies to interpret time-course data frommarker and polyester-bag experiments, where the functional form of the solu-tion is fitted to the data using a curve-fitting procedure. This enables biologicalmeasures, such as mean retention time in the rumen prior to escape and theextent of ruminal degradation, to be calculated from the estimated parameters.

For the more complex cases, only numerical solutions to the rate:stateequations can be obtained. This can be conveniently achieved by using one ofthe many computer software packages available for tackling such problems.Such models are used to simulate complex digestive and metabolic systems.They are normally used as tactical research tools to evaluate current under-standing for adequacy and, when current understanding is inadequate, helpidentify critical experiments. Thus, they play a useful role in hypothesis evalu-ation and in the identification of areas where knowledge is lacking, leading toless ad hoc experimentation. Also, a mechanistic simulation model is likely tobe more suitable for extrapolation than an empirical model, as its biologicalcontent is generally far richer.

Further discussion of these issues can be found in Thornley and France(2005).

Acknowledgement

We are pleased to acknowledge Dr Graham Faichneys contribution to Fig. 1.2and related material.

Further Reading

Textbooks

Baldwin, R.L. (1995) Modelling Ruminant Digestion and Metabolism. Chapman &Hall, London.

Blaxter, K.L. (1989) Energy Metabolism in Animals and Man. Cambridge UniversityPress, Cambridge.

Church, D.C. (ed.) (1993) The Ruminant Animal: Digestive Physiology and Nutri-tion. Waveland Press, Inc., Englewood Cliffs, New Jersey.

Introduction 9

Czerkawski, J.W. (1986) An Introduction to Rumen Studies. Pergamon Press,Oxford, UK.

Food and Agriculture Organization (2004) FAOSTAT Data, 2004. FAO, Rome.Forbes, J.M. (1995) Voluntary Food Intake and Diet Selection in Farm Animals, 1st

edn. CAB International, Wallingford, UK.Getty, R. (ed.) (1975) Sisson and Grossmans Anatomy of the Domestic Animals, 5th

edn. W.B. Saunders Co, Philadelphia, Pennsylvania.Hobson, P.N. and Stewart, C.S. (eds) (1997) The Rumen Microbial Ecosystem, 2nd

edn. Blackie Academic & Professional, London.Hungate, R.E. (1966) The Rumen and Its Microbes. Academic Press, New York.McDonald, P., Edwards, R.A., Greenhalgh, J.F.D. and Morgan, C.A. (2002) Animal

Nutrition. Prentice-Hall, Englewood Cliffs, New Jersey.Monod, J. (1975) Chance and Necessity: An Essay on the Natural Philosophy of

Modern Biology. Collins, London.Reece, W.O. (ed.) (2004) Dukes Physiology of Domestic Animals, 12th edn. Com-

stock Publishing, Ithaca, New York.Theodorou, M.K. and France, J. (eds) (2000) Feeding Systems and Feed Evaluation

Models. CAB International, Wallingford, UK.Thornley, J.H.M. and France, J. (2005) Mathematical Models in Agriculture, 2nd

edn. CAB International, Wallingford, UK.Thornley, J.H.M. and Johnson, I.R. (1989) Plant and Crop Modelling. Oxford Uni-

versity Press, Oxford, UK.Van Soest, P.J. (1994) Nutritional Ecology of the Ruminant, 2nd edn. Cornell

University Press, Ithaca, New York.

Proceedings of symposia

Baker, S.K., Gawthorne, J.M., Mackintosh, J.B. and Purser, D.B. (eds) (1985) Rumin-ant Physiology: Concepts and Consequences. School of Agriculture, University ofWestern Australia, Perth, Western Australia.

Cronje, P. (ed.) (2000) Ruminant Physiology: Digestion, Metabolism, Growth andReproduction. CAB International, Wallingford, UK.

Dobson, A. and Dobson, M.J. (eds) (1988) Aspects of Digestive Physiology in Rumin-ants. Comstock, Ithaca, New York.

Kebreab, E., Mills, J.A.N. and Beever, D.E. (eds) (2004) Dairying Using Science toMeet Consumers Needs. Nottingham University Press, Nottingham, UK.

Kebreab, E., Dijkstra, J., Gerrits, W.J.J., Bannink, A. and France, J. (eds) (2005)Nutrient Digestion and Utilization in Farm Animals: Modelling Approaches.CAB International, Wallingford, UK.

McNamara, J.P., France, J. and Beever, D.E. (eds) (2000) Modelling Nutrient Utiliza-tion in Farm Animals. CAB International, Wallingford, UK.

Milligan, L.P., Grovum, W.L. and Dobson, A. (eds) (1986) Control of Digestion andMetabolism in Ruminants. Prentice-Hall, Englewood Cliffs, New Jersey.

Tsuda, T., Sasaki, Y. and Kawashima, R. (eds) (1991) Physiological Aspects of Diges-tion and Metabolism in Ruminants. Academic Press, San Diego, California.

Von Engelhardt, W., Leonhard-Marek, S., Breves, G. and Giesecke, D. (1995) Rumin-ant Physiology: Digestion, Metabolism, Growth and Reproduction. FerdinandEnke Verlag, Stuttgart, Germany.

10 J. Dijkstra et al.

Digestion

2 Rate and Extent of DigestionD.R. Mertens

USDA Agricultural Research Service, US Dairy Forage Research Center,Madison, WI 53706, USA

Introduction

Digestion in ruminants is the result of two competing processes: digestion andpassage. Rate of passage determines the time feed is retained in the alimentarytract for digestive action and the rate and potential extent of degradationdetermines the digestion that can occur during the retention time. To predictdynamic flows of nutrients or static estimates of digestibility at various levels ofperformance, the processes of digestion and passage must be described incompatible mathematical terms and integrated. This chapter will focus on themathematical description or modelling of digestion, especially fermentativedigestion in the rumen because it typically represents the largest proportionof total tract digestibility and is the first step in the digestive process forruminants that influences the processes that follow.

The digestive process involves the time-dependent degradation or hydroly-sis of complex feed components into molecules that can be absorbed by theanimal as digesta passes through the alimentary tract. Conceptually, digestionand passage can be described as multi-step processes using compartmentalmodels (Blaxter et al., 1956; Waldo et al., 1972; Baldwin et al., 1977, 1987;Mertens and Ely, 1979; Black et al., 1980; Poppi et al., 1981; France et al.,1982). Because feed components do not digest or pass through the digestivetract similarly (Sutherland, 1988), an understanding about the nature of pas-sage in ruminants provides an important framework for developing compatibledigestion models.

In ruminants, passage of digesta through the alimentary tract is a complexprocess that involves selective retention, mixing, segregation, and escape ofparticles and liquid from the rumen before they pass into and through the smalland large intestines. Mechanistically, the reticulorumen, small intestine andlarge intestine differ in mixing and flow. The rumen operates as an imperfectlystirred, continuous-flow reactor, whereas the small and large intestines act

CAB International 2005. Quantitative Aspects of Ruminant Digestionand Metabolism, 2nd edition (eds J. Dijkstra, J.M. Forbes and J. France) 13

more like plugged-flow reactors (Levenspiel, 1972; Penry and Jumars, 1987).Furthermore, ruminal contents act as though there were at least three differentsubcompartments with different flow characteristics: liquid, escapable particlesand retained particles. Soluble feed components dissolve and pass out at therate of ruminal liquids. Ground concentrates and forages pass out of the rumenmore quickly than large fibre particles, which are retained selectively andruminated. Models of digestion must be compatible with these differences inpassage rates and processes.

Separate compartments are needed to represent the distinct digestive andpassage processes of the reticulorumen, small intestine and large intestine. Theunique digestive kinetics of feed components should be described by dividingfeed into rapidly digested, slowly digested and indigestible compartments. Thevariety of compartments needed to model digestion and passage illustrates animportant principle. Model compartments are defined by their kinetic proper-ties and may not necessarily correspond to anatomical, physiological, chemicalor physical compartments in the real system. Thus, non-escapable and escap-able particles should be described as separate compartments, though both arein the ruminal environment. The kinetic property of escapability rather thanparticle size is used to define particles because small particles trapped in thelarge particle ruminal mat pass differently from those located in the reticularzone of escape (Allen and Mertens, 1988). Particles are uniquely definedbecause they have different kinetic parameters and require separate equationsto describe the processes of digestion and passage. Similarly, digestible andindigestible matter may be contained in the same feed particle, yet eachrequires a separate compartment to describe their unique kinetics of digestionand passage.

Current models describe digestion as a function of the mass of substratethat is available in a compartment, i.e. they are mass-action models. Generally,digestion is described as a first-order process with respect to substrate (Waldoet al., 1972; Mertens and Ely, 1979); however, some models describe it as asecond-order process that depends on the pools of substrate and microorgan-isms present in the system (France et al., 1982; Baldwin et al., 1987).Regardless of the model used, it appears that rate and extent of digestion arecritical variables in the description of the digestion process. Kinetic parametersof digestion are important because they not only describe digestion, but alsothey characterize the intrinsic properties of feeds that limit their availability toruminants.

To be useful, models based on mechanistic assumptions must replicate thereal system with an acceptable degree of accuracy. The number of differentmechanistic models that can predict a set of observations may be large, per-haps infinite (Zierler, 1981). Thus, accuracy in predicting a specific set of datacannot prove that a model is uniquely valid, but only indicates that it is oneplausible explanation of reality. To be universally applicable, models should bevalid in extreme situations and under varied experimental conditions, ratherthan predicting the average accurately, even if it is from a large data set.

The goal of this chapter is to present the theoretical development and useof models for quantifying rate and extent of the digestion process in the rumen.

14 D.R. Mertens

To accomplish this goal, methods used to collect kinetic data will be analysed,the background of simple models for measuring rate and extent of fermentativedigestion will be discussed, mathematical models will be proposed that moreaccurately describe the methods used to obtain kinetic data, and methods offitting data to models for estimating kinetic parameters will be reviewed.

Terminology

Before proceeding, some terminology that will be used in the remainder of thechapter needs to be defined. Considerable confusion results from incorrect orundefined use of terms. Even the most common terms such as rate or extentare often defined or interpreted differently by authors. All too often mathemat-ical formulations used to generate coefficients are not provided explicitly,adding further confusion to the discussion of factors affecting digestion kinetics.For example, in one paper rate may be defined as the starting amount ofmaterial minus the ending amount of material divided by the interval allowedfor digestion (an absolute rate). In another paper, rate is determined as thefraction of the potentially digestible material that disappears per hour (a frac-tional or relative rate). Analysing the same data in these two different ways canlead to opposite conclusions about which treatment has the faster rate (Table2.1). Caution is advised when reviewing literature on digestion kinetics becauseof non-standardized and ambiguous use of terminology. Valuable time andresources have been wasted in explaining discrepancies that were only afunction of fuzzy definitions or contradictions between verbal concepts andmodels.

Table 2.1. Effect of using different definitions of rate (absolute versusfractional) on the comparison of digestion kinetics from two treatments.

Variable Treatment 1 Treatment 2

Time (h) Residue remaining (mg)0 100.0 100.0

12 63.9 63.024 44.1 48.848 27.3 41.372 22.2 40.2

Absolute ratea (mg/h) 2.33 2.13Fractional rateb (per h) 0.05 0.08Potential digestibilityb (mg) 80 60

aAbsolute rate determined by taking the difference in residue weights at 0 and 24 h

and dividing by 24.bFractional rate (Kd) and potential digestibility (D0) determined using the model

R(t)D0 exp(Kdt)I0, where I0 is indigestible residue.

Rate and Extent of Digestion 15

The following are definitions of terms used in this chapter:

Aggregation: Combining entities or attributes in a model that have similarkinetic properties to reduce detail and complexity.

Assumptions: Implicit or explicit relationships or attributes of a model that areaccepted a priori.

Attributes: Coefficients of parameters and variables used to describe theentities in a model.

Compartment: Boundaries of an entity that is distributed in an environmentthat is assumed to have homogeneous dynamic or static properties. Com-partments are typically represented in diagrams by solid-lined boxes.

Dynamic: Systems, reactions or processes that change over time.Entities: Independent, complete units or substances that have uniquely defined

chemical or physical properties in a system.Environment: Physical location of an entity in a system.Extent of digestion: A digestion coefficient that represents the proportion of a

feed component that has disappeared as a result of digestion after aparticular time in a specified system. It is a function of the time allowedfor digestion and the digestion rate. Units are fractions or percentages.Extent of digestion is a more general term that is not equal to either thepotentially digestible fraction or potential extent of digestion.

Flux or flow: Amount of material per unit of time that is transferred to or froma compartment. In non-steady-state conditions, fluxes vary over time.Although they may have the same mathematical form in some cases, fluxesare not the same concept as the derivative of the pool size. Fluxes typicallyare represented in diagrams by arrows.

Flux ratio: Proportion of a flux that is transferred to or from a compartment.Flux ratios differ conceptually from fractional rates because ratios partitionfluxes, whereas rates are proportions of pools that are transferred. Fluxratios typically are represented in mathematical equations by lower case rwith a subscript.

Indigestible residue: Residue of feed that remains after an infinite time ofdigestion in a specified system. It is often approximated by measuring thedisappearance of matter after long times of digestion.

Kinetics, mass-action: Systems in which material is transferred between com-partments in proportion to the mass of material in each compartment.

Kinetics, MichaelisMenten (or HenriMichaelisMenten): Kinetics derivedfrom a reversible second-order mass-action system in which the flux ofproduct formation is proportional to the concentration of substrate andenzyme (or microbial mass). With respect to substrate, the reaction variesfrom zero-order when enzyme is limiting, to first-order when enzyme (ormicrobial mass) is in excess.

Models: Representations of real-world systems. Models do not duplicate thereal world because they always contain assumptions about, and aggrega-tions of, components of the real-world system. Mathematical models useexplicit equations to describe a system.

16 D.R. Mertens

Models, deterministic: Assume the system can be simulated with certaintyfrom known or assumed principles or relationships.

Models, dynamic: Simulate the change in the system over time.Models, empirical: Based on relationships derived directly from observations

about the system. These data-driven models are sometimes called black boxor inputoutput models.

Models, kinetic: Kinetics refers to movement and the forces affecting it. Inchemical and biological systems, kinetic models are related to the molecu-lar movement associated with chemical or physical systems.

Models, mechanistic: Are based on known or assumed biological, chemical orphysical theories or principles about the system. These concept-drivenmodels are sometimes called white box models.

Models, static: Represent time-invariant systems or processes. The steady-state solution of dynamic systems is a specific type of static model.

Models, stochastic: Assume that the system operates on probabilistic prin-ciples or contains random elements that cannot be known with certainty.

Order of reaction: The combined power terms of the pools in mass-actionkinetic systems. For example, in first-order systems the flux of reaction isrelated to the amount or concentration of a single pool raised to the power1. In second-order systems, flux is related to a single pool raised to thepower 2 or the product of two pools raised to the power 1.

Parameters: Constants in equations that are not affected by the operation ofthe model.

Pool: Mass, weight or volume of material in a compartment. Pools are typicallyrepresented by upper case letters in mathematical equations.

Potentially digestible fraction: Inverse of the indigestible fraction (1.0 indigestible fraction). It is the proportion of feed that can disappear dueto digestion given an infinite time in a specified system. The potentiallydigestible fraction is the same as the potential extent of digestion ormaximal extent of digestion.

Processes: Activities or mechanisms that connect entities within a system anddetermine flows or fluxes between compartments.

Rate: Change per unit of time, which can be expressed in many different units;therefore, it is important to indicate the specific type of rate being dis-cussed, preferably with a mathematical description.

Rate, absolute: Has the units of mass per unit of time. Absolute rates andfluxes are the same, but the term flux is preferred because it preventsconfusion associated with the unqualified use of the term rate.

Rate, first-order: Fractional rates that are proportional to a single pool.Rate, fractional (or relative): Proportion of mass in a pool that changes per

unit of time. This rate has no mass units and is usually a constant that doesnot vary over time. First-order fractional rate constants are usually repre-sented in mathematical equations by a lower case k with subscripts.

Simulation: Operation of a model to predict a result expected in the real-worldsystem.

Rate and Extent of Digestion 17

Sinks: Irreversible end-point compartments of entities that are outside opensystems. Sinks are typically represented in diagrams by clouds with enter-ing arrows.

Sources: Initial locations of materials that are supplied from outside opensystems. Sources are typically represented in diagrams as clouds withexiting arrows.

State, quasi-steady: Occurs when pools within compartments in a dynamicsystem do not change significantly. Under natural situations, the timeneeded to attain quasi-steady-state is relative. True steady state cannot beachieved in perturbed systems because small changes are occurring con-tinuously. Quasi-steady-state is sometimes called the steady-state approxi-mation.

State, steady: Occurs when pools within compartments in a dynamic systemdo not change. True steady state is a mathematical construct that occurswhen the derivative of a pool with respect to time equals zero.

Systems: Organized collections of entities that interact through various pro-cesses. Open systems can accept or return material outside the system,whereas all material must originate and be retained in a closed system.

Time, retention: Is the average time an entity is retained in a compartment.Time, turnover: Is the time needed for a compartment to transfer an amount

of material equal to its pool size.Validation: Evaluating the credibility or reliability of a model by comparing it to

real-world observations. No model can be validated completely because allof the infinite possibilities cannot be evaluated. Some modellers prefer theterm evaluation rather than validation.

Variables: Coefficients that change during or among model simulations. Vari-ables can be external or internal to the model. External or exogenousvariables are inputs that affect or interact with the system that is modelled,but are controlled outside of it. Internal or endogenous variables are calcu-lated within the model during its operation.

Variables, state: Define the level, mass or concentration within the pools of thesystem.

Verification: Checking the accuracy by which a model is described mathemat-ically and implemented.

Requirements for Quantifying Rate and Extent of Digestion

Robust quantitative description of the rate and extent of digestion requiresthree components:

1. Appropriate biological data measured in a defined, representative systemusing an optimal experimental design.2. Proper mathematical models that reflect biological principles.3. Accurate fitting procedures for parameter estimation.

The validity of digestion kinetics depends on data that are accurately collected ina relevant system.Once the biology of the system for collecting data is described,

18 D.R. Mertens

models should be developed that correctly reflect the system. Only then can avalid fitting procedure be used to accurately estimate rate and extent of digestion.

Kinetic Data

Accurate biological data, generated by a method that is consistent with themathematical model and its assumptions, is a necessary first step in quantifyingdigestion kinetics. Subtle differences among measurements can have substan-tial effects on the parameterization and interpretation of digestion kinetics.Three characteristics of the data have critical impact on modelling and theinterpretation of kinetic properties:

1. The method used to measure kinetic changes.2. The specific component on which kinetic information is measured.3. The design of sampling times and replications.

Kinetic data can be collected using either in vitro or in situ methods, and thecomponent measured can vary from specific polysaccharides to total dry matter(DM). Reported end-point sampling times have varied from as little as 6 h tomore than 40 days.

Data collection method

Both in vitro and in situ techniques use time-series sampling to obtain kineticdata. In vitro methods involve the incubation of samples in tubes or flasks witha buffer solution and ruminal fluid or enzymes. In situ techniques require theincubation of samples in porous bags that are suspended in the rumens offistulated cows. Either method may be appropriate for measuring digestionkinetics, depending on research objectives. However, both methods haveadvantages and disadvantages that influence their suitability for a given appli-cation, affect the mathematical model that is needed, and alter interpretation ofresults. Regardless of the model used to describe digestion, kinetic parameterscan be determined only on the assumption that they are constant during thetime data are collected, and the component that is reacting can be measuredaccurately and unambiguously.

In vitro methodsModels to measure digestion kinetics in vitro are less complex than thoseneeded to measure in situ kinetics because the environment of the system iseasier to control and measurements are not affected by infiltration or loss ofmaterials from the fermentation vessel. However, not all in vitro systems usedto measure 48-h digestibility are acceptable methods for measuring kineticdata. Many in vitro systems fail to include adequate inocula, buffers, reagentsor equipment to guarantee that pH, anaerobiosis, redox potential, microbialnumbers, essential nutrients for microbes, etc. do not limit digestion duringsome or all of the time that kinetic data are collected. Furthermore, it is

Rate and Extent of Digestion 19

important that particle size of the sample does not inhibit digestion if theresearch objective is to measure the intrinsic rate of digestion of chemicalcomponents and for this purpose samples are typically ground to pass througha 1 mm screen.

If some characteristic of the in vitro system limits digestion, it is obviousthat kinetic parameters intrinsic to the substrate are not measured. Besidesensuring that factors affecting rate and extent of digestion do not changesignificantly during fermentation, any in vitro system used for kinetic analysisalso must ensure that conditions in early and late fermentation do not limitdigestion. Many in vitro procedures shock microbes during inoculum prepar-ation or at inoculation because the sample-containing media is inadequatelyreduced and anaerobic. These systems will cause biased estimates of digestionkinetics because digestion during early fermentation is low. If non-substratecharacteristics of the in vitro technique limit digestion kinetics, it may bedifficult to detect underlying mechanisms or measure differences among treat-ments. Differences in in vitro systems can create a two- to threefold differencein kinetic parameter estimates.

The primary disadvantage of the in vitro method for generating kineticdata is that it may differ from the in vivo environment. Yet, this deficiency canbe an advantage when the research objective is to study intrinsic properties ofthe substrate. Conditions in vitro can be controlled to prevent fluctuations inpH, dilution, fermentation pattern, etc., that occur in vivo. In addition, in vitromethods can be adjusted to ensure that the characteristic of interest in thesubstrate is the only factor limiting fermentation. For example, if the intrinsiccharacteristics of fibre are to be investigated, the in vitro method can bemodified to ensure that particle size, nitrogen, trace nutrients, pH, etc. arenot the factors limiting rate and extent of fibre digestion.

If the goal is to assess effects of extrinsic factors on rate and extent ofdigestion, the in vitro method can be modified to maintain constant fermenta-tion conditions that do not violate assumptions needed to estimate kineticparameters. For example, pH of the buffer can be varied in vitro to determineits direct and interacting effects on digestion kinetics. If the objective is tomeasure the digestion kinetics of a feed when fed to an animal as the solediet, the substrate should be fermented in an in vitro system that contains nosupplemental nitrogen or trace nutrient sources that would not be available byrecycling in the animal.

In situ methodsIf the research objective is to determine the combined effects of the intrinsicproperties of the feed and the extrinsic characteristics of the fermentationpattern in the animal on digestion kinetics, the in situ method may be appro-priate, biologically. Justification for using the in situ method is based on theconcept that dynamic animaldiet interactions are important. Consequently,kinetics of digestion measured in situ are valid only when the feed in the bag isalso the feed fed to the host animal. However, if in situ data are to be used toestimate kinetic parameters, an additional constraint is required. Conditionsof fermentation in the rumen must be constant, i.e. the animal must be in

20 D.R. Mertens

quasi-steady-state to meet the restriction that compartments have homoge-neous kinetic properties during the time kinetic data are collected.

Usually, the objective of kinetic experiments is to measure the intrinsic rateand extent of digestion of the test material. In these situations, the in situmethod has disadvantages that affect the interpretation of rate and extentparameters. Kinetic results obtained under non-steady-state conditions maybe biased by the time samples were placed in the rumen because fermentationpatterns vary relative to the animals feeding time. In addition, kinetic param-eters may be related more to the type of diet that the host animal is fed (andresulting ruminal conditions) than to the intrinsic properties of the substrate. Ifrate of digestion varies because of factors that are extrinsic to the substrate,interpretation of kinetic parameters is complex, and their general applicabilityis questionable. Even if all samples are included in the same animal simultan-eously, it is difficult, if not impossible, to attribute differences between treat-ments to intrinsic differences in substrates, unless interactions between intrinsicand extrinsic factors are known not to exist.

In situ kinetic data also is hampered by losses of DM and contaminationfrom incoming material. In situ bags are porous to allow infiltration of microbesfor fermentation of residues inside the bag. Unfortunately, these same poresallow escape of undigested, fine particles, and infiltration of fine particles fromruminal contents. France et al. (1997) suggested models and mathematics forcorrecting in situ disappearance for particle losses and variable fractional ratesduring the initial period of digestion. However, these models do not account forthe possibility that material may also enter bags while they are in the rumen, butnot be completely washed out after fermentation. Because much of the finematter in the rumen is indigestible or extensively digested, influx contaminationcan result in high estimates of the indigestible fraction, which in turn can bias thepotentially digestible fraction and the fractional digestion rate.

An obvious solution to fine particle infiltration is to either physically removefine-particle mass by washing the bags or arithmetically subtracting an estimateof particle contamination of the residues using blank bags (Weakley et al.,1983; Cherney et al., 1990). The first option has the disadvantage thatextensive washing can cause loss of substrate from the bags (especially atearly fermentation times) that is not due to digestion. In addition, it is notpossible to confirm that the washing technique is adequate without first includ-ing blanks. Blank bags probably should contain ground inert material of a masssimilar to that of the samples to prevent them from collapsing and preventingthe infiltration of fine particles. Alternatively, a model can be developed thatrepresents migration of residues into and out of in situ bags. Similarly, modelscan be developed that account for the initial solubilization of matter that occursin both the in vitro and in situ systems.

Component

Determining kinetics of fibre digestion is the least complex of any feedcomponent because fibre should not be affected by initial solubilization or

Rate and Extent of Digestion 21

contamination by microbial debris. Models are often developed to account forinitial solubilization of feed components such as DM or protein (rskov andMcDonald, 1979). However, without careful design of the experiment it isdifficult, if not impossible, to separate solubilization from lag phenomena. Ifkinetic analysis of feed components that solubilize is desired, samples must betaken at zero time to measure solubilization directly.

For compounds that are contaminated by microbial residues, the determin-ation and interpretation of digestion kinetics is more complex. Digestion of DMand protein, uncorrected for microbial contamination, does not represent truedigestion kinetics of feed components, rather it represents the kinetics of netdigestion, which is analogous to apparent digestibility coefficients. Not only is ituncertain that microbial contamination will be similar in other situations wherethe kinetic parameters are used, but also the moderating effect of microbialresidues on disappearance of DM and protein may mask true differencesamong feeds. If the goal of the research is to relate digestion kinetics to intrinsicproperties of the feed, the use of net residues, contaminated by microbialdebris, is questionable.

Theoretically, simple models of digestion are inappropriate for measuringintrinsic kinetic properties of DM or protein. One solution to this problem is tomeasure and subtract the contamination associated with microbial debris usingmicrobial markers (Nocek, 1988; Huntington and Givens, 1995; Vanzantet al., 1998). Fractional rates of protein degradation were changed dramatic-ally by removing contamination, thereby providing empirical evidence thatmicrobial residues can result in biased estimates of kinetic parameters. Alter-natively, the digestion model can be modified to include microbial residues asdescribed later in this chapter. These models can assess potential errors asso-ciated with the use of simple models and provide analytical solutions that canestimate more appropriately the intrinsic rates and extents of digestion of DMand protein.

Design

Regardless of the method used to generate kinetic data, the experimentaldesign must be consistent with the objective of obtaining accurate estimatesof parameters. Biological, statistical, kinetic and resource management consid-erations should be used to adequately and efficiently collect kinetic data. Bio-logically, variation in both in vitro and in situ experiments is greater betweenruns than within runs. Therefore, to estimate universally valid kinetic param-eters the experimental design should replicate substrate between runs ratherthan within runs. Replicated measures within a run are repeated measures, likereplicated laboratory analyses, and do not qualify as independent measureswhen doing statistical tests or estimating standard errors. Replicated data fromdifferent runs provide additional information about run by substrate interactionsand are useful in estimating lack-of-fit statistics.

For most efficient use of resources, more measurements should be made atadditional fermentation times instead of replicating measurements at fewer

22 D.R. Mertens

fermentation times within a run. Statistical concepts indicate that regressioncoefficients are determined more accurately when the same number of obser-vations are collected once at more times rather than multiple observationscollected at fewer times. Deviation from regression is a good estimate ofreplicate variation, thereby making duplicate sampling at each time statisticallyredundant. Although there is no statistical rule, experience suggests that thereshould be at least three observations for each parameter to be estimated in themodel. Most digestion models contain three independent parameters, indicat-ing that at least nine fermentation times are needed to estimate parameters ofsimple digestion models adequately and accurately.

Spacing of fermentation times is important in optimizing the design ofkinetic experiments. When nothing is known about the process, it is best toevenly space observations for regression analysis. However, a priori informa-tion about digestion kinetics can be used to improve the efficiency of regressionanalysis. In general, variance in kinetic data is proportional to the absolute rateof reaction that occurs between 6 and 18 h of fermentation. Therefore, obser-vations should be taken more often between 3 and 30 h than during otherperiods of fermentation to offset the greater variation that occurs during thisperiod of rapid fermentation. Optimal and minimal sampling times suggestedfor collecting kinetic data are given in Table 2.2. Also, it is desirable to recordthe exact time samples are taken to the nearest 0.1 h because regressionanalysis assumes that the independent variable (time) is measured withouterror and inaccurate time measurements can significantly affect results.

Table 2.2. Recommended sampling times to obtain accurate parameter estimates fordigestion kinetics.

Number of samplesRapidly digesting component

(hours after inoculation)Slowly digesting component(hours after inoculation)

Optimalsamplinga

Minimalsamplingb

Optimalsampling

Minimalsampling

Optimalsampling

Minimalsampling

1 0 02 1 0 0 0 03 2 2 2 3 34 3 4 4 6 65 4 8 8 9 96 5 12 12 12 127 16 188 6 20 20 24 249 24 30

10 7 32 32 36 3611 40 4812 8 48 48 72 7213 9 64 64 96 96

aOptimal sampling strategy for digestion models containing three parameters.bMinimal sampling strategy for digestion models containing three parameters.

Rate and Extent of Digestion 23

Observations at the beginning and end of fermentation also are criticalbecause they establish initial solubilization/lag and potential extent of digestion,respectively. Accurate zero-time measurement is needed to distinguish solubil-ization from digestion and estimate the lag effect. Thus, it is important to makeextra observations during the lag phenomenon and to duplicate measurementswhen time equals zero. Replicated measurements are also valuable in estimat-ing the potential extent of digestion.

Models of Digestion

The mathematics for describing first-order dynamic systems is rather simple.Too often it is assumed that rigorous mathematical training is required to modela biological system. Typically, biological conceptualization of the system is themost difficult part of the modelling process. Fear of mathematics has createdtoo much dependence on the selection of equations from those reported in theliterature and has inhibited many scientists from formally describing theirconceptual model in precise mathematical terms that accurately describe thebiological process being investigated. The focus of this section will be thedevelopment of simple models that demonstrate the principles of relatingbiology to the mathematical model and thereby stimulate the reader to generateother suitable models for describing kinetic data.

First-order digestion models can be classified into four types, depending onthe number of compartments and the number and type of reactions (Fig. 2.1).In simultaneous systems, flows from compartments occur simultaneously andindependently. In sequential systems, flow from some compartments becomes

Single compartmentSingle reaction

A . ka

Single compartmentMultiple simultaneous reactions

A . k 1

A . k2

A A

A . kaA

Multiple compartmentsSingle simultaneous reactions

B . kbB

Multiple compartmentsSingle sequential reactions

A . kaAB . kbB

Fig. 2.1. Illustrations of the various types of first-order models used to describe digestion.

24 D.R. Mertens

the input to other compartments, which creates a time dependency for thesecond compartment. Because the models are first-order, they will have anexponential function in the equation for each compartment in the system. Eachtype of model has a distinct set of linear and semi-logarithmic plots of theirdifferential and integral functions that can be used to identify the type ofdigestive process being investigated.

Comments about rates of digestion first appeared in the literature in the1950s, but development of digestion kinetics was hampered by the lack of abiological concept of the digestion process that could be described by a math-ematical formula. Description of the process was difficult because digestioncurveswerenon-linear, differed in asymptote anddid not appear to fit the kineticsof typical chemical reactions. Waldo (1970) was the first to suggest a conceptualbreakthrough that serves as the basis for our current viewof digestion kinetics.Hesuggested that digestion curves are combinations of digestible and indigestiblematerial.His hypothesis that somematter is indigestiblewas based on thework ofWilkins (1969) who observed that some cellulose was undigested in the rumenafter 7 days. Waldo speculated that if the indigestible residue was subtracted, thepotentially digestible fractionmight follow first-order,mass-action kinetics. Inter-estingly, nutritionists would have arrived at this same conclusion if they had usedclassical curve peeling approaches to analyse and interpret digestion curves inwhich fermentation was extended to more than 72 h.

Model 1: Simple first-order digestion with an indigestible fraction

The concept that all feed components are not potentially digestible not onlysimplifies the mathematical description of digestion, but also clarifies the bio-logical framework for explaining digestion. However, the problem in describingdigestion kinetics is that residues remaining at any digestion time are a mixtureof undigested and indigestible matter. The model proposed by Waldo (1970) isillustrated in Fig. 2.2. It assumes that the indigestible residue does not disap-pear, whereas the potentially digestible residue disappears at a rate that isproportional to its mass at any time. It is intuitive that rates of digestion areonly valid for potentially digestible components, i.e. indigestible componentshave rates of digestion of zero. Equations for this model are:

D . kdD

D = potentially digestible fractionkd = fractional rate of digestionI = indigestible fraction

0I

Digestedsink

Fig. 2.2. Model 1: Simple first-order model of digestion with an indigestible fraction.

Rate and Extent of Digestion 25

dD=dt kdD (2:1)dI=dt 0 (2:2)

where t represents time, I the indigestible residue, D the potentially digestibleresidue and kd the fractional rate constant of digestion.

Although derivatives of time describe the system elegantly, we seldommeasure fluxes under steady-state conditions, instead we measure amounts orconcentrations in a system at specified times. Thus, to describe the data usuallycollected, the above equations must be integrated over time to derive equationsthat correspond to observed data. The integrated equations are:

D(t) Di exp (kdt) (2:3)I(t) I0 (2:4)

R(t) D(t) I(t) Di exp (kdt) I0 (2:5)

where I0 and Di are the indigestible and potentially digestible residues at t 0and R(t) is the total undigested residue at any time.

The implicit assumptions of this first-order model are:

1. The potentially digestible and indigestible pools act as distinct compart-ments with homogeneous kinetic characteristics.2. The fractional rate of digestion is constant and is an intrinsic function ofthe digestive system and the substrate.3. Digestion begins instantly at time zero and continues indefinitely.4. Enzyme or microbial concentrations are not limiting.5. Flux or absolute rate is strictly a function of the amount of potentiallydigestible substrate present at any time.

The equation for D(t) can be transformed into a linear function by naturallogarithmic transformation (ln) and substitution:

ln [D(t)] ln [Di] kdt (2:6)D(t) R(t) I0 (2:7)

ln [R(t) I0] ln [Di] kdt (2:8)

By estimating I0 using long-term fermentations and regressing ln [R(t) I0] ontime, the intercept can be used to estimate Di and the slope or regressioncoefficient estimates the fractional rate constant of digestion (kd), which isdescribed on page 42. The true indigestible fraction can be reached only afterinfinite time, and any fermentation end-point is an overestimation of the trueasymptote. A practical estimate of the asymptote (I0) can be obtained whendigestion is >99% complete. The time at which a pool declines to 1% of itsoriginal value can be approximated by dividing 4.6 by the fractional rate of thepool. For a rate of 0.10/h it will take 46 h to decline to 1% of its original valuecompared with 92 h for a fractional rate of 0.05/h.

Van Milgen et al. (1992) observed differences in the indigestible acid deter-gent fibre fraction when measured after 42 days in situ when host animals were

26 D.R. Mertens

fed diets differing in the proportion of concentrate. They concluded that theindigestible fraction is not an intrinsic characteristic of the feed because it wasaffected by the diet of the animal. However, it could be argued that the intrinsicindigestibility of a feed can only be measured under optimal ruminal conditionsthat result in maximal digestion. Any perturbation of fermentation that does notallow maximal digestion results in indigestible residues that are contaminated byundigested potentially digestible matter. Although indigestibility may not be aconstant intrinsic characteristic of the feed, it may be more appropriate tomeasure the intrinsic indigestibility of the feed using an optimal system andthen modelling the extrinsic factors that cause incomplete digestion, even afterlong fermentation times, as a function of the fermentation system.

The classical test for the appropriateness of the first-order mass-actionmodel is to plot the natural logarithm of the potentially digestible residue versustime. If the plot is linear, the flux or absolute rate of reaction is constant andproportional to the amount of the potentially digestible pool; therefore the first-order, fractional rate constant model is a plausible description of the digestiveprocess. Although most researchers have used R2 to assess linearity, the mostpowerful statistical test is a lack-of-fit test comparing linear and quadraticfunctions of time using multiple samples each measured once in replicatedin vitro or in situ trials. Several scientists (Gill et al., 1969; Smith et al.,1972; Lechtenberg et al., 1974) evaluated the first-order model for potentiallydigestible matter, using either 48- or 72-h fermentations as the end-point forestimating I0. Their results indicated that first-order, mass-action kineticswith an indigestible fraction was an acceptable model of digestion for neutraldetergent fibre (NDF) and cellulose.

Model 2: Simple first-order digestion with indigestible and soluble fractions

For feed components that contain a significant soluble fraction, such as proteinand DM, the simple first-order model must be modified to include an additionalparameter to describe the digestive process. At the beginning of digestion,there can be disappearance of residue due to solubilization that should not beconfounded with rate of digestion (rskov and McDonald, 1979). This solubil-ization is so rapid compared with degradation that it can be considered instant-aneous. Except for the instant of solubilization, the differential equations forthis model (Fig. 2.3) are:

dD=dt kdD (2:9)dI=dt 0 (2:10)dS=dt 1 (2:11)

where S is the soluble fraction of the feed component and all other variables arethe same as defined for Model 1.

The integral equations for this system are the same as the simple first-ordermodel except:

Rate and Extent of Digestion 27

at t 0,

S(0) S0 (2:12)

and

R(0) D(0) I(0) S0 Di I0 S0 (2:13)

at t > 0,

S(t) 0 (2:14)

and

IR(t) D(t) I(t) Di exp (kdt) I0 (2:15)

where IR(t) is insoluble residue at any time t.The last equation, similar to that for the simple first-order model, can be

used to estimate instantaneous solubilization, assuming no lag effect, by ex-trapolating the potentially digestible fraction to t 0 and comparing (Di I0)to R0. If (Di I0) is less than R0, the difference is an estimate of S0, assumingno lag. Because the assumption of no lag effect is uncertain, it is necessary tomeasure insoluble residue at time zero (IR0), which allows estimation of bothS0 R0 IR0 and lag effects.

Model 3: Simple first-order digestion with discrete lag time and an indigestible fraction

The simple first-order model indicates that digestion begins instantaneously attime zero. Mertens (1977) observed that logarithmically transformed digestion

Fig. 2.3. Model 2: Simple first-ordermodel of digestion with soluble andindigestible fractions.

D . kdD

S = soluble fraction = infinite fractional rate indicating instantaneous transferD = potentially digestible fractionkd = fractional rate of digestionI = indigestible fraction

0I

S

Digestedsink

28 D.R. Mertens

curves typically exhibited non-linearity before 6 h of fermentation, which sug-gests a lag phenomenon. The potentially digestible pool (Di) estimated as theintercept of the simple model at t 0 usually exceeded 100% of that possiblebecause the actual potentially digestible pool (D0) at t 0 must be equal to totalresidue at time zero minus indigestible residue. Mertens (1977) proposed thatthe lag phenomenon could be easily quantified by including a discrete lag timein the simple first-order model (Fig. 2.4). Discrete lag time was defined as thetime at which the first-order equation derived for a data set equals the actualpotentially digestible fraction at zero time. The discrete lag model assumes thatno digestion occurs until lag time, when digestion begins instantaneously. Aftera discrete lag time, the differential equations and integral solutions are similar toModel 1. Differential equations for this model are:

at t < L:

dD=dt 0 (2:16)

and

dI=dt 0 (2:17)

at t L:

dD=dt kdD (2:18)

and

dI=dt 0 (2:19)

where L is discrete lag time.The integral equations for the discrete lag model are:

at t < L:

D(t) D0 (2:20)

D . kdD

D = potentially digestible fractionkd = fractional rate of digestionI = indigestible fraction

0I

0D

0I

At t < discrete lag time At t = or > discrete lag time

Digestedsink

Fig. 2.4. Model 3: Simplefirst-order model of digestionwith a discrete lag time beforedigestion and an indigestiblefraction.

Rate and Extent of Digestion 29

and

I(t) I0 (2:21)R(t) D0 I0 (2:22)

at t L:D(t) Di exp (kd[t L]) (2:23)

and

I(t) I0 (2:24)and

R(t) D(t) I(t) Di exp (kd[t L]) I0 (2:25)At t L:

R0 I0 D0 Di exp (kd[L]) (2:26)and

L [ ln (D0) ln (Di)]=(kd) (2:27)This model can be modified easily to incorporate the digestion kinetics of feedcomponents that exhibit initial solubilization (Dhanoa, 1988). However, toestimate lag time for these components, there must be a measure of the amountof insoluble residue at t 0 to provide an estimate of IR0 that must equal(D0 I0). Although the discrete lag model may not adequately describe lagphenomena for use in dynamic simulation models, it provides a simple andquantitative measure of the lag effect that can be used to compare feeds.Although Lopez et al. (1999) concluded that discrete lag models are difficultto justify biologically because some digestion occurs before lag time, theyobserved that the simple exponential model with discrete lag was only rankedbelow generalized exponential and inverse polynomial models for lack-of-fit,rank of residual mean of squares (RMS) and average RMS when used to describein situ DM, NDF and protein degradation. However, generalized exponentialand inverse polynomial models also have difficult biological interpretations.

When the intercept (Di) is greater than D0 clearly some type of lag phe-nomenon has occurred (see Fig. 2.9 in the Curve Peeling section). WhenDi < D0, the discrete lag time L is negative, which implies that digestion beginsbefore t 0, a result that is difficult, if not impossible, to accept biologically.However, there is a biological explanation for negative lag times because theysimply indicate that instantaneous solubilization has occurred, which equalsD0 Di. However, both solubilization and lag can occur when initial solubiliza-tion is greater than that indicated by the difference betweenD0 andDi, but theireffects cannot be separated unless IR0 is measured at time zero so that D0 canbe estimated. Setting bounds on discrete lag to prevent it from being less thanzero is not appropriate because it eliminates the possibility for detecting solu-bilization and can result in biased estimates of kinetic parameters.

30 D.R. Mertens

Model 4: Sequential first-order reaction for lag and digestion with an indigestiblefraction

Other models of digestion have been proposed that describe digestion as asequential compartmental process (Allen and Mertens, 1988; Mertens, 1990;Van Milgen et al., 1991). In these models, the digestive process is described bya two-step mechanism (Fig. 2.5). In the first stage, lag is modelled as a first-order process involving the change in the substrate from an unavailable form toone that is available for digestion. Biologically, this step could represent hydra-tion of substrate, removal of digestion inhibitors, or attachment or close asso-ciation of microorganisms with the substrate. The second stage is also first-order and represents actual degradation of the substrate. This model exhibits asmooth curvilinear transition from no digestion at t 0 to maximum absolutedigestion rate at the inflection point of the digestion curve. Differential equa-tions for this model are:

dU=dt klU (2:28)dA=dt klU kdA (2:29)

dI=dt 0 (2:30)

where U is the unavailable potentially digestible pool, A is the potentiallydigestible pool that is available for digestion, I is the indigestible residue, kl isthe fractional rate constant for lag and kd is the fractional rate constant fordigestion.

The integral equations for this digestive process are:

U(t) U0 exp (klt) (2:31)A(t) U0[kl=(kd kl)][ exp (klt) exp (kdt)] (2:32)

A

U = unavailable potentially digestible fractionk l = fractional rate of availability (lag phenomena)A = available potentially digestible fractionkd = fractional rate of digestionI = indigestible fraction

U . k l A . kdU

0I

Digestedsink

Fig. 2.5. Model 4: Sequential multi-compartmental model of digestion and lag with anindigestible fraction.

Rate and Extent of Digestion 31

I(t) I0 (2:33)R(t) U(t)A(t) I(t) (2:34)

R0 U0 I0 (2:35)

because A0 0 at t 0

Given

R(t) U(t) A(t) I0 (2:36)

at t > 0:

R(t) [U0=(kd kl)][kd exp (klt) kl exp (kdt)] I0 (2:37)

Although this model does not contain a discrete lag, Mertens (1990) observedthat a discrete lag term was a necessary addition to the model for it toadequately describe digestion processes with prolonged lag effects.

Model 5: Second-order digestion based on substrate and enzyme concentrations

Previous models assume that rate and extent of digestion are limited only byintrinsic properties of the substrate. However, it may be possible that extrinsicfactors, such as microbial mass or enzymatic activity, limit the rate of reaction(France et al., 1982; Baldwin et al., 1987). A more complex model used todescribe digestion is based on the HenriMichaelisMenten (HMM) kineticsdeveloped for enzyme reactions. The complete model of HMM kinetics is areversible, four-compartment system with both first- and second-order reac-tions (see p. 20 in Segel, 1975). Using quasi-steady-state approximation, theseries of differential equations used to describe the complete system can besolved as a function of substrate concentration (Segel, 1975). If we assume thatmicrobial mass acts like an enzyme and the substrate is potentially digestiblefibre, the final differential equations are:

dD=dt [Vmax=(Km D)]D (2:38)dI=dt 0 (2:39)

where Vmax is the maximal rate of reaction when all microbial mass is activelydigesting substrate, Km is proportional to the rates of degradation (kmd) andformation (kf) of the active complex, i.e. (kf kmd)=kf, and other variables asdefined previously.

This model assumes that microbial mass can limit digestion instead ofassuming, as in all previous models, that only intrinsic properties of the sub-strate limit digestion. In the HMMmodel, the fractional rate of digestion relativeto the amount of potentially digestible fibre is not a constant, but is proportionalto the total amount of microbial ma