Journal of Theoretical Biology 228 (2004) 271–289

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*Correspond

fax: +519-836-

E-mail addr

0022-5193/$ - se

doi:10.1016/j.jtb

An integrative model of amino acid metabolism in the liverof the lactating dairy cow

M.D. Hanigana, L.A. Cromptonb, C.K. Reynoldsc, D. Wray-Cahend,M.A. Lomaxe, J. Francef,*

aPurina Mills LLC, P.O. Box 66812, St. Louis, MO 63166-6812, USAbSchool of Agriculture, Policy & Development, The University of Reading, Whiteknights, P.O. Box 237, Reading, Berkshire RG6 6AR, UK

cDepartment of Animal Sciences, Ohio State University, OARDC, 1680 Madison Avenue, Wooster, OH 44691-4076, USAdCenter for Devices and Radiological Health, Food and Drug Administration, OST DLS/HSB, 8401 Muirkirk Road, Laurel, MD 20708, USA

eDepartment of Agricultural Sciences, Imperial College London, Wye Campus, Ashford, Kent TN25 5AH, UKfDepartment of Animal & Poultry Science, University of Guelph, 50 Gordon Street, Guelph, Ont., Canada N1G 2W1

Received 2 January 2003; received in revised form 13 January 2004; accepted 16 January 2004

Abstract

The objective of this work was to construct a dynamic model of hepatic amino acid metabolism in the lactating dairy cow that

could be parameterized using net flow data from in vivo experiments. The model considers 22 amino acids, ammonia, urea, and 13

energetic metabolites, and was parameterized using a steady-state balance model and two in vivo, net flow experiments conducted

with mid-lactation dairy cows. Extracellular flows were derived directly from the observed data. An optimization routine was used

to derive nine intracellular flows. The resulting dynamic model was found to be stable across a range of inputs suggesting that it can

be perturbed and applied to other physiological states. Although nitrogen was generally in balance, leucine was in slight deficit

compared to predicted needs for export protein synthesis, suggesting that an alternative source of leucine (e.g. peptides) was utilized.

Simulations of varying glucagon concentrations indicated that an additional 5mol/d of glucose could be synthesized at the reference

substrate concentrations and blood flows. The increased glucose production was supported by increased removal from blood of

lactate, glutamate, aspartate, alanine, asparagine, and glutamine. As glucose output increased, ketone body and acetate release

increased while CO2 release declined. The pattern of amino acids appearing in hepatic vein blood was affected by changes in amino

acid concentration in portal vein blood, portal blood flow rate and glucagon concentration, with methionine and phenylalanine

being the most affected of essential amino acids. Experimental evidence is insufficient to determine whether essential amino acids are

affected by varying gluconeogenic demands.

r 2004 Published by Elsevier Ltd.

Keywords: Model; Liver; Metabolism; Amino acid; Dairy cow; Lactation

1. Introduction

Ruminants, including lactating dairy cattle, convertdietary energy to energy in products (meat and milk)with greater efficiency than they convert dietary nitro-gen to productive nitrogen (Bequette et al., 2003; NRC,2001). This low nitrogen conversion efficiency contri-butes to the environmental challenges associated withdairy production systems (Howarth et al., 2002;Tamminga, 1996). A better quantitative understandingof ruminant nitrogen metabolism in general and amino

ing author. Tel.: +519-824-4120 x52209;

9873.

ess: jfrance@uoguelph.ca (J. France).

e front matter r 2004 Published by Elsevier Ltd.

i.2004.01.010

acid (AA) metabolism in particular may help identifystrategies to reduce nitrogen intake while maintainingproduction levels, thereby reducing nitrogen excretionand improving the efficiency of utilization. Significantefforts have been devoted to defining relationshipsbetween AA supply to the mammary glands andremoval and use for milk production (Bequette et al.,1998; France et al., 1995; Hanigan et al., 2001b; Maaset al., 1998). Both endocrine regulation of mammarymetabolism (Bequette et al., 2001; McGuire et al., 1995)and AA supply to the udder (Bequette et al., 1996, 2000;Hanigan et al., 1998, 2000) affect removal and use formilk protein synthesis. While increased total AA supplyto the udder does not always result in increased milkprotein, reductions in supply of one or more AA can

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Extracellular Compartment

Intracellular

Compartment

Py As

nCd

Ak

Oa

nGlu

nAm

nCit

nArg

nAsp

+

-

eTrp eBueFa eLeueIle

eKb

eLys

eTyr

ePhe

eAc

eGly

eCd

eGlu

eHis

ePro

eGln

eAm

eCit

eUr

Amino

AcidsxPrt

eOrn

eArg

eIle

eVal

eThr

eMet

eAsn

eAsp

ePr

eGl

eAla

eLa

eVa

eCys eSer

Hepatic Artery

Blood

Portal Vein

Blood

eGy

eTg

Gluc +

Ins -

+

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289272

result in a loss of milk protein production (Bequette et al.,2000). Thus mammary utilization of AA and subsequentproduction of milk protein are determined, energyavailability not withstanding, by the balance of AAarriving at the mammary glands (Hanigan et al., 2000).Although apparent hepatic affinities for AA are

relatively low (Hanigan et al., 1998), the combinationof large hepatic blood flow rates and the passage of allabsorbed AA through the liver via the portal vein resultsin clearance of significant quantities (around two-thirds)of AA from circulation on a net daily basis (Reynoldset al., 1988b; Wray-Cahen et al., 1997). However,removal of individual AA differs (Hanigan et al.,1998) resulting in significant alterations in the patternof AA in peripheral circulation as compared to thepattern absorbed from the gastro-intestinal tract (Wray-Cahen et al., 1997). Therefore, if the pattern andquantity of AA presented to the udder is to be predicted,models of hepatic AA metabolism are needed.Models of liver metabolism have been constructed

previously. The models of Waghorn (1982) and Freetlyet al. (1993) were built primarily to describe energytransactions within the liver. As such, descriptions ofhepatic AA metabolism were highly aggregated. Ad-ditionally, the paucity of data on hepatic AA metabo-lism by the dairy cow prevented a less aggregateddescription. The model of Danfaer (1990) was con-structed as a submodel of a whole animal model; thus isrelatively aggregated in form with no consideration ofindividual AA flow. A single AA model was constructedby France et al. (1999) to interpret isotopic leucine data.Since the construction of the earlier models, addi-

tional in vivo hepatic AA data from non-lactating (Bachet al., 2000; Wray-Cahen et al., 1997) and lactating(Caton et al., 2001; Reynolds et al., 1995, 1999, 2000,2001; and data presented herein) dairy cows havebecome available . An objective of the present workwas to construct a model of hepatic AA metabolism thatcould be defined by and used to interpret trans-hepaticobservations. Therefore, the effort includes a lessdetailed description of energy metabolism. The latterwas required to represent catabolic exchange of nitrogenand the glucogenic use of AA. Nitrogenous entitiesother than ammonia, urea, and AA were also notconsidered due to their minimal inpact on nitrogenbalance. The effort herein was intended to complement,not replicate, efforts by Waghorn (1982), Freetly et al.(1993), and France et al. (1999).

Hepatic Vein Blood

Fig. 1. Flow diagram of the model of liver metabolism in the dairy

cow, illustrating the primary metabolic flows. Dashed lines represent

compartment boundaries and boxes enclosed by solid lines represent

pools. Solid arrows represent flows and dashed arrows regulatory

effects. Double arrowheads denote bidirectional flows.

2. Model description

2.1. General considerations

The model represents the liver of a cow with emphasison predicting net removal from or appearance of

metabolites in blood perfusing the organ. The organwas assumed to be constant in size with intracellular andextracellular compartments of fixed volume, and thus norepresentation of protein turnover was included. Themodel is based on a set of dynamic differential equationscoded in Advanced Continuous Simulation Language(ACSL; Aegis Technologies Group, Inc., Huntsville,AL) and contains 14 state variables. Units for mass,volume, concentration, and time were moles, litres,moles/litre, and days, respectively, unless specifiedotherwise.An overview of the model is depicted in Fig. 1. Pools

are depicted as boxes with solid lines, compartmentboundaries by boxes with dashed lines, and flows asarrows. Double arrowheads denote bidirectional flows.Abbreviations used in Fig. 1 and throughout the paperare given in Table 1. Three letter codes denotingAA were assigned according to previous convention(ACS, 1997). Abbreviations for non-AA metabolites aresimilar to those used by Freetly et al. (1993). For

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Table 1

Abbreviations used in the model

Abbreviation Designation Abbreviation Designation

AA Amino Acid La Lactate

Ac Acetate Lys Lysine

Ad ADP Kb Ketone body

Ak a-ketoglutarate Met Methionine

Ala Alanine Nd NAD+

Am Ammonia Ndh NADH

Arg Arginine O2 Oxygen

As Acetyl-CoA Oa Oxaloacetate

Asn Asparagine Orn Ornithine

Asp Aspartate Ox Oxidation

At ATP Phe Phenylalanine

Bu Butyrate Pr Propionate

Cd CO2 Pro Proline

Cit Citrulline Py Pyruvate

Cys Cysteine Ser Serine

Fa Fatty Acids Tg Triacylglycerol

Fdh FADH Thr Threonine

Gl Glucose TNd NAD+ and NADH

Gln Glutamine Trp Tryptophan

Glu Glutamate Tyr Tyrosine

Gluc Glucagon Und Undefined Use

Gly Glycine Ur Urea

Gy Glycerol Val Valine

His Histidine Va Valerate and isobutyrate

Ile Isoleucine Vol Volume

Ins Insulin xPrt Export Protein

Leu Leucine

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 273

metabolites present in more than one compartment, apreceding lower case a, p, e, or n was used to designatearterial, portal, extracellular or intracellular entities,respectively. The extracellular compartment was as-sumed to comprise vascular and interstitial spaces, i.e.that space not bounded by extracellular membrane. Theintracellular compartment was assumed to comprise allspace within the organ that was bounded by extra-cellular membrane. This would include all cell typespresent in the organ. Potential heterogeneity of meta-bolism due to intracellular compartmentalization wasnot considered.Inputs to the model are arterial and portal venous

blood and outflow is hepatic venous blood. Themetabolites in blood considered include the 20 primaryAA plus ornithine (eOrn), citrulline (eCit), ammonia(eAm), urea (eUr), glucose (eGl), lactate (eLa), propio-nate (ePr), butyrate (eBu), acetate (eAc), valerate (eVa;valerate plus isobutyrate and isovalerate), ketone bodies(eKb; b-hydroxybutyrate plus acetoacetate plus acet-one), non-esterified fatty acids (eFa), triacylglycerol(eTg), glycerol (eGy), oxygen (O2), and CO2 (eCd).Some simplification of the diagram was required duesolely to complexities associated with flows for somepools. Pools not illustrated in Fig. 1 include NADH(Ndh), NAD+ (Nd), ATP (At), and ADP (Ad). Thesepools are described fully in the text.

2.2. Basic equations

Model flows are denoted in general terms as UA,B

(mol/d) where the subscript A, B refers to substrate andproduct, respectively. Equation forms used were pri-marily mass-action type. The general form was:

UA;B ¼ kACA

CC

KC

CD

KD

� �fD JE

CE

; ð1Þ

where kA represents the primary rate parameter forconversion of A to B (l/d), and CA represents thetransient concentration of A (mol/l), CC and CD

represent transient concentrations (mol/l) of secondarysubstrates or activators C and D, respectively, KC andKD represent the constants for the secondary substrates(mol/l), CE represents the transient concentration of aninhibitor (mol/l), and JE represents the constant for theinhibitor (mol/l). In all cases, KC, KD, and JE were setequal to reference concentrations of C, D, and E,respectively, yielding ratios of 1 in the reference state(described below). The effects of secondary substrateswere represented as a ratio to their reference concentra-tions so that the primary rate constants retained unitsand parameter values pertinent to the primary substrate.Additionally, consideration as a ratio whose value is 1 inthe reference state facilitates application of a sensitivityexponent fD. When applied in this manner, fD willhave no effect in the reference state, and as effecterconcentrations move away from the reference state, thesensitivity to such changes can be altered withoutaffecting the value of kA. Additionally, fD can beconverted to a function representing an endocrine orregulatory signal where such regulation is required.Although only illustrated for activator D, this approachapplies to the other activators where designated.Differential equations describing pools were defined as

dQA

dt¼ UB;A þ?� UA;B �?; ð2Þ

where dQA/dt represents the change in metabolite Awith respect to time (mol/d). The quantity of metabolitepresent at any point in time (QA, mol) was determinedby numerical integration of Eq. (2) using an initial mass(iQA). Transient concentrations were defined as

CA ¼QA

Vol; ð3Þ

where Vol (l) represents the volume of the intracellularcompartment. iQA was calculated from the reference CA

(Table 2) by rearrangement of Eq. (3).

2.3. Blood–tissue exchange

Eq. (1) with no secondary substrates, activators orinhibitors included, i.e. its simple mass-action form, wasused to describe removal of metabolites from blood

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Table 2

Reference intracellular metabolite concentrations (mmol/l) from

Freetly et al. (1993)

Metabolite Concentration

Ad 0.236

Ak 0.124

Am 75.0

Arg 0.097

As 0.107

Asp 1.05

At 0.236

Cd 31.0

Cit 0.359

Glu 4.17

Nd 0.236

Ndh 0.236

Oa 0.544

Py 0.038

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289274

where CA represented extracellular concentrations (CeA)of each metabolite.As extracellular space is small relative to the volume

of blood flowing through the tissue per unit time(Reynolds et al., 1988b; Waghorn, 1982; Wray-Cahenet al., 1997) and exchange between capillary andinterstitial space is rapid (Goresky, 1980), a steady-stateexpression can be used as an approximation for theextracellular compartment provided predictions ofchanges in removal were not made for time frames lessthan 5–10 s (Hanigan et al., 1998, 2001a):

CeA ¼CaAFa þ CpAFp þ UB;A

kA þ Fa þ Fp

; ð4Þ

where Fa and Fp represent arterial and portal bloodflows (l/d), respectively, and UB,A represents any outflowfrom the tissue when bidirectional blood–tissue ex-change is considered explicitly or when net synthesis bythe liver occurs. In this representation, CeA is equivalentto hepatic vein concentrations (Hanigan et al., 1998).Bidirectional exchange was included for eAc, asparagine(eAsn), aspartate (eAsp), eCit, glutamine (eGln), gluta-mate (eGlu), eLa, and proline (ePro) (Casse et al., 1994;Freetly and Ferrell, 1998; Hanigan et al., 1998; Lomaxand Baird, 1983; Reynolds and Tyrrell, 1991; Wray-Cahen et al., 1997). Additionally, Brockman and Berg-man (1975) observed significant changes in alanine(eAla) net removal when glucagon was infused. Thismay occur as a result of pyruvate (Py) depletion due toinducement of pyruvate carboxylase (see below) causinga reduction in unidirectional Ala outflow from theorgan. Therefore, bidirectional exchange was includedfor eAla. Exchange of lactate is apparently driven byenergy status of the tissue (Freetly and Ferrell, 1998;Reynolds, 1995), and thus the secondary substrate Ndwas included. This secondary substrate was reflected in

Eq. (4) by replacing kA with the term kA�CNd=KNd in thedenominator.Although ornithine can be taken up by liver under

conditions of acute hyperammonemia (Milano andLobley, 2001), it is normally released and thus bidirec-tional flows were not considered. Gl, eCd, eUr, and eKbare also generally released on a net basis by the liver ofa lactating cow (Baird et al., 1974, 1975; Lomax andBaird, 1983; Reynolds et al., 1988a, b). For thesemetabolites, kA was set to zero thus allowing outflowequations to determine solely rates of output. Whencalculating the outflow of eGl, a stoichiometry of 0.5was applied to UOa,eGl to reflect the requirement of2moles of oxaloacetate (Oa) to synthesize 1mol of eGl.Outflow of eKb included contributions from UAs,eKb andUeVa,Oa (with stoichiometric coefficients of 0.5 and0.778, respectively) and from the incomplete oxidationof AA associated with UeLeu,As, UeLys,eKb, UePhe,Oa,UeTrp,Py, and UeTyr,Oa (stoichiometric coefficients of 1).

2.4. Intracellular metabolism

Explicit representation of intracellular metabolismwas restricted to the major metabolites and thoseinvolved in metabolic regulation. These pools includedacetyl-CoA (As), a-ketoglutarate (Ak), and Oa torepresent the tri-carboxylic acid (TCA) cycle; nAspand nGlu due to their integral roles in transaminationand shuttling of AA carbon skeletons to and from theTCA cycle; nAm, nAsp, nCit, and arginine (nArg) torepresent the Ur cycle; and Ndh, Nd, At and Ad torepresent the energy status of the tissue allowingregulation of key metabolic steps. Carbon dioxide wasalso represented. Glycolysis was represented by catabo-lism of various metabolites to Py and gluconeogenesisby conversion of Oa to eGl. The differential equationsdescribing intracellular pools are listed in Appendix A,and flow equations and associated parameters are listedin Table 3.Intracellular pools were not defined for most AA. To

accommodate use of AA for synthesis of extracellularprotein (xPrt) when defining transfers into intracellularpools, Eq. (1) was amended to reflect such use:

UA;B ¼ kACeA � UA;xPrt; ð5Þ

where UA,xPrt was defined as

UA;xPrt ¼ fAUAA;xPrt: ð6Þ

UAA,xPrt represents the flow of total AA to exportprotein and fA represents the molar fraction of eachrespective AA in liver export protein. UAA,xPrt wasassumed to be 0.71mol AA/d based on previous work(Freetly et al., 1993; Raggio et al., 2002). The molar AAproportions were adopted from Freetly et al. (1993) andare listed in Table 4. To ensure that export protein didnot exceed supply of essential AA (EAA), UAA,xPrt was

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Table 3

Flow equations and associated parameter values. Parameters and their estimates (see text for details) are ordered with respect to substrates. Rate

parameter notation follows the equation form listed

Flow Equation Substrates Parameters Parameter values (Units:

k ¼ l=d ;K ¼ mol=l; J ¼ mol=l)

Reference

flow (mol/d)

UAA,xPrt (7) eHis, eIle, eLeu, eLys,

eMet, ePhe, eThr, eTrp,

eVal

VAA,xPrt 0.713 (mol/d) 0.713

UAk,nGlu (1) Ak, nAm, Ndh kA, KC, KD 92,119, 7.50E�02, 2.36E�04 11.4

UAk,Oa (1) Ak, Ad, Nd kA, KC, KD 211,318, 2.36E�04, 2.36E�04 26.2

UAs,eAc (1) As kA 94,910 10.2

UAs,eKb (1) As kA 62,323 6.67

UAt,Und At kA 376,811 356

UeAc,As (1) eAc kA 781 2.0

UeAla,Py (5) eAla kA 34,758 7.80

UeAm,nAm (1) eAm kA 156,915 13.0

UeArg,nArg (1) eArg kA 5181 0.474

UeAsn,nAsp (5) eAsn kA 16,797 1.86

UeAsp,nAsp (1) eAsp kA 279,497 4.96

UeBu,As (1) eBu kA 71,012 3.78

UeCit,nCit (1) eCit kA 3731 1.29

UeCys,Py (5) eCys kA 1000 0.10

UeFa,As (1) eFa kA 4596 1.13

UeFa,eTg (1) eFa kA 1224 0.30

UeGln,nGlu (5) eGln kA 5707 0.815

UeGlu,nGlu (1) eGlu kA 17,570 1.80

UeGly,nCd (5) eGly kA 8976 2.69

UeGy,Py (1) eGy kA 5104 1.53

UeHis,nGlu (5) eHis kA 4456 0.174

UeIle,Oa (5) eIle kA 339 0.044

UeLa,Py (1) eLa, Nd kA, KC 34,084, 2.36E�04 10.2

UeLeu,As (5) eLeu kA 332 0.058

UeLys,eKb (5) eLys kA 2339 0.226

UeMet,Oa (5) eMet kA 7908 0.143

UeO2,H2O (29) 98.2

UePhe,Oa (5) ePhe kA 7413 0.439

UePr,Oa (1) ePr kA 153,989 18.0

UePro,nGlu (5) ePro kA 6797 0.655

UeSer,Py (5) eSer kA 16,390 1.40

UeTg,As (1) eTg kA 6281 0.595

UeThr,Oa (5) eThr kA 4126 0.554

UeTrp,Py (5) eTrp kA 3524 0.183

UeTyr,Oa (5) eTyr kA 8393 0.430

UeVa,As (1) eVa kA 163,397 3.84

UeVa,Oa (1) eVa kA 163,397 3.84

UeVal,Oa (5) eVal kA 787 0.188

UFdh,Ox (28) 69.7

UKb,eKb (30) 7.50

UnArg,eOrn (1) nArg kA 426 0.041

UnArg,nCit (1) nArg, nAm, nGlu, At kA, KC, KD,

KE

135,769, 7.50E�02, 4.17E�03,2.36E�04

13.2

UnArg,nGlu (1) nArg, Ak kA, KC 12,353, 1.24E�04 1.20

UnAsp,eAsn (1) nAsp kA 476 0.50

UnAsp,eAsp (1) nAsp kA 4762 5.0

UnAsp,Oa (1) nAsp, Ak kA, KC 5946, 1.24E�04 6.24

UnCd,eCd (1) nCd kA 1722 53.4

UnCit,eCit (1) nCit kA 1393 0.50

UnCit,nArg (1) nCit, nAsp kA, KC 38,892, 1.05E�03 14.0

UNdh,Ox (1) Ndh kA 517,752 122

UnGlu,Ak (1) nGlu, Nd, nAm, At kA, KC, JD,

JE

1199, 2.36E�04, 7.50E�02,2.36E�04

5.0

UnGlu,eGln (1) nGlu kA 120 0.50

UnGlu,eGlu (1) nGlu kA 719 3.0

UnGlu,ePro (1) nGlu kA 47.96 0.20

UnUr,eUr (31) 14.4

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 275

ARTICLE IN PRESS

Table 3 (continued)

Flow Equation Substrates Parameters Parameter values (Units:

k ¼ l=d ;K ¼ mol=l; J ¼ mol=l)

Reference

flow (mol/d)

UOa,Ak (1) Oa, Ad, Nd kA, KC, KD 46,813, 2.36E�04, 2.36E�04 25.5

UOa,eGl (1) Oa, Gluc, Ins kA, KC, JD 50,672, 1, 1 27.6

UOa,nAsp (1) Oa, nGlu kA, KC 9191, 4.17E�03 5.0

UPy,As (1) Py, Nd, As kA, KC, JD,

fD

169,759, 2.36E�04, 1.07E�04, 2 6.5

UPy,eAla (1) Py kA 131,579 5.0

UPy,eLa (1) Py, Ndh kA, KC 131,579, 2.36E�04, 5.0

UPy,Oa (1) Py, As, At, Gluc kA, KC, fC,

KD, KE

134,642, 1.07E�04, 2, 2.36E�04,1

5.12

Table 4

Amino acid composition (mol/mol total AA) of export protein as

derived by Freetly et al. (1993)

Amino acid Molar proportion Amino acid Molar proportion

Alanine 0.0396 Leucine 0.0773

Arginine 0.0379 Lysine 0.0618

Asparagine 0.0550 Methionine 0.0229

Aspartate 0.0587 Phenylalanine 0.0360

Cysteine 0.0209 Proline 0.1014

Glutamate 0.0827 Serine 0.0660

Glutamine 0.0827 Threonine 0.0397

Glycine 0.0293 Tryptophan 0.0070

Histidine 0.0220 Tyrosine 0.0383

Isoleucine 0.0512 Valine 0.0696

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289276

defined asUAA;xPrt ¼minimum

VAA;xPrt; #UA1;xPrt; #UA2;xPrt; ::: ; #UAn;xPrt

� �;

ð7Þ

where VAA,xPrt represents the maximal rate of conver-sion and #UAi ;xPrtthe potential rate of protein synthesisgiven the prevailing rate of removal:#UAi ;xPrt ¼ kAi

CAi: ð8Þ

A1 to An represents histidine (eHis), isoleucine (eIle),leucine (eLeu), lysine (eLys), methionine (eMet), pheny-lalanine (ePhe), threonine (eThr), tryptophan (eTrp),and valine (eVal), i.e. the primary EAA. In this manner,as removal of any single EAA approaches zero, the usefor export protein synthesis will also approach zero forall EAA ensuring that Eq. (5) does not assume negativevalues for those AA that cannot be synthesized in theliver. UA,B in Eq. (5) thus represents the catabolism ofAA not utilized for export protein synthesis and islimited to the difference between removal and use forprotein export.Metabolic regulation was represented at several key

points. The flow UnArg,nCit (Table 3) represents thecarbamoyl phosphate synthetase reaction with nAmserving as a substrate and nGlu and At as activators ofthe reaction (Newsholme and Leech, 1986). IntracellularGlu serves as a proxy for the actual regulator N-acetyl-glutamate, which was not represented. Exchange of

Oa and nAsp is coupled via transamination with theUAk,nGlu, hence the inclusion of these latter entities assubstrates.As Gl absorbed from the digestive tract in ruminants

is inadequate to meet demand (Reynolds et al., 1988b;Reynolds, 1995), significant quantities of Gl must besynthesized. Propionate is the primary substrate for Glsynthesis (Judson et al., 1968), and Gl output by theliver (Baird et al., 1980; Brockman and Bergman, 1975)and plasma Gl concentrations (Peters and Elliot, 1984)have been observed to respond to increased propionatesupply, although this is not always the case (Baird et al.,1980; Casse et al., 1994). However, as both insulin andglucagon release are responsive to propionate supply(Peters and Elliot, 1984; Sano et al., 1993), it is not clearwhether increased Gl production is strictly a substrateresponse or a combined substrate and endocrineresponse. The latter seems more likely as gluconeogen-esis is responsive to glucagon concentrations (Bobe et al.,2002; Brockman et al., 1975), while provision ofgluconeogenic precursors not associated with gluconeo-genic signals (insulin and glucagon), such as alanine, donot result in additional Gl production (Reynolds andTyrrell, 1991). Finally, infusion of Gl has been shown toresult in reductions in Gl output (Freetly and Klindt,1996). She et al. (1999) also observed inducement ofpyruvate carboxylase mRNA when glucagon wasinfused with no down-regulation as insulin concentra-tions increased. Based on these observations, provisionsfor the regulation of UOa,eGl by insulin and glucagon andUPy,Oa by glucagon were included (Table 3). Insulin andglucagon levels in plasma were treated as input variableswhose reference values are unity.As noted above, eLa removal can range from positive

to negative (Freetly and Ferrell, 1998). This range isdetermined largely by energy and carbon needs withinthe cell with Ndh status being the energy determinant(Newsholme and Leech, 1986; Reynolds, 1995). There-fore, the additional substrates Ndh and Nd wereconsidered.Pyruvate dehydrogenase kinase is inhibited by high

ratios of At/Ad, As/Co-A and Ndh/Nd, which enhancephosphorylation of the enzyme to an inactive state, thus

ARTICLE IN PRESSM.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 277

reducing the conversion of Py to As. It has also beenobserved that As acts as a feedback inhibitor on thisreaction. Regulation of pyruvate carboxylase appears tobe associated with concentrations of As and the At/Adratio within the cell, whereby an increase in the ratioactivates pyruvate carboxylase. Increased concentra-tions of As appear to activate pyruvate carboxylase(Newsholme and Leech, 1986). Therefore, the regulatoryeffects of As, At, and Nd on flows describing conversionof Py to As and Oa were included (Table 3). Suchregulation is critical in the model to maintain TCA cyclestability and At and Ndh supply. Similarly, Nd and Adwere required as secondary substrates in UOa,Ak and Ndin UAk,Oa to provide TCA cycle stability (Table 3)(Newsholme and Leech, 1986). Finally, the inclusion ofNdh, Nd, and At as regulatory elements and nAm as asubstrate in the Ak-nGlu exchanges (Table 3) werebased on the observation of Saradambal et al. (1981),where At substituted for the observed dependance uponGTP.

3. Model parameterization

The data used for model parameterization were takenfrom a series of experiments conducted as part of aproject carried out at The University of Reading and theRowett Research Institute [Experiment C7 (a briefdescription is provided in Appendix A)] and workreported by Reynolds et al. (1988a, b). The referencesteady state was derived from the mean values of thiscombined data set. Thus reference concentrations arethe mean observed concentrations. Reference bloodflows are the mean observed flows with adjustments toC7 flows as discussed below. Reference flows weredetermined using reference concentrations and bloodflows.

3.1. Blood–tissue exchange parameters

Estimates for net removal of metabolites from bloodwere derived from the reference data by fitting the modelof Hanigan et al (1998) [Eq. (4) with UB,A omitted] tothe arteriovenous uptake data using the NLIN proce-dure of SAS (1988). The results of this analysis showingderived rate constants and associated statistics arepresented in Table 5. Plasma values were used for AAconcentrations and adjusted to a whole blood basis asdescribed previously (Hanigan et al., 2001b), based onthe observations of Heitmann and Bergman (1980). Glwas assumed to be delivered only in the plasma portionof whole blood. Observations of cysteine were notavailable; thus removal was assumed to be 0.1mol/d.The Tg observations of Reynolds et al. (2003) were used,as comparable observations were not taken in either ofthe experiments described herein.

As it was assumed that eCd, eGl, eKb and eOrn wereonly released from the liver, kA in Eq. (4) was set to zero.Rate parameters (l/d) for the outflows from tissue toblood [UB,A in Eq. (4)] were derived from determinedmean flow values:

rate parameter ¼UB;A

iCB

; ð9Þ

where iCB represents the reference concentration of B.The equation used for eKb, obtained from Eq. (30) bysubstituting simple mass-action forms for UKb,eKb andUAs,eKb then re-arranging, was:

kAs;eKb ¼

�2keKbiCeKb � UeLeu;As � UeLys;eKb � UePhe;Oa

�UeTrp;Py � UeTyr;Oa � 0:778UeVa;Oa

!

iCAs

:

ð10Þ

Equation (10) reflects a correction for production ofacetoacetate or acetoacetyl-CoA through the incompleteoxidation of eLeu, eLys, ePhe, eTrp and eTyr.

3.2. Balance model

Having derived the estimates of net substrate removalfrom blood described in Section 3.1, a balance modelwas set up by assuming all the derivatives of thedynamic model were zero, i.e. the animals were insteady-state during the experiments. Initial inputs to thebalance model were obtained using the equations foruptake, associated parameter values (Section 3.1 andTable 5), and mean observed values for arterial andportal vein concentrations and blood flows (Table 6).A value of 0.71moles total AA/d was used for export

protein synthesis based on previous observations(Freetly et al., 1993; Raggio et al., 2002), however, theremoval of Leu by the liver was inadequate to supportthis rate of export protein synthesis. Therefore, anadditional flow of Leu from undefined sources (e.g.peptides) was added to support export protein synthesis.The rate of export protein synthesis in the absence of aknown limitation in EAA cannot be determined usingthe balance model necessitating specification of exportsynthesis. Given that specification, catabolism of eachAA can be uniquely identified allowing calculation ofrate constants.The model contained bidirectional exchanges that

could not be resolved numerically in the absence ofisotopic data. The assumed flow values for UeAc,As,UeFa,eTg, UnAsp,eAsn, UnAsp,eAsp, UnCit,eCit, UnGlu,Ak,UnGlu,eGln, UnGlu,eGlu, UnGlu,ePro, UOa,nAsp and UPy,eLa

and the associated reverse flows are given in Table 3.The remaining unknown values of flow in the balance

model were obtained using an optimization routinewhere the objective function was an unweighted sum ofsquares of the difference between predicted and ob-served uptakes or release of Tg, Gl, O2, CO2, Ur and

ARTICLE IN PRESS

Table 5

Rate constants (kA) and hepatic venous blood concentrations (CH) for the data of Wray-Cahen et al. (Experiment C7) and Reynolds et al. (1988a, b).

Rate constants were derived using the model of Hanigan et al. (1998)

Mean CH

Amino acid N kA (l/d) SE of kA P-value regression Calculated (mmol/l) Observed (mmol/l)

Alanine 9 12,467 1246 0.0001 226 226

Arginine 9 5181 921 0.0001 92 91

Asparagine 9 12,290 2253 0.0001 111 111

Aspartate 9 �2219 2578 0.0001 17.8 17.9

Citrulline 9 2288 922 0.0001 347 349

Glutamine 9 2204 1284 0.0001 143 144

Glutamate 9 �11,684 951 0.0001 102 102

Glycine 9 8976 1071 0.0001 303 300

Histidine 9 4456 665 0.0001 39.0 38.9

Isoleucine 9 339 976 0.0001 131 130

Leucine 9 �984 1013 0.0001 174 173

Lysine 9 2339 1398 0.0001 97.1 96.3

Methionine 9 7908 1705 0.0001 18.1 18.0

Ornithine 9 �880 904 0.0001 47.0 46.9

Phenylalanine 9 7413 1686 0.0001 59.5 59.2

Proline 9 4722 843 0.0001 102 102

Serine 9 16,390 1616 0.0001 86.4 86.0

Threonine 9 4126 844 0.0001 135 134

Tryptophan 9 3524 816 0.0001 51.9 51.8

Tyrosine 9 8393 1537 0.0001 51.4 51.0

Valine 9 787 619 0.0001 239 238

Other metabolite: mmol/l mmol/l

Acetate 7 �3186 733 0.0001 2.55 2.58

Ammonia 9 159,031 28275 0.0001 0.086 0.086

BHBA �7175 829 0.0001 1.07 1.08

Butyrate 7 71,012 9106 0.0001 0.052 0.052

CO2 7 �2204 242 0.0001 17.5 16.8

Glucose 7 �5588 724 0.0001 2.76 2.79

Lactate 7 17,303 5123 0.0001 0.280 0.269

NEFA 7 5820 1086 0.0001 0.240 0.241

O2 7 34,217 4059 0.0001 2.17 2.17

Propionate 7 153,989 18525 0.0001 0.110 0.109

Urea 9 �6338 626 0.0001 2.16 2.18

Valerate+Isobutyrate 7 163,397 46529 0.0038 0.020 0.019

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289278

Am (see Appendix A). These last five metabolites wereincluded in the objective function because there was notan original solution to the balance model that providedappropriate quantities of precursors or products tomatch predicted inputs and outputs to the observedvalues for these metabolites. In particular, there was asignificant deficit of carbon as seen in previous work(Freetly et al., 1993). Given the magnitude of the deficit,it was thought that Tg uptake was the most likelycontributor, therefore, a Tg flow was included in theobjective function to ensure carbon balance across theliver.Although the data appeared to be of high quality

overall, it seemed likely that the split between arterialand portal supplies for Experiment C7 was notconsistent with previous observations. In particular,arterial flow represented over 29% of the total hepaticflow where values in the range of 10–20% are more

appropriate for lactating animals (Reynolds, 1995).Therefore, the balance model was resolved using theoptimization routine with arterial proportions rangingfrom 5% to 29%. Results are presented in Table 7.Decreasing flow from 29% to 5% of total hepatic flowresulted in an increase in the residual error, however, thedependence on an undefined source of carbon asindicated by UeTg,As was reduced. Based on previousobservations (Reynolds, 1995) and the results of thesensitivity analyses, a reference value for arterial flow of20% of total hepatic flow was adopted for observationsin Experiment C7. Using these revised blood flowestimates, the model was rebalanced and the revisedestimates of metabolite flow were used for subsequentanalysis. Finally, the derived values of metabolite flowwere used to determine rate parameters using respectiveextracellular or intracellular concentrations. Thus, thefinal derived parameter set for substrate removal from

ARTICLE IN PRESS

Table 6

Reference values for steady state arterial and portal venous blood concentrations for the data of Wray-Cahen et al. (Experiment C7) and Reynolds

et al. (1988a, b). Values are the mean observations (SE)

Amino acid N Arterial concentration (mmol/l) Portal venous concentration (mmol/l)

Alanine 9 228 (11) 282 (12)

Arginine 9 85 (7) 103 (9)

Asparagine 9 89 (14) 145 (14)

Aspartate 9 17 (1) 17 (1)

Citrulline 9 339 (26) 365 (29)

Glutamine 9 144 (7) 149 (7)

Glutamate 9 82 (5) 83 (3)

Glycine 9 322 (37) 352 (37)

Histidine 9 37 (5) 43 (6)

Isoleucine 9 117 (8) 135 (9)

Leucine 9 149 (9) 179 (10)

Lysine 9 83 (6) 105 (7)

Methionine 9 15 (1) 22 (1)

Ornithine 9 48 (4) 47 (4)

Phenylalanine 9 52 (2) 70 (4)

Proline 9 101 (8) 112 (9)

Serine 9 89 (5) 114 (5)

Threonine 9 125 (9) 148 (10)

Tryptophan 9 52 (2) 56 (2)

Tyrosine 9 47 (3) 61 (4)

Valine 9 225 (12) 246 (14)

Other metabolite: mmol/l mmol/l

Acetate 7 1.54 (0.07) 2.64 (0.11)

Ammonia 9 0.099 (0.012) 0.355 (0.024)

BHBA 7 0.79 (0.19) 0.95 (0.20)

Butyrate 7 0.030 (0.004) 0.137 (0.015)

CO2 7 16.2 (0.8) 16.8 (0.9)

Glucose 7 2.47 (0.14) 2.43 (0.13)

Glycerola 0.0041 0.0049

Lactate 7 0.274 (0.032) 0.412 (0.036)

NEFA 7 0.257 (0.028) 0.272 (0.031)

O2 7 4.83 (0.12) 3.33 (0.09)

Propionate 7 0.090 (0.015) 0.501 (0.022)

Triacylglycerola 0.0094 0.0097

Urea 9 2.14 (0.25) 1.89 (0.24)

Valerate+Isobutyrate 7 0.015 (0.004) 0.106 (0.007)

Blood flow (l/d) 16 12,503 (6 1 7) 47,134 (1880)

aFrom Reynolds et al. (2003).

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 279

blood deviates from the initial estimates presented inTable 5. Flow and parameter values are summarized inTable 3.

4. Model application

Having formulated and parameterized the dynamicmodel, the time constants were found to vary from7.2� 10�6 d to 0.03 d, indicating that the model was stiffand that the maximum integration interval with a fixedstep integration algorithm should be approximately7� 10�6 d. Monitoring the step size chosen when usingthe Adams–Moulton variable step algorithm and amaximum integration interval of 0.005 d, indicated thata step size as small as 2� 10�6d was often chosen by the

algorithm (Fig. 2). Using a fixed-step integrationalgorithm with a maximum integration interval of7� 10�6 d, a 1 day simulation required 4.5 s to completeon an AMD 700MHz processor. Using a variable stepalgorithm such as the Runge–Kutta–Fehlberg fifth-order method, marginally reduced the time required to3.1 s. Choosing Gear’s stiff algorithm resulted in thesame run being completed in 0.3 s when the maximumstep size was set to 1� 10�4 d supporting the timeconstant observations with respect to model stiffness.These latter settings were used for all subsequent workunless specified otherwise.Model stability was examined by perturbing inputs

tenfold in the positive and negative directions andobserving pool sizes of the state variables. Exampleresults from these analyses are presented in Fig. 3.

ARTICLE IN PRESS

Table 7

Flow values and adjustments required to maintain model balance as arterial blood flow was changed from the observed 29% of total hepatic flow to

lesser values. As arterial flow was changed offsetting adjustments were made in portal flow to maintain total hepatic flow equal to the observed value

of 59,637 l/d

29% Arterial 25% Arterial 20% Arterial 15% Arterial 10% Arterial 5% Arterial

Flow estimates, (mol/d)

UAk,nGlu 11.0 11.2 11.4 11.7 11.9 12.2

UAk,Oa 27.4 26.9 26.2 25.5 24.8 24.0

UeAm,nAm 12.2 12.5 13.0 13.5 14.0 14.5

UeTgAs 0.78 0.71 0.59 0.48 0.37 0.25

UnArg,nCit 12.6 12.8 13.2 13.5 13.8 14.2

UnArg,nGlu 1.19 1.19 1.20 1.21 1.21 1.22

UnAsp,Oa 6.20 6.21 6.24 6.27 6.30 6.33

UnCd,eCd 54.0 53.7 53.4 53.0 52.7 52.3

UnCit,nArg 13.4 13.6 14.0 14.3 14.6 15.0

UnUr,eUr 13.9 14.1 14.4 14.7 15.1 15.4

UOa,Ak 26.7 26.2 25.5 24.7 24.0 23.3

UOa,eGl 28.3 28.0 27.6 27.1 26.7 26.3

UPy,As 4.07 5.00 6.45 7.90 9.35 10.8

UPy,Oa 7.44 6.53 5.12 3.70 2.29 0.87

Flow adjustments, % of observed flow

UeAm,nAm �0.84 �1.05 �1.35 �1.63 �1.88 �2.12UeO2,H2O 1.45 1.68 2.04 2.41 2.80 3.20

UeTg,As 84.6 83.0 79.7 74.9 67.0 52.0

UnCd,eCd �3.88 �4.45 �5.35 �6.26 �7.18 �8.11UnUr,eUr �0.33 1.61 4.52 7.30 9.96 12.5

UOa,eGl 6.69 7.69 9.29 10.9 12.6 14.4

Residual error (mol/d)2

Objective function 10.66 13.73 19.68 27.03 35.78 45.94

0.0E+00

4.0E-06

8.0E-06

1.2E-05

1.6E-05

2.0E-05

0 0.5 1 1.5 2

Time, d

Inte

grat

ion

Inte

rval

, d

Fig. 2. Integration interval chosen by the Adams–Moulton algorithm

versus time for the model when using a maximum step size of 0.005 d.

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289280

Regardless of whether inputs were perturbed in apositive or a negative direction, the model achieved anew steady state within 0.05 d, and the new state wasmaintained thereafter (Figs. 3a–d) leading to theconclusion that the model is stable over a range ofinputs. When inputs were returned to their referencevalues after a perturbation, the model flows returned totheir reference values in the same time period.Sensitivity analysis to varying glucagon concentra-

tions are summarized in Fig. 4. Gl output ranged from9.25 to 19mol/d as the reference concentration ofglucagon was varied from 0.25 to 4. Carbon requiredto support Gl synthesis was derived primarily from

increased rates of La removal and decreased rates ofAsp and Glu release with lesser contributions from Ala,Asn, Pro, and Gln. It was surprising that Alacontributed so much less than La given the commonintermediate, Py. This occurred as a result of a greaterthan expected removal of La, due to a decrease in Ndhconcentration as glucagon concentrations increased.Release of Kb (Fig. 4a) and Ac (data not shown)increased as glucagon concentration increased due to adecrease in Oa and an increase in As. The increasedrates of carbon release in the form of Ac and Kb, causeda decrease in O2 removal and CO2 release (data notshown).Release of Ur was increased in response to changes in

deamination rates of AA used for gluconeogenesis asglucagon was increased. As hormonal regulation ofEAA flows was not included in the model, removal ofthese AA in support of Gl synthesis was not observed.Consequently, the pattern of AA released in the hepaticvein would deviate from portal and arterial concentra-tions, but those deviations would result from changes innon-essential AA (NEAA) (Fig. 5) and other metaboliteconcentrations. Hepatic release of total AA declined dueto the increased removal of AA in support of Glsynthesis.The effects of glucagon and portal blood flow on

predicted patterns of AA in the hepatic vein are

ARTICLE IN PRESS

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

0 0.02 0.04 0.06 0.08 0.1

Time (d)

Poo

l Siz

e (m

ol)

OaAspGlu

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

0 0.2 0.4 0.6 0.8 1

Time (d)

Poo

l Siz

e (m

ol)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Asp

and

Glu

(m

ol)

AkOaNdhGluAsp

1.0E-04

6.0E-04

1.1E-03

1.6E-03

2.1E-03

2.6E-03

3.1E-03

3.6E-03

0 0.2 0.4 0.6 0.8 1

Time (d)

Poo

l Siz

e (m

ol)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Glu

(m

ol) Py

AkNdhAtGlu

0.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-031.4E-031.6E-031.8E-032.0E-03

0 0.2 0.4 0.6 0.8 1

Time (d)A

rg a

nd A

k (m

ol)

00.10.20.30.40.50.60.70.80.9

Poo

l Siz

e (m

ol)

AkArgGluAspAm

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

0 0.02 0.04 0.06 0.08 0.1

Time (d)

Poo

l Siz

e (m

ol)

OaAspGlu

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

0 0.2 0.4 0.6 0.8 1

Time (d)

Poo

l Siz

e (m

ol)

0

0.04

0.06

0.08

0.1

0.12

0.14

Asp

and

Glu

(m

ol)

-

-

.2 . 0.6 .

Time (d)

Poo

l Siz

e (m

ol)

0.06

0.

PyAkNdhAtGlu

0.0E+002.0E-044.0E-046.0E-048.0E-041.0E-031.2E-031.4E-031.6E-031.8E-032.0E-03

0 0.2 0.4 0.6 0.8 1

Time (d)A

rg a

nd A

k (m

ol)

0

oo

(a) (b)

(d)(c)

Fig. 3. Model responses to a 10-fold increase (a) and decrease (b) in portal propionate concentration, a 10-fold increase in portal blood flow (c), and

a 10-fold increase in portal glutamate concentration (d). Changes were relative to the reference state as defined in the model parameterization section.

Abbreviations used are defined in Table 1.

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 281

presented in Fig. 5. As noted above, EAA were notgenerally responsive to glucagon changes, however theyare responsive to changes in portal flow. The predictedpattern of EAA changes, whereby Met and Phe areaffected to a greater degree than are Ile, Leu, and Valwith intermediate changes in His, Thr, Arg, and Lys.Hepatic vein concentrations were predicted to changemore dramatically for NEAA and these changes wereresponsive to both glucagon and portal flow. Relativelylarge changes were observed in Asp and Glu concentra-tions with very marginal responses for Cys and Gln.The sensitivity of hepatic vein release and hepatic

tissue outflows to independent variations in portalconcentrations of metabolites was evaluated, and theresults are presented in Table 8. Most outflows were notvery responsive to changes in portal concentrations.However, those AA involved in transammination werechanged significantly with changes in other nitrogendonors, i.e. Asp and Glu responded to changes in Ala,and also responded to changes in Cit, La, Tg and Pr.Similarly, metabolites sharing common intermediateswere affected by changes in either donor, i.e. La andAla release each responded to changes in the othermetabolite. Gl was most responsive to Pr and Tgconcentration changes, with little response to othermetabolites. However, this response was small relativeto that elicited by changes in glucagon. Changes in Citconcentration decreased the release of Asn, Asp, Gln,Glu, Pro, Ur and La.

5. Discussion

The work described herein provides an integrative andquantitative representation of AA metabolism in theliver of a lactating cow. As the data used to parameterizethe model were derived from mid-lactation cows, itseems prudent to restrict use of the dynamic model tothe same physiological state. However, the model wasfound to be stable over a broad range of inputs,suggesting that it could be applied to cattle in similarphysiological states after appropriate evaluation.The large deficit in carbon (about 13mol/d), also

observed previously by Freetly et al. (1993), is troublingas it represents either an unknown substrate or isindicative of a deficiency in the measurements com-monly made. It seems unlikely that it could be caused bybias in any of the measurements currently undertaken.For example, it represents almost a third of the observedcarbon outflow in Gl and a quarter of the observed CO2

release. Such large errors in measurement would havebeen identified. It appears more likely that the deficit isdue to failure to measure all the inputs to the organ.Freetly et al. (1993) hypothesized that the deficit incarbon was due to the absence of Tg flow measurements.Hepatic Tg flows have been measured by Reynolds et al.(2003), and those values were used herein. AlthoughReynolds et al. (2003) observed a positive removal ofTg, the flow was not adequate to cover the apparentdeficit.

ARTICLE IN PRESS

Fig. 5. Responses in hepatic vein non-essential (a) and essential (b)

amino acid concentrations to changes in glucagon concentration

(Gluc) or portal blood flow (PorBF). Values are expressed relative to

concentrations in the reference state.

-20

-15

-10

-5

0

5

10

0 0.5 1 1.5 2 2.5 3 3.5 4

Hep

atic

Rem

oval

(m

ol/d

)

Ala

Gl

Glu

Kb

La

Ur

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Hep

atic

Rem

oval

(m

ol/d

)

Asn

Asp

Cit

Gln

Orn

Pro

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

Rel

ativ

e C

once

ntra

tion

Arg

As

At

Ndh

Oa

Py

Relative Glucagon Concentration

(a)

(b)

(c)

Fig. 4. Model responses (a–c) to varying glucagon concentrations.

Glucagon and intracellular concentrations are expressed relative to the

reference state as defined in the model parameterization section.

Abbreviations used are defined in Table 1.

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289282

Hepatic nitrogen balance would be affected if aportion of the carbon deficit comprises nitrogen contain-ing compounds such as peptides (Koeln et al., 1993).Given that there is slightly more nitrogen being removedby liver than could be accounted for in Ur output(Table 7), additional nitrogen removal associated withuptake and catabolism of peptides seems unlikely unlesscatabolized nitrogen is being lost from the liver in formsother than those considered herein. The branched chainAA may be an exception to this conclusion, as theirremoval is nearly zero and can be negative (Reynoldset al., 1995, 1999, 2000). Removal of the correspondingketo-acids is not adequate to provide the missingcarbon, suggesting that peptides may represent thedeficit.It is interesting that a large amount of carbon leaves

the liver in the form of Ac and Kb as opposed to CO2,resulting in unusual respiratory quotients (RQ; definedas CO2/O2). The normal RQ range for an animal is from0.7 to 1.2 (Kleiber, 1975). The RQ derived for the dataused herein was 0.56. However, it was reduced to 0.42when glucagon concentrations were increased.

Gl synthesis was responsive to hormonal regulation,as is observed in vivo, with the potential for increasingGl production by at least 5mol/d when arterial andportal inputs were held constant. Changes in substratesupply in the absence of a change in hormonal signalselicited very little change in Gl production (Table 8).In response to a glucagon signal, La was the largestcontributor to increases in gluconeogenesis. As most Lais derived from Gl oxidation in peripheral tissues, it isnot clear if a 5mol/d increase is sustainable when thedriving factor behind the glucagon signal is anirreversible loss of Gl in products such as lactose, i.e.La concentrations in blood may not remain constant.However, Bobe et al. (2002) observed sustained in-creases in blood Gl concentration during 14 d infusionsof glucagon in lactating cows.The predicted decline in Ndh concentrations asso-

ciated with increasing glucagon is not supported byprevious observations (Ayuso et al., 1986). The pre-dicted reduction appears to be due to a decline in Oaconcentrations (Fig. 4), which in turn leads to a decreasein Ndh generation from the tri-carboxylic acid cycle.This inconsistency with observed data suggests that thedescription of glucogenic carbon entry into the tri-carboxylic acid cycle was not adequate. Regulation ofpyruvate carboxylase (UPy,As) by As was included in themodel as represented by fC, however, the available datawere not adequate to define fC accurately and thereforean assumed value of two was adopted. Increasing the

ARTICLE IN PRESS

Table 8

Model responses to increases in portal concentrations of various metabolites. The concentration of each metabolite was varied independently. Values

are standardized to a 10% change in total hepatic input (arterial plus portal) to allow comparisons across metabolites

Hepatic release Changes in hepatic outflows relative to the reference flow %

Input %a Ala Asn Asp Gln Glu Pro Ur CO2 O2 Gl La Kb Ac

Ac 98.99 0.01 0.00 0.08 0.01 0.05 0.00 0.00 0.01 �0.03 0.01 0.02 0.03 9.37

Ala 69.75 8.42 0.14 1.74 0.16 1.08 0.09 0.19 �0.02 �0.13 0.04 0.76 0.03 0.02

Am 27.43 0.04 0.04 0.51 0.04 0.26 0.02 0.46 �0.06 �0.04 �0.03 0.07 �0.01 �0.01Arg 92.02 �0.02 �0.07 �0.83 �0.06 �0.41 �0.04 0.09 0.01 �0.11 0.03 0.08 0.00 �0.01Asn 79.35 �0.01 9.50 2.17 0.04 0.25 0.02 0.11 0.01 �0.08 0.04 0.08 �0.01 �0.01Asp 29.24 �0.01 0.08 2.52 0.02 0.10 0.01 0.03 0.01 �0.04 0.02 0.03 �0.01 �0.01Bu 45.64 0.01 0.03 0.34 0.04 0.25 0.02 �0.01 0.00 �0.12 0.07 0.09 0.14 0.13

Cit 95.07 �0.05 �0.19 �2.39 �0.18 �1.22 �0.11 0.18 0.04 �0.32 0.08 0.21 �0.01 �0.01Cys 98.34 0.02 0.00 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00

Fa 92.77 �0.04 0.03 0.46 0.05 0.35 0.03 �0.01 �0.02 �0.49 0.10 0.08 0.20 0.19

Gln 91.73 �0.01 0.02 0.26 9.42 0.30 0.03 0.05 0.00 �0.06 0.02 0.04 0.00 �0.01Glu 82.82 �0.01 0.02 0.34 0.06 6.24 0.03 0.03 0.01 �0.09 0.02 0.05 0.00 �0.01Gly 86.91 0.00 0.01 0.15 0.01 0.08 0.01 0.09 0.04 �0.04 0.00 0.06 0.00 0.00

Gy 45.47 0.19 0.01 0.15 0.02 0.11 0.01 0.00 0.00 �0.06 0.02 0.22 0.01 0.01

His 93.04 0.00 0.00 0.06 0.01 0.07 0.01 0.01 0.00 �0.01 0.01 0.01 0.00 0.00

Ile 99.42 0.00 0.00 0.02 0.00 0.01 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00

La 72.17 1.06 0.06 0.72 0.08 0.50 0.04 0.01 0.02 �0.26 0.09 9.27 0.04 0.04

Leu 99.41 0.00 0.00 0.02 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00

Lys 96.22 0.00 0.01 0.11 0.01 0.06 0.01 0.02 0.00 �0.02 0.00 0.01 0.04 0.00

Met 88.33 0.00 0.00 0.03 0.00 0.02 0.00 0.00 0.00 �0.01 0.01 0.01 0.00 0.00

Phe 88.91 0.00 0.01 0.15 0.01 0.10 0.01 0.01 0.00 �0.13 0.02 0.02 0.07 �0.01Pr 27.95 �0.18 0.18 2.29 0.26 1.73 0.15 �0.07 0.10 �1.63 0.44 0.63 �0.16 �0.15Pro 89.97 �0.01 0.01 0.20 0.04 0.24 9.66 0.01 0.00 �0.05 0.02 0.04 0.00 0.00

Ser 78.44 0.19 0.01 0.16 0.01 0.10 0.01 0.05 0.00 �0.02 0.02 0.16 0.00 0.00

Tg 89.41 0.04 0.06 0.73 0.08 0.56 0.05 �0.02 �0.02 �0.74 0.15 0.21 0.30 0.28

Thr 93.54 �0.01 0.01 0.11 0.01 0.07 0.01 0.02 0.00 �0.05 0.01 0.03 0.00 �0.01Trp 94.43 0.03 0.00 0.06 0.01 0.04 0.00 0.01 0.00 �0.05 0.01 0.03 0.03 0.00

Tyr 87.65 0.00 0.01 0.15 0.01 0.10 0.01 0.01 0.00 �0.09 0.02 0.02 0.07 �0.01Va 26.78 �0.03 0.03 0.39 0.04 0.29 0.02 �0.01 0.02 �0.27 0.07 0.11 0.35 0.04

Val 98.73 0.00 0.01 0.07 0.01 0.05 0.00 0.01 0.00 �0.03 0.01 0.01 0.00 0.00

aPercentage of the incremental increase in input that is released into the hepatic vein.

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 283

sensitivity to As (via increasing the numerical value offC, from two) should prevent the decline in Oaconcentration and the assocatiated decline in Ndhconcentrations.The decrease in Oa concentrations, associated with

increasing glucagon concentrations, also results in anincrease in predicted As concentrations, which leads toan increase in Kb generation from As via mass action(Fig. 4). Increasing blood propionate concentrationshad the opposite effects, causing an increase in Oa(Fig. 3) and a reduction in Kb release (Table 8). Areduction in ketogenesis in association with the provi-sion of additional propionate is consistent with bothin vivo (Grohn, 1985) and in vitro (Lomax et al., 1983)observations. Glucagon infusions in lactating cows havebeen shown to result in a decrease in the bloodconcentration of both b-hydroxybutyrate and NEFA.The decease in NEFA concentrations also results in adecline in their rate of removal by the liver, whichreduces the supply of Kb precursor As (Bobe et al.,2002). Increasing the sensitivity of pyruvate carboxylate

to As will also decrease the rate of Kb generation byallowing more As to enter the tri-carboxylic acid cycle.Therefore, the combination of decreasing blood NEFAconcentrations and increasing sensitivity of pyruvatecarboxylase to As should result in better predictions ofKb release when simulating glucagon infusions.Gluconeogenesis appears to place significant demands

on AA supply (Fig. 5). However, the largest impact is onNEAA, which are not thought to be critical in terms ofmaintaining productive capacity in the lactating cow.The possibility that EAA removal by liver is regulatedby endocrines such as insulin or glucagon cannot beruled out. A number of EAA including Met, Thr andHis are glucogenic. Removal of Phe, Arg and His hasbeen observed to increase during glucagon infusionusing a perfused rat liver model (Mallette et al., 1969).If transport activity was stimulated by glucagon in theruminant liver, there could be a significant impact onperipheral availability of these AA. This is of particularinterest in early lactation, when intake is not increasingas rapidly as milk production. During this time, EAA

ARTICLE IN PRESSM.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289284

supply may be compromised if removal of EAA by theliver is enhanced by gluconeogenic signals. However,such a mechanism was not included in the currentmodel, as there is no firm evidence to support it at thistime. Although EAA removal for gluconeogenic pur-poses could impact EAA supply significantly, it seemslikely that it would have little impact on total Gl flow asthe mass of carbon available from the EAA is smallrelative to that available from other sources such as Pr,La and NEAA.Changes in blood flow, such as those that occur

following a meal, would be expected to affect thecomposition of AA in blood flowing from the liver(Fig. 5). Concentrations of Met and Phe were affected tothe greatest degree. Met is of particular concern, as it isbelieved to often limit production (NRC, 2001). It is notclear whether the increased flow of absorbed EAA thatwould normally be associated with a meal contains apattern of AA that is complimentary to the predictedchanges associated with changes in portal flow. Perhaps,hepatic removal of Met and Phe are offset by anabsorption pattern that contains greater proportions ofMet and Phe than are needed.Metabolite removal by the liver in response to

increases in input concentrations ranges from about1% for Ac, Ile, Leu and Val to about 70% for Asp, Am,Pr and Va. Although Gl release is predicted to respondto substrate supply, the response to a 10% change inportal propionate was slightly less than 0.5%, andresponses to similar changes in other substrates resultedin changes in Gl release of less than 0.1% (Table 8).Consequently, it would appear that Gl release is drivenprimarily by endocrine status.The balance model is useful for interpreting experi-

mental results. Measurement of arterial, portal andhepatic vein concentrations of AA, and energeticsubstrates provides enough information to allow calcu-lation of a number of intracellular flows with aminimum of assumptions. This model might be usedwith individual animal data and the estimatesderived subjected to similar statistical analyses as theobserved values. In this manner, the effects of treat-ments such as increasing glucagon concentrations couldbe evaluated with respect to extracellular and intracel-lular flows and rate parameters. Changes in flowpatterns, associated with a particular treatment, mightbe used to distinguish between regulatory effects andmetabolic responses.The dynamic model appears to be fairly robust with

respect to input variation and can be defined largely byarterio-venous difference data. Once tested thoroughlyagainst independent data, it could be used to predictchanges in metabolite removal and release. Addition-ally, it provides insight as to simplifications that couldbe tolerated when incorporating metabolic knowledgeinto a more aggregated whole animal model.

Acknowledgements

This work was funded by a consortium of govern-mental and industrial organizations, including theDepartment for Environment, Food and Rural Affairs,the Biotechnology and Biological Science ResearchCouncil, Purina Mills Inc., the Milk DevelopmentCouncil of England and Wales, and NUTRECO Inc.Additional aspects of the work were funded throughDEFRA contract LS3608. The comments and supportof Dr N. E. Smith are greatly appreciated.

Appendix A

A.1. Differential and associated equations

The following are the differential equations of themodel. Flows used are defined in Table 3. Numericalcoefficients preceding a flow variable representstandard stoichiometries for conversion of moles ofsubstrate to moles of product (Freetly et al., 1993;Newsholme and Leech, 1986; Salway, 1994).Stoichiometries for eVa metabolism were derived byassuming a 1.0:0.8:1.8 mixture of valerate, isobutyrate,and isovalerate, respectively, based on standard stoi-chiometries and the observations of Reynolds et al.(1988a)

dQPy

dt¼UeAla;Py þ UeCys;Py þ UeGy;Py

þ UeLa;Py þ UeSer;Py þ UeTg;As

þ UeTrp;Py � 0:33UeFa;eTg � UPy;As

� UPy;eAla � UPy;eLa � UPy;Oa; ðA:1Þ

dQOa

dt¼UAk;Oa þ UnAsp;Oa þ UeIle;Oa

þ UeMet;Oa þ UePhe;Oa þ UePr;Oa

þ UPy;Oa þ UeThr;Oa þ UeTyr;Oa

þ 0:222UeVa;Oa þ UeVal;Oa

� UOa;Ak � UOa;eGl � UOa;nAsp; ðA:2Þ

dQAs

dt¼UeAc;As þ 2UeBu;As þ 8UeFa;As

þ 24UeTg;As þ UeIle;As þ UeLeu;As

þ 0:778UeVa;As þ UPy;As

� UAs;eAc � UAs;Ak � UAs;eKb; ðA:3Þ

where

UeVa;As ¼ UeVa;Oa; ðA:4Þ

UAs;Ak ¼ UOa;Ak; ðA:5Þ

UeIle;As ¼ UeIle;Oa; ðA:6Þ

ARTICLE IN PRESSM.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 285

dQnAsp

dt¼UeAsn;nAsp þ UeAsp;nAsp þ UOa;nAsp

� UnAsp;eAsp � UnAsp;xPrt

� UnAsp;eAsn � UnAsp;Oa; ðA:7Þ

dQAk

dt¼UnCit;nArg þ UnGlu;Ak þ UOa;Ak þ UOa;nAsp

þ UPy;eAla � UAk;nGlu � UAk;Oa � UeAla;Py

� UeCys;Py � UeIle;Oa � UeLeu;As � 2UeLys;eKb

� UePhe;Oa � UeTrp;Py � UeTyr;Oa

� UeVal;Oa � UnArg;nGlu � UnAsp;Oa; ðA:8Þ

dQnGlu

dt¼UAk;nGlu þ 2UnArg;nGlu þ UeAla;Py þ UeCys;Py

þ UeGln;nGlu þ UeGlu;nGlu þ UeHis;nGlu

þ UeIle;Oa þ UeLeu;As þ 2UeLys;eKb þ UePhe;Oa

þ UePro;nGlu þ UeTrp;Py þ UeTyr;Oa

þ UeVal;Oa þ UnAsp;Oa � UnCit;nArg � UnGlu;Ak

� UnGlu;eGln � UnGlu;eGlu � UnGlu;ePro

� UnGlu;xPrt � UOa;nAsp � UPy;eAla: ðA:9Þ

Multipliers of 2 and 2 as applied to UnArg,nGlu andUeLys,eKb in Eq. (A.9) reflect the stoichiometries asso-ciated with conversion of nArg and eLys to nGlu andAs, respectively.

dQnCit

dt¼UeCit;nCit þ UnArg;nCit

� UnCit;eCit � UnCit;nArg; ðA:10Þ

dQnArg

dt¼UeArg;nArg þ UnCit;nArg � UnArg;eOrn

� UnArg;xPrt � UnArg;nCit � UnArg;nGlu; ðA:11Þ

dQnAm

dt¼UeAm;nAm þ UeAsn;nAsp þ UeGln;nGlu

þ UeGly;nCd þ UeHis;nGlu þ UeMet;Oa

þ UeSer;Py þ UeThr;Oa þ UeTrp;Py

þ UnGlu;Ak � UAk;nGlu � UnArg;nCit

� UnAsp;eAsn � UnGlu;eGln; ðA:12Þ

dQnCd

dt¼UAk;Oa þ 2UeGly;nCd þ UeHis;nGlu

þ UeIle;Oa þ 2UeLys;eKb þ 2UeMet;Oa

þ 4UeTrp;Py þ UeVal;Oa þ UOa;Ak

þ UOa;eGl þ UPy;As � UePr;Oa

� 0:778UeVa;Oa � UnArg;nCit

� UnCd ;eCd � UPy;Oa: ðA:13Þ

Multipliers of 2, 2, 2 and 4 as applied to UeGly,nCd,UeLys,eKb, UeMet,Oa, and UeTrp,Py in Eq. (A.13) corre-spond with the stoichiometries associated with conver-

sion of eGly, eLys, eMet and eTrp to nCd, respectively.

dQNdh

dt¼ 2UAk;Oa þ UeBu;As þ 7UeFa;As

þ UeGly;nCd þ 2UeGy;Py þ 3UeIle;Oa

þ UeLa;Py þ UeLeu;As þ 3UeLys;eKb

þ 2UeMet;Oa þ UePhe;Oa þ UePr;Oa

þ UePro;nGlu þ 23UeTg;As þ 2UeThr;Oa

þ 2UeTrp;Py þ UeTyr;Oa þ 0:444UeVa;Oa

þ 4UeVal;Oa þ UnArg;nGlu þ UnCit;nArg

þ UnGlu;Ak þ UOa;Ak þ UPy;As

� UAk;nGlu � 0:5UAs;eKb

RNdh;TNd CNdh

iCNdh

� 0:67UeFa;eTg � UNdh;Ox

� UnGlu;ePro � UOa;eGl � UPy;eLa; ðA:14Þ

dQNd

dt¼ �

dQNdh

dt: ðA:15Þ

Multipliers in Eq. (A.14) reflect the stoichiometries ofNdh production associated with metabolism. Thefactor 0.67 associated with UeFa,eTg represents thecost of converting Py to phosphorylated glycerol foruse in eTg synthesis from eFa. The term RNdh,TNdCNdh/

iCNdh is used to calculate the mixture of b-hydroxybu-tyrate and acetoacetate that is released from liver aspredicted from Ndh concentrations, where RNdh,TNd

represents the observed Ndh:(Ndh+Nd) ratio forhepatic tissue in vivo (Baird et al., 1975), and CNdh/

iCNdh is used to reference RNdh,TNd to model concentra-tions of Ndh.

dQAt

dt¼UAk;Oa þ UeGy;Py þ UeTrp;Py þ 2UFdh;Ox

þ 3UNdh;Ox þ 3UnpH;Ox � 5UAA;xPrt

� UAt;Und � 2UeAc;As � UeBu;As

� 0:67UeFa;eTg � UeLeu;As � 2UeMet;Oa

� 2UePr;Oa � UeTg;As � UeVa;Oa

� 2UnArg;nCit � 2UnAsp;eAsn � 2UnCit;nArg

� UnGlu;eGln � 2UOa;eGl � UPy;Oa; ðA:16Þ

dQAd

dt¼ �

dQAt

dt; ðA:17Þ

where UnpH,Ox was used to correct a slight imbalance ofintracell pH, with an assumed energetic equivalency ofNdh. UFdh,Ox was assumed to be equal to its rate ofsynthesis (France et al., 1992):

UFdh;Ox ¼UAk;Oa þ 7UeFa;As þ 2UeIle;Oa

þ UeLeu;As þ UeLys;eKb þ UeMet;Oa

þ UePr;Oa þ 21UeTg;As þ UeThr;Oa

þ UeTrp;Py þ UeVa;Oa þ 2UeVal;Oa; ðA:18Þ

ARTICLE IN PRESS

Table 9

Production data from Experiment C7. Treatments were low, medium,

and high dietary crude protein levels

Low Medium High SED PTreatment

Yield (kg/d)

Milk 27.8 29.6 32.1 0.84 0.02

Fat 1.052 1.124 1.242 0.025 0.004

Protein 0.862 0.944 0.963 0.032 0.07

Lactose 1.357 1.431 1.556 0.057 0.06

Concentration (g/kg)

Fat 37.76 38.48 38.62 0.692 0.47

Protein 31.09 32.24 30.17 0.535 0.04

Lactose 48.64 48.45 48.73 0.377 0.77

M.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289286

and UAt,Und represented undefined uses of At such asthat used for protein turnover, maintenance of sodiumand potassium balance, synthesis and maintenance ofDNA and RNA, etc. Multipliers used in Eq. (A.16)reflect the stoichiometries of At production associatedwith metabolism of FADH (Fdh), Ndh, nArg, nCit,eMet, ePr, Oa and synthesis of export protein.Mulipliers in Eq. (A.18) correspond with stoichiometriesof Fdh production associated with metabolism of eFa,eIle, eTg, eVa and eVal.Oxygen removal by the tissue was calculated from

oxidative flows:

UeO2;H2O ¼ 0:5UFdh;Ox þ 0:5UNdh;Ox

þ 3UePhe;Oa þ 3UeTrp;Py þ 2UeTyr;Oa: ðA:19Þ

Multipliers in Eq. (A.19) reflect stoichiometries of O2

use associated with ePhe, eTrp, eTyr, Fdh, and Ndhoxidation.Ketone body (UKb,eKb) and Ur (UnUr,eUr) production

were calculated as

UKb;eKb ¼ 0:5UAs;eKb þ UeLeu;As þ UeLys;eKb þ UePhe;Oa

þ UeTrp;Py þ UeTyr;Oa þ 0:778UeVa;Oa; ðA:20Þ

UnUr;eUr ¼ UnArg;eOrn þ UnArg;nCit þ UnArg;nGlu; ðA:21Þ

where nUr is a zero pool, since the model assumes thatall intracellularly formed urea is instanteously pumpedout of the cell into the extracellular pool.

A.2. Methods for experiment C7

Four multiparous Holstein–Friesian cows were fittedwith catheters in the hepatic-portal vein, hepatic vein,mesenteric vein, and an artery (mesenteric or intercostal)10 to 14 days post-calving. This surgical preparationallowed measurement of net metabolite uptake andproduction across the portal-drained viscera (PDV),hepatic and splanchnic beds. The portal catheter of onecow did not give consistent results, therefore only threeanimals were used to determined PDV and hepaticuptakes; four were used for all other determinations.Beginning week 9 of lactation, cows were fed three dietsof grass silage (170 g crude protein (CP)/kg DM):con-centrates in a 40:60 ratio (DM basis). The three barley-based concentrates were formulated to contain 105(L),154(M), or 204(H) g CP/kg; the additional protein wasprovided by a protected soya [Sopralin, BP Nutrition(UK), Ltd]. Silage and concentrates were given hourlyvia automated feeders. Diets were fed to meet the energyrequirements of the cows at their level of milkproduction. A 3� 3 Latin square design with 3-weekperiods was employed with one additional sequence.Samples were taken simultaneously from the portal vein,hepatic vein and artery over a 9 h period on the last dayof each period. Blood flow was determined using the dye

dilution technique (with para-amino hippuric acid) toenable measurement of net metabolite flow across thePDV and liver. A summary of the results is provided inTable 9.

A.3. Parameter estimation from the balance model

Measurements of arterio-venous difference and bloodflow were adequate to parameterize the majority offlows in the model although some inconsistencies wereobserved. The removal of eHis, eIle, eLeu and eVal wasnegative, which is inconsistent with known biology, andsuggests the presence of alternative unmeasured sourcesfor these AA. Therefore, an undefined source wasincluded in the balance model. A number of internalflows could not be defined directly from the availabledata. There also was a shortage of carbon entering thetissue relative to measured outputs and slightly less Uroutput than was required to dispose of the observednitrogen inputs from AA and Am. In an attempt both toderive the remaining unknown flows and to balance themodel as closely as possible to observed values so thatintracellular parameters could be calculated, an optimi-zation problem was formulated and solved. Theoptimization problem included the following equations,which were obtained from the balance model. Thedifferential equations for Ak, As, nAm, nArg, nAsp,nCd, nCit, Ndh, nGlu, nUr, Oa and Py were set to zeroand rearranged with known flows (denoted by U) on theright and unknown flows (denoted by #U) on the left toyield the following, respectively.

#UAk;nGlu þ #UAk;Oa þ #UnArg;nGlu þ #UnAsp;Oa

� #UnCit;nArg � #UOa;Ak

¼ UnGlu;Ak þ UOa;nAsp þ UPy;eAla

� UeAla;Py � UeCys;Py � UeIle;Oa

� UeLeu;As � 2UeLys;eKb � UePhe;Oa

� UeTrp;Py � UeTyr;Oa � UeVal;Oa; ðA:22Þ

ARTICLE IN PRESSM.D. Hanigan et al. / Journal of Theoretical Biology 228 (2004) 271–289 287

#UOa;Ak � #UPy;As � 24 #UeTg;As

¼ UeAc;As þ 2UeBu;As þ 8UeFa;As

þ UeIle;As þ UeLeu;As

þ 0:778UeVa;As � UAs;eAc � UAs;eKb; ðA:23Þ

#UAk;nGlu þ #UnArg;nCit � #UeAm;nAm

¼ UeAsn;nAsp þ UeGln;nGlu þ UeGly;nCd

þ UeHis;nGlu þ UeMet;Oa þ UeSer;Py þ UeThr;Oa

þ UeTrp;Py þ UnGlu;Ak � UnAsp;eAsn � UnGlu;eGln; ðA:24Þ

#UnArg;nCit þ #UnArg;nGlu � #UnCit;nArg

¼ UeArg;nArg � UnArg;eOrn � UnArg;xPrt; ðA:25Þ

#UnAsp;Oa ¼UeAsn;nAsp þ UeAsp;nAsp þ UOa;nAsp

� UnAsp;eAsn � UnAsp;eAsp � UnAsp;xPrt; ðA:26Þ

#UnArg;nCit þ #UnCd;eCd þ #UPy;Oa � #UAk;Oa

� #UOa;Ak � #UOa;eGl � #UPy;As

¼ 2UeGly;Cd þ UeHis;nGlu

þ UeIle;Oa þ 2UeLys;eKb þ 2UeMet;Oa

þ 4UeTrp;Py þ UeVal;Oa � UePr;Oa

� 0:778UeVa;Oa; ðA:27Þ

#UnCit;nArg � #UnArg;nCit

¼ UeCit;nCit � UnCit;eCit; ðA:28Þ

#UAk;nGlu þ #UNdh;Ox þ #UOa;eGl � 2 #UAk;Oa

� #UnArg;nGlu � #UnCit;nArg � #UOa;Ak

� #UPy;As � 23 #UeTg;As

¼ UeBu;As þ 7UeFa;As þ UeGly;nCd

þ 2UeGy;Py þ 3UeIle;Oa þ UeLa;Py

þ UeLeu;As þ 3UeLys;eKb þ 2UeMet;Oa

þ UePhe;Oa þ UePr;Oa þ UePro;nGlu

þ 2UeThr;Oa þ 2UeTrp;Py þ UeTyr;Oa

þ 1:22UeVa;Oa þ 4UeVal;Oa þ UnGlu;Ak

� 0:5UAs;eKbRNdh;TNdCNdh=iCNdh

� 0:67UeFa;eTg � UnGlu;ePro � UPy;eLa; ðA:29Þ

#UnCit;nArg � #UAk;nGlu � 2 #UnArg;nGlu � #UnAsp;Oa

¼ UeAla;Py þ UeCys;Py þ UeGln;nGlu

þ UeGlu;nGlu þ UeHis;nGlu þ UeIle;Oa

þ UeLeu;As þ 2UeLys;eKb þ UePhe;Oa

þ UePro;nGlu þ UeTrp;Py þ UeTyr;Oa

þ UeVal;Oa � UnGlu;Ak � UnGlu;eGlu

� UnGlu;eGln � UnGlu;ePro � UnGlu;ePrt

� UOa;nAsp � UPy;eAla; ðA:30Þ

#UUr;eUr � #UnArg;nCit � #UnArg;nGlu ¼ UnArg;eOrn; ðA:31Þ

#UOa;Akþ #UOa;eGl � #UAk;Oa � #UnAsp;Oa � #UPy;Oa

¼UeIle;Oa þ UeMet;Oa þ UePhe;Oa

þ UePr;Oa þ UeThr;Oa þ UeTyr;Oa

þ 0:222UeVa;Oa þ UeVal;Oa � UOa;nAsp; ðA:32Þ

#UPy;As þ #UPy;Oa � #UeTg;As

¼UeAla;Py þ UeCys;Py þ UeGy;Py

þ UeLa;Py þ UeSer;Py þ UeTrp;Py

� 0:33UeFa;eTg � UPy;eAla � UPy;eLa: ðA:33Þ

UeTg;Aswas included as the unknown source of carbon,as it was not measured in the experiments used forparameter estimation, and previous work indicates thatit may be used by the liver (Reid et al., 1979; Reynoldset al., 2003), though the rates of use would appear to beinadequate to supply all of the missing carbon.Although UeAm,nAm, UnCd,eCd, UnUr,eUr and UOa,eGl weremeasured, they were included in the above equations asunknowns to allow manipulation to achieve carbon andnitrogen balance, which could not be achieved whenthey were constrained to the observed values. As theabove flows were manipulated, At and Ndh balance wasmaintained by automatic adjustments of the flowUAt,Und. To minimize substrate cycling, the problemwas originally constrained to solutions where observedand predicted uptakes of O2 were equal. However, sucha constraint prevented the problem from converging.The use of 3.0 and 2.0 instead of 2.5 and 1.5 (Nelson andCox, 2000) for the stoichiometries of At synthesis fromNdh and Fdh would account for some of the error inpredicting O2 consumption. Consequently, the con-straint was converted to a residual error representationand added to the objective function yielding thefollowing equation:

obj ¼ #UeAm;nAm � UeAm;nAm

� �2þ #UeO2;H2O � UeO2;H2O

� �2þ #UeTg;As � UeTg;As

� �2þ #UnCd;eCd � UnCd ;eCd

� �2þ #UnUr;eUr � UnUr;eUr

� �2þ #UOa;eGl � UOa;eGl

� �2;

ðA:34Þ

where U represented the experimentally observed flowsand #U the predicted ones. Additional assumptions werethat nVol was 10.4l (Freetly et al., 1993) and RNdh,TNd

was 0.91. The problem was solved by minimizingEq. (A.34) subject to Eqs. (A.22)–(A.33) and the restric-tion that all #U were nonnegative.

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