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ARTICLE IN PRESS

Journal of Theoretical Biology 228 (2004) 271–289

An integrative model of amino acid metabolism in the liver of the lactating dairy cow M.D. Hanigana, L.A. Cromptonb, C.K. Reynoldsc, D. Wray-Cahend, M.A. Lomaxe, J. Francef,* a Purina Mills LLC, P.O. Box 66812, St. Louis, MO 63166-6812, USA School of Agriculture, Policy & Development, The University of Reading, Whiteknights, P.O. Box 237, Reading, Berkshire RG6 6AR, UK c Department of Animal Sciences, Ohio State University, OARDC, 1680 Madison Avenue, Wooster, OH 44691-4076, USA d Center for Devices and Radiological Health, Food and Drug Administration, OST DLS/HSB, 8401 Muirkirk Road, Laurel, MD 20708, USA e Department of Agricultural Sciences, Imperial College London, Wye Campus, Ashford, Kent TN25 5AH, UK f Department of Animal & Poultry Science, University of Guelph, 50 Gordon Street, Guelph, Ont., Canada N1G 2W1 b

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 5 mol/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, convert dietary energy to energy in products (meat and milk) with greater efficiency than they convert dietary nitrogen to productive nitrogen (Bequette et al., 2003; NRC, 2001). This low nitrogen conversion efficiency contributes to the environmental challenges associated with dairy production systems (Howarth et al., 2002; Tamminga, 1996). A better quantitative understanding of ruminant nitrogen metabolism in general and amino *Corresponding author. Tel.: +519-824-4120 x52209; fax: +519-836-9873. E-mail address: [email protected] (J. France). 0022-5193/$ - see front matter r 2004 Published by Elsevier Ltd. doi:10.1016/j.jtbi.2004.01.010

acid (AA) metabolism in particular may help identify strategies to reduce nitrogen intake while maintaining production levels, thereby reducing nitrogen excretion and improving the efficiency of utilization. Significant efforts have been devoted to defining relationships between AA supply to the mammary glands and removal and use for milk production (Bequette et al., 1998; France et al., 1995; Hanigan et al., 2001b; Maas et al., 1998). Both endocrine regulation of mammary metabolism (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 for milk protein synthesis. While increased total AA supply to the udder does not always result in increased milk protein, reductions in supply of one or more AA can

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

result in a loss of milk protein production (Bequette et al., 2000). Thus mammary utilization of AA and subsequent production of milk protein are determined, energy availability not withstanding, by the balance of AA arriving at the mammary glands (Hanigan et al., 2000). Although apparent hepatic affinities for AA are relatively low (Hanigan et al., 1998), the combination of large hepatic blood flow rates and the passage of all absorbed AA through the liver via the portal vein results in clearance of significant quantities (around two-thirds) of AA from circulation on a net daily basis (Reynolds et al., 1988b; Wray-Cahen et al., 1997). However, removal of individual AA differs (Hanigan et al., 1998) resulting in significant alterations in the pattern of AA in peripheral circulation as compared to the pattern absorbed from the gastro-intestinal tract (WrayCahen et al., 1997). Therefore, if the pattern and quantity 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 Freetly et al. (1993) were built primarily to describe energy transactions within the liver. As such, descriptions of hepatic AA metabolism were highly aggregated. Additionally, the paucity of data on hepatic AA metabolism by the dairy cow prevented a less aggregated description. The model of Danfaer (1990) was constructed as a submodel of a whole animal model; thus is relatively aggregated in form with no consideration of individual AA flow. A single AA model was constructed by France et al. (1999) to interpret isotopic leucine data. Since the construction of the earlier models, additional in vivo hepatic AA data from non-lactating (Bach et 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 have become available . An objective of the present work was to construct a model of hepatic AA metabolism that could be defined by and used to interpret trans-hepatic observations. Therefore, the effort includes a less detailed description of energy metabolism. The latter was required to represent catabolic exchange of nitrogen and the glucogenic use of AA. Nitrogenous entities other than ammonia, urea, and AA were also not considered due to their minimal inpact on nitrogen balance. The effort herein was intended to complement, not replicate, efforts by Waghorn (1982), Freetly et al. (1993), and France et al. (1999).

2. Model description 2.1. General considerations The model represents the liver of a cow with emphasis on predicting net removal from or appearance of

metabolites in blood perfusing the organ. The organ was assumed to be constant in size with intracellular and extracellular compartments of fixed volume, and thus no representation of protein turnover was included. The model is based on a set of dynamic differential equations coded 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 specified otherwise. An overview of the model is depicted in Fig. 1. Pools are depicted as boxes with solid lines, compartment boundaries by boxes with dashed lines, and flows as arrows. Double arrowheads denote bidirectional flows. Abbreviations used in Fig. 1 and throughout the paper are given in Table 1. Three letter codes denoting AA were assigned according to previous convention (ACS, 1997). Abbreviations for non-AA metabolites are similar to those used by Freetly et al. (1993). For Hepatic Artery Blood

Portal Vein Blood

Extracellular Compartment

eCys

eSer

eTrp

eTg

eFa

eBu eIle

eLeu eLys eTyr

eVa eGy

ePhe

eLa eAla

Py

eKb

As

eAc

+

+

Gluc

-

eGly

+ eGl Ins

Oa

-

nCd

ePr eAsp

eCd eHis

Ak

nAsp

ePro nGlu

eAsn

eGlu

eMet nAm

eThr

eGln eAm

eIle nCit

eVal

eArg

nArg

Amino Acids

Intracellular Compartment

eOrn

eCit

xPrt

eUr

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.

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

273

Table 1 Abbreviations used in the model

2.2. Basic equations

Abbreviation Designation

Abbreviation Designation

AA Ac Ad Ak Ala Am Arg As Asn Asp At Bu Cd Cit Cys Fa Fdh Gl Gln Glu Gluc Gly Gy His Ile Ins Leu

La Lys Kb Met Nd Ndh O2 Oa Orn Ox Phe Pr Pro Py Ser Tg Thr TNd Trp Tyr Und Ur Val Va Vol xPrt

Model flows are denoted in general terms as UA,B (mol/d) where the subscript A, B refers to substrate and product, respectively. Equation forms used were primarily mass-action type. The general form was: CC CD fD JE ; ð1Þ UA;B ¼ kA CA KC KD CE

Amino Acid Acetate ADP a-ketoglutarate Alanine Ammonia Arginine Acetyl-CoA Asparagine Aspartate ATP Butyrate CO2 Citrulline Cysteine Fatty Acids FADH Glucose Glutamine Glutamate Glucagon Glycine Glycerol Histidine Isoleucine Insulin Leucine

Lactate Lysine Ketone body Methionine NAD+ NADH Oxygen Oxaloacetate Ornithine Oxidation Phenylalanine Propionate Proline Pyruvate Serine Triacylglycerol Threonine NAD+ and NADH Tryptophan Tyrosine Undefined Use Urea Valine Valerate and isobutyrate Volume Export Protein

metabolites present in more than one compartment, a preceding lower case a, p, e, or n was used to designate arterial, portal, extracellular or intracellular entities, respectively. The extracellular compartment was assumed to comprise vascular and interstitial spaces, i.e. that space not bounded by extracellular membrane. The intracellular compartment was assumed to comprise all space within the organ that was bounded by extracellular membrane. This would include all cell types present in the organ. Potential heterogeneity of metabolism due to intracellular compartmentalization was not considered. Inputs to the model are arterial and portal venous blood and outflow is hepatic venous blood. The metabolites in blood considered include the 20 primary AA plus ornithine (eOrn), citrulline (eCit), ammonia (eAm), urea (eUr), glucose (eGl), lactate (eLa), propionate (ePr), butyrate (eBu), acetate (eAc), valerate (eVa; valerate plus isobutyrate and isovalerate), ketone bodies (eKb; b-hydroxybutyrate plus acetoacetate plus acetone), non-esterified fatty acids (eFa), triacylglycerol (eTg), glycerol (eGy), oxygen (O2), and CO2 (eCd). Some simplification of the diagram was required due solely to complexities associated with flows for some pools. Pools not illustrated in Fig. 1 include NADH (Ndh), NAD+ (Nd), ATP (At), and ADP (Ad). These pools are described fully in the text.

where kA represents the primary rate parameter for conversion of A to B (l/d), and CA represents the transient concentration of A (mol/l), CC and CD represent transient concentrations (mol/l) of secondary substrates or activators C and D, respectively, KC and KD represent the constants for the secondary substrates (mol/l), CE represents the transient concentration of an inhibitor (mol/l), and JE represents the constant for the inhibitor (mol/l). In all cases, KC, KD, and JE were set equal to reference concentrations of C, D, and E, respectively, yielding ratios of 1 in the reference state (described below). The effects of secondary substrates were represented as a ratio to their reference concentrations so that the primary rate constants retained units and parameter values pertinent to the primary substrate. Additionally, consideration as a ratio whose value is 1 in the reference state facilitates application of a sensitivity exponent fD. When applied in this manner, fD will have no effect in the reference state, and as effecter concentrations move away from the reference state, the sensitivity to such changes can be altered without affecting the value of kA. Additionally, fD can be converted to a function representing an endocrine or regulatory signal where such regulation is required. Although only illustrated for activator D, this approach applies to the other activators where designated. Differential equations describing pools were defined as dQA ¼ UB;A þ ? UA;B ?; dt

ð2Þ

where dQA/dt represents the change in metabolite A with respect to time (mol/d). The quantity of metabolite present at any point in time (QA, mol) was determined by numerical integration of Eq. (2) using an initial mass (iQA). Transient concentrations were defined as CA ¼

QA ; Vol

ð3Þ

where Vol (l) represents the volume of the intracellular compartment. 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 or inhibitors included, i.e. its simple mass-action form, was used to describe removal of metabolites from blood


274

Table 2 Reference intracellular metabolite concentrations (mmol/l) from Freetly et al. (1993) Metabolite

Concentration

Ad Ak Am Arg As Asp At Cd Cit Glu Nd Ndh Oa Py

0.236 0.124 75.0 0.097 0.107 1.05 0.236 31.0 0.359 4.17 0.236 0.236 0.544 0.038

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-Cahen et al., 1997) and exchange between capillary and interstitial space is rapid (Goresky, 1980), a steady-state expression can be used as an approximation for the extracellular compartment provided predictions of changes in removal were not made for time frames less than 5–10 s (Hanigan et al., 1998, 2001a): CeA ¼

CaA Fa þ CpA Fp þ UB;A ; kA þ Fa þ Fp

ð4Þ

where Fa and Fp represent arterial and portal blood flows (l/d), respectively, and UB,A represents any outflow from the tissue when bidirectional blood–tissue exchange is considered explicitly or when net synthesis by the liver occurs. In this representation, CeA is equivalent to hepatic vein concentrations (Hanigan et al., 1998). Bidirectional exchange was included for eAc, asparagine (eAsn), aspartate (eAsp), eCit, glutamine (eGln), glutamate (eGlu), eLa, and proline (ePro) (Casse et al., 1994; Freetly and Ferrell, 1998; Hanigan et al., 1998; Lomax and Baird, 1983; Reynolds and Tyrrell, 1991; WrayCahen et al., 1997). Additionally, Brockman and Bergman (1975) observed significant changes in alanine (eAla) net removal when glucagon was infused. This may occur as a result of pyruvate (Py) depletion due to inducement of pyruvate carboxylase (see below) causing a reduction in unidirectional Ala outflow from the organ. Therefore, bidirectional exchange was included for eAla. Exchange of lactate is apparently driven by energy status of the tissue (Freetly and Ferrell, 1998; Reynolds, 1995), and thus the secondary substrate Nd was included. This secondary substrate was reflected in

Eq. (4) by replacing kA with the term kA CNd =KNd in the denominator. Although ornithine can be taken up by liver under conditions of acute hyperammonemia (Milano and Lobley, 2001), it is normally released and thus bidirectional flows were not considered. Gl, eCd, eUr, and eKb are also generally released on a net basis by the liver of a lactating cow (Baird et al., 1974, 1975; Lomax and Baird, 1983; Reynolds et al., 1988a, b). For these metabolites, kA was set to zero thus allowing outflow equations to determine solely rates of output. When calculating the outflow of eGl, a stoichiometry of 0.5 was applied to UOa,eGl to reflect the requirement of 2 moles of oxaloacetate (Oa) to synthesize 1 mol of eGl. Outflow of eKb included contributions from UAs,eKb and UeVa,Oa (with stoichiometric coefficients of 0.5 and 0.778, respectively) and from the incomplete oxidation of 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 metabolism was restricted to the major metabolites and those involved in metabolic regulation. These pools included acetyl-CoA (As), a-ketoglutarate (Ak), and Oa to represent the tri-carboxylic acid (TCA) cycle; nAsp and nGlu due to their integral roles in transamination and shuttling of AA carbon skeletons to and from the TCA cycle; nAm, nAsp, nCit, and arginine (nArg) to represent the Ur cycle; and Ndh, Nd, At and Ad to represent the energy status of the tissue allowing regulation of key metabolic steps. Carbon dioxide was also represented. Glycolysis was represented by catabolism of various metabolites to Py and gluconeogenesis by conversion of Oa to eGl. The differential equations describing intracellular pools are listed in Appendix A, and flow equations and associated parameters are listed in Table 3. Intracellular pools were not defined for most AA. To accommodate use of AA for synthesis of extracellular protein (xPrt) when defining transfers into intracellular pools, Eq. (1) was amended to reflect such use: UA;B ¼ kA CeA UA;xPrt ;

ð5Þ

where UA,xPrt was defined as UA;xPrt ¼ fA UAA;xPrt :

ð6Þ

UAA,xPrt represents the flow of total AA to export protein and fA represents the molar fraction of each respective AA in liver export protein. UAA,xPrt was assumed to be 0.71 mol AA/d based on previous work (Freetly et al., 1993; Raggio et al., 2002). The molar AA proportions were adopted from Freetly et al. (1993) and are listed in Table 4. To ensure that export protein did not exceed supply of essential AA (EAA), UAA,xPrt was


275

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)

UAA,xPrt

(7)

VAA,xPrt

0.713 (mol/d)

UAk,nGlu UAk,Oa UAs,eAc UAs,eKb UAt,Und UeAc,As UeAla,Py UeAm,nAm UeArg,nArg UeAsn,nAsp UeAsp,nAsp UeBu,As UeCit,nCit UeCys,Py UeFa,As UeFa,eTg UeGln,nGlu UeGlu,nGlu UeGly,nCd UeGy,Py UeHis,nGlu UeIle,Oa UeLa,Py UeLeu,As UeLys,eKb UeMet,Oa UeO2,H2O UePhe,Oa UePr,Oa UePro,nGlu UeSer,Py UeTg,As UeThr,Oa UeTrp,Py UeTyr,Oa UeVa,As UeVa,Oa UeVal,Oa UFdh,Ox UKb,eKb UnArg,eOrn UnArg,nCit

(1) (1) (1) (1)

eHis, eIle, eLeu, eLys, eMet, ePhe, eThr, eTrp, eVal Ak, nAm, Ndh Ak, Ad, Nd As As At eAc eAla eAm eArg eAsn eAsp eBu eCit eCys eFa eFa eGln eGlu eGly eGy eHis eIle eLa, Nd eLeu eLys eMet

kA, KC, KD kA, KC, KD kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA kA, KC kA kA kA

92,119, 7.50E02, 2.36E04 211,318, 2.36E04, 2.36E04 94,910 62,323 376,811 781 34,758 156,915 5181 16,797 279,497 71,012 3731 1000 4596 1224 5707 17,570 8976 5104 4456 339 34,084, 2.36E04 332 2339 7908

ePhe ePr ePro eSer eTg eThr eTrp eTyr eVa eVa eVal

kA kA kA kA kA kA kA kA kA kA kA

7413 153,989 6797 16,390 6281 4126 3524 8393 163,397 163,397 787

nArg nArg, nAm, nGlu, At

UnArg,nGlu UnAsp,eAsn UnAsp,eAsp UnAsp,Oa UnCd,eCd UnCit,eCit UnCit,nArg UNdh,Ox UnGlu,Ak

(1) (1) (1) (1) (1) (1) (1) (1) (1)

nArg, Ak nAsp nAsp nAsp, Ak nCd nCit nCit, nAsp Ndh nGlu, Nd, nAm, At

UnGlu,eGln UnGlu,eGlu UnGlu,ePro UnUr,eUr

(1) (1) (1) (31)

nGlu nGlu nGlu

kA kA, KE kA, kA kA kA, kA kA kA, kA kA, JE kA kA kA

(1) (5) (1) (1) (5) (1) (1) (1) (5) (1) (1) (5) (1) (5) (1) (5) (5) (1) (5) (5) (5) (29) (5) (1) (5) (5) (1) (5) (5) (5) (1) (1) (5) (28) (30) (1) (1)

KC, KD, KC

KC

KC KC, JD,

426 135,769, 7.50E02, 4.17E03, 2.36E04 12,353, 1.24E04 476 4762 5946, 1.24E04 1722 1393 38,892, 1.05E03 517,752 1199, 2.36E04, 7.50E02, 2.36E04 120 719 47.96

Reference flow (mol/d) 0.713

11.4 26.2 10.2 6.67 356 2.0 7.80 13.0 0.474 1.86 4.96 3.78 1.29 0.10 1.13 0.30 0.815 1.80 2.69 1.53 0.174 0.044 10.2 0.058 0.226 0.143 98.2 0.439 18.0 0.655 1.40 0.595 0.554 0.183 0.430 3.84 3.84 0.188 69.7 7.50 0.041 13.2 1.20 0.50 5.0 6.24 53.4 0.50 14.0 122 5.0 0.50 3.0 0.20 14.4


276 Table 3 (continued) Flow

Equation

Substrates

Parameters

Parameter values (Units: k ¼ l=d; K ¼ mol=l; J ¼ mol=l)

UOa,Ak UOa,eGl UOa,nAsp UPy,As

(1) (1) (1) (1)

Oa, Ad, Nd Oa, Gluc, Ins Oa, nGlu Py, Nd, As

46,813, 2.36E04, 2.36E04 50,672, 1, 1 9191, 4.17E03 169,759, 2.36E04, 1.07E04, 2

UPy,eAla UPy,eLa UPy,Oa

(1) (1) (1)

Py Py, Ndh Py, As, At, Gluc

kA, KC, kA, KC, kA, KC kA, KC, fD kA kA, KC kA, KC, KD, KE

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 Arginine Asparagine Aspartate Cysteine Glutamate Glutamine Glycine Histidine Isoleucine

0.0396 0.0379 0.0550 0.0587 0.0209 0.0827 0.0827 0.0293 0.0220 0.0512

Leucine Lysine Methionine Phenylalanine Proline Serine Threonine Tryptophan Tyrosine Valine

0.0773 0.0618 0.0229 0.0360 0.1014 0.0660 0.0397 0.0070 0.0383 0.0696

defined as UAA;xPrt ¼ minimum VAA;xPrt ; U# A1 ;xPrt ; U# A2 ;xPrt ; ::: ; U# An ;xPrt ; ð7Þ where VAA,xPrt represents the maximal rate of conversion and U# Ai ;xPrt the potential rate of protein synthesis given the prevailing rate of removal: ð8Þ U# Ai ;xPrt ¼ kAi CAi : A1 to An represents histidine (eHis), isoleucine (eIle), leucine (eLeu), lysine (eLys), methionine (eMet), phenylalanine (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 use for export protein synthesis will also approach zero for all EAA ensuring that Eq. (5) does not assume negative values for those AA that cannot be synthesized in the liver. UA,B in Eq. (5) thus represents the catabolism of AA not utilized for export protein synthesis and is limited to the difference between removal and use for protein export. Metabolic regulation was represented at several key points. The flow UnArg,nCit (Table 3) represents the carbamoyl phosphate synthetase reaction with nAm serving as a substrate and nGlu and At as activators of the reaction (Newsholme and Leech, 1986). Intracellular Glu serves as a proxy for the actual regulator N-acetylglutamate, which was not represented. Exchange of

KD JD JD,

fC,

131,579 131,579, 2.36E04, 134,642, 1.07E04, 2, 2.36E04, 1

Reference flow (mol/d) 25.5 27.6 5.0 6.5 5.0 5.0 5.12

Oa and nAsp is coupled via transamination with the UAk,nGlu, hence the inclusion of these latter entities as substrates. 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 be synthesized. Propionate is the primary substrate for Gl synthesis (Judson et al., 1968), and Gl output by the liver (Baird et al., 1980; Brockman and Bergman, 1975) and plasma Gl concentrations (Peters and Elliot, 1984) have been observed to respond to increased propionate supply, although this is not always the case (Baird et al., 1980; Casse et al., 1994). However, as both insulin and glucagon release are responsive to propionate supply (Peters and Elliot, 1984; Sano et al., 1993), it is not clear whether increased Gl production is strictly a substrate response or a combined substrate and endocrine response. The latter seems more likely as gluconeogenesis is responsive to glucagon concentrations (Bobe et al., 2002; Brockman et al., 1975), while provision of gluconeogenic precursors not associated with gluconeogenic signals (insulin and glucagon), such as alanine, do not result in additional Gl production (Reynolds and Tyrrell, 1991). Finally, infusion of Gl has been shown to result in reductions in Gl output (Freetly and Klindt, 1996). She et al. (1999) also observed inducement of pyruvate carboxylase mRNA when glucagon was infused with no down-regulation as insulin concentrations increased. Based on these observations, provisions for the regulation of UOa,eGl by insulin and glucagon and UPy,Oa by glucagon were included (Table 3). Insulin and glucagon levels in plasma were treated as input variables whose reference values are unity. As noted above, eLa removal can range from positive to negative (Freetly and Ferrell, 1998). This range is determined largely by energy and carbon needs within the cell with Ndh status being the energy determinant (Newsholme and Leech, 1986; Reynolds, 1995). Therefore, the additional substrates Ndh and Nd were considered. Pyruvate dehydrogenase kinase is inhibited by high ratios of At/Ad, As/Co-A and Ndh/Nd, which enhance phosphorylation of the enzyme to an inactive state, thus


reducing the conversion of Py to As. It has also been observed that As acts as a feedback inhibitor on this reaction. Regulation of pyruvate carboxylase appears to be associated with concentrations of As and the At/Ad ratio within the cell, whereby an increase in the ratio activates pyruvate carboxylase. Increased concentrations of As appear to activate pyruvate carboxylase (Newsholme and Leech, 1986). Therefore, the regulatory effects of As, At, and Nd on flows describing conversion of Py to As and Oa were included (Table 3). Such regulation is critical in the model to maintain TCA cycle stability and At and Ndh supply. Similarly, Nd and Ad were required as secondary substrates in UOa,Ak and Nd in UAk,Oa to provide TCA cycle stability (Table 3) (Newsholme and Leech, 1986). Finally, the inclusion of Ndh, Nd, and At as regulatory elements and nAm as a substrate in the Ak-nGlu exchanges (Table 3) were based on the observation of Saradambal et al. (1981), where At substituted for the observed dependance upon GTP.

277

As it was assumed that eCd, eGl, eKb and eOrn were only released from the liver, kA in Eq. (4) was set to zero. Rate parameters (l/d) for the outflows from tissue to blood [UB,A in Eq. (4)] were derived from determined mean flow values: UB;A rate parameter ¼ ; ð9Þ iCB where iCB represents the reference concentration of B. The equation used for eKb, obtained from Eq. (30) by substituting simple mass-action forms for UKb,eKb and UAs,eKb then re-arranging, was: 2

kAs;eKb ¼

keKb iCeKb UeLeu;As UeLys;eKb UePhe;Oa UeTrp;Py UeTyr;Oa 0:778UeVa;Oa iCAs

!

:

ð10Þ Equation (10) reflects a correction for production of acetoacetate or acetoacetyl-CoA through the incomplete oxidation of eLeu, eLys, ePhe, eTrp and eTyr. 3.2. Balance model

3. Model parameterization The data used for model parameterization were taken from a series of experiments conducted as part of a project carried out at The University of Reading and the Rowett Research Institute [Experiment C7 (a brief description is provided in Appendix A)] and work reported by Reynolds et al. (1988a, b). The reference steady state was derived from the mean values of this combined data set. Thus reference concentrations are the mean observed concentrations. Reference blood flows are the mean observed flows with adjustments to C7 flows as discussed below. Reference flows were determined using reference concentrations and blood flows. 3.1. Blood–tissue exchange parameters Estimates for net removal of metabolites from blood were derived from the reference data by fitting the model of Hanigan et al (1998) [Eq. (4) with UB,A omitted] to the arteriovenous uptake data using the NLIN procedure of SAS (1988). The results of this analysis showing derived rate constants and associated statistics are presented in Table 5. Plasma values were used for AA concentrations and adjusted to a whole blood basis as described previously (Hanigan et al., 2001b), based on the observations of Heitmann and Bergman (1980). Gl was assumed to be delivered only in the plasma portion of whole blood. Observations of cysteine were not available; thus removal was assumed to be 0.1 mol/d. The Tg observations of Reynolds et al. (2003) were used, as comparable observations were not taken in either of the experiments described herein.

Having derived the estimates of net substrate removal from blood described in Section 3.1, a balance model was set up by assuming all the derivatives of the dynamic model were zero, i.e. the animals were in steady-state during the experiments. Initial inputs to the balance model were obtained using the equations for uptake, associated parameter values (Section 3.1 and Table 5), and mean observed values for arterial and portal vein concentrations and blood flows (Table 6). A value of 0.71 moles total AA/d was used for export protein synthesis based on previous observations (Freetly et al., 1993; Raggio et al., 2002), however, the removal of Leu by the liver was inadequate to support this rate of export protein synthesis. Therefore, an additional 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 a known limitation in EAA cannot be determined using the balance model necessitating specification of export synthesis. Given that specification, catabolism of each AA can be uniquely identified allowing calculation of rate constants. The model contained bidirectional exchanges that could not be resolved numerically in the absence of isotopic 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 routine where the objective function was an unweighted sum of squares of the difference between predicted and observed uptakes or release of Tg, Gl, O2, CO2, Ur and


278

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)

Alanine Arginine Asparagine Aspartate Citrulline Glutamine Glutamate Glycine Histidine Isoleucine Leucine Lysine Methionine Ornithine Phenylalanine Proline Serine Threonine Tryptophan Tyrosine Valine

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

12,467 5181 12,290 2219 2288 2204 11,684 8976 4456 339 984 2339 7908 880 7413 4722 16,390 4126 3524 8393 787

7 9

3186 159,031 7175 71,012 2204 5588 17,303 5820 34,217 153,989 6338 163,397

Other metabolite: Acetate Ammonia BHBA Butyrate CO2 Glucose Lactate NEFA O2 Propionate Urea Valerate+Isobutyrate

7 7 7 7 7 7 7 9 7

SE of kA

P-value regression

Calculated (mmol/l)

Observed (mmol/l)

1246 921 2253 2578 922 1284 951 1071 665 976 1013 1398 1705 904 1686 843 1616 844 816 1537 619

0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

226 92 111 17.8 347 143 102 303 39.0 131 174 97.1 18.1 47.0 59.5 102 86.4 135 51.9 51.4 239

226 91 111 17.9 349 144 102 300 38.9 130 173 96.3 18.0 46.9 59.2 102 86.0 134 51.8 51.0 238

733 28275 829 9106 242 724 5123 1086 4059 18525 626 46529

0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0038

mmol/l 2.55 0.086 1.07 0.052 17.5 2.76 0.280 0.240 2.17 0.110 2.16 0.020

mmol/l 2.58 0.086 1.08 0.052 16.8 2.79 0.269 0.241 2.17 0.109 2.18 0.019

Am (see Appendix A). These last five metabolites were included in the objective function because there was not an original solution to the balance model that provided appropriate quantities of precursors or products to match predicted inputs and outputs to the observed values for these metabolites. In particular, there was a significant 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 likely contributor, therefore, a Tg flow was included in the objective function to ensure carbon balance across the liver. Although the data appeared to be of high quality overall, it seemed likely that the split between arterial and portal supplies for Experiment C7 was not consistent with previous observations. In particular, arterial flow represented over 29% of the total hepatic flow where values in the range of 10–20% are more

appropriate for lactating animals (Reynolds, 1995). Therefore, the balance model was resolved using the optimization routine with arterial proportions ranging from 5% to 29%. Results are presented in Table 7. Decreasing flow from 29% to 5% of total hepatic flow resulted in an increase in the residual error, however, the dependence on an undefined source of carbon as indicated by UeTg,As was reduced. Based on previous observations (Reynolds, 1995) and the results of the sensitivity analyses, a reference value for arterial flow of 20% of total hepatic flow was adopted for observations in Experiment C7. Using these revised blood flow estimates, the model was rebalanced and the revised estimates of metabolite flow were used for subsequent analysis. Finally, the derived values of metabolite flow were used to determine rate parameters using respective extracellular or intracellular concentrations. Thus, the final derived parameter set for substrate removal from


279

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 Arginine Asparagine Aspartate Citrulline Glutamine Glutamate Glycine Histidine Isoleucine Leucine Lysine Methionine Ornithine Phenylalanine Proline Serine Threonine Tryptophan Tyrosine Valine

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

228 (11) 85 (7) 89 (14) 17 (1) 339 (26) 144 (7) 82 (5) 322 (37) 37 (5) 117 (8) 149 (9) 83 (6) 15 (1) 48 (4) 52 (2) 101 (8) 89 (5) 125 (9) 52 (2) 47 (3) 225 (12)

282 (12) 103 (9) 145 (14) 17 (1) 365 (29) 149 (7) 83 (3) 352 (37) 43 (6) 135 (9) 179 (10) 105 (7) 22 (1) 47 (4) 70 (4) 112 (9) 114 (5) 148 (10) 56 (2) 61 (4) 246 (14) mmol/l 2.64 (0.11) 0.355 (0.024) 0.95 (0.20) 0.137 (0.015) 16.8 (0.9) 2.43 (0.13) 0.0049 0.412 (0.036) 0.272 (0.031) 3.33 (0.09) 0.501 (0.022) 0.0097 1.89 (0.24) 0.106 (0.007) 47,134 (1880)

Other metabolite: Acetate Ammonia BHBA Butyrate CO2 Glucose Glycerola Lactate NEFA O2 Propionate Triacylglycerola Urea Valerate+Isobutyrate

9 7

mmol/l 1.54 (0.07) 0.099 (0.012) 0.79 (0.19) 0.030 (0.004) 16.2 (0.8) 2.47 (0.14) 0.0041 0.274 (0.032) 0.257 (0.028) 4.83 (0.12) 0.090 (0.015) 0.0094 2.14 (0.25) 0.015 (0.004)

Blood flow (l/d)

16

12,503 (6 1 7)

a

7 9 7 7 7 7 7 7 7 7

From Reynolds et al. (2003).

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

4. Model application Having formulated and parameterized the dynamic model, the time constants were found to vary from 7.2 106 d to 0.03 d, indicating that the model was stiff and that the maximum integration interval with a fixed step integration algorithm should be approximately 7 106 d. Monitoring the step size chosen when using the Adams–Moulton variable step algorithm and a maximum integration interval of 0.005 d, indicated that a step size as small as 2 106d was often chosen by the

algorithm (Fig. 2). Using a fixed-step integration algorithm with a maximum integration interval of 7 106 d, a 1 day simulation required 4.5 s to complete on an AMD 700 MHz processor. Using a variable step algorithm such as the Runge–Kutta–Fehlberg fifthorder method, marginally reduced the time required to 3.1 s. Choosing Gear’s stiff algorithm resulted in the same run being completed in 0.3 s when the maximum step size was set to 1 104 d supporting the time constant observations with respect to model stiffness. These latter settings were used for all subsequent work unless specified otherwise. Model stability was examined by perturbing inputs tenfold in the positive and negative directions and observing pool sizes of the state variables. Example results from these analyses are presented in Fig. 3.


280

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

Flow estimates, (mol/d) UAk,nGlu UAk,Oa UeAm,nAm UeTgAs UnArg,nCit UnArg,nGlu UnAsp,Oa UnCd,eCd UnCit,nArg UnUr,eUr UOa,Ak UOa,eGl UPy,As UPy,Oa

29% Arterial

25% Arterial

20% Arterial

15% Arterial

10% Arterial

5% Arterial

11.0 27.4 12.2 0.78 12.6 1.19 6.20 54.0 13.4 13.9 26.7 28.3 4.07 7.44

11.2 26.9 12.5 0.71 12.8 1.19 6.21 53.7 13.6 14.1 26.2 28.0 5.00 6.53

11.4 26.2 13.0 0.59 13.2 1.20 6.24 53.4 14.0 14.4 25.5 27.6 6.45 5.12

11.7 25.5 13.5 0.48 13.5 1.21 6.27 53.0 14.3 14.7 24.7 27.1 7.90 3.70

11.9 24.8 14.0 0.37 13.8 1.21 6.30 52.7 14.6 15.1 24.0 26.7 9.35 2.29

12.2 24.0 14.5 0.25 14.2 1.22 6.33 52.3 15.0 15.4 23.3 26.3 10.8 0.87

1.05 1.68 83.0 4.45 1.61 7.69

1.35 2.04 79.7 5.35 4.52 9.29

1.63 2.41 74.9 6.26 7.30 10.9

1.88 2.80 67.0 7.18 9.96 12.6

2.12 3.20 52.0 8.11 12.5 14.4

13.73

19.68

27.03

35.78

45.94

Flow adjustments, % of observed flow UeAm,nAm 0.84 1.45 UeO2,H2O UeTg,As 84.6 UnCd,eCd 3.88 UnUr,eUr 0.33 6.69 UOa,eGl Residual error (mol/d)2 Objective function 10.66

Integration Interval, d

2.0E-05 1.6E-05 1.2E-05 8.0E-06 4.0E-06 0.0E+00 0

0.5

1

1.5

2

Time, 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.

Regardless of whether inputs were perturbed in a positive or a negative direction, the model achieved a new steady state within 0.05 d, and the new state was maintained thereafter (Figs. 3a–d) leading to the conclusion that the model is stable over a range of inputs. When inputs were returned to their reference values after a perturbation, the model flows returned to their reference values in the same time period. Sensitivity analysis to varying glucagon concentrations are summarized in Fig. 4. Gl output ranged from 9.25 to 19 mol/d as the reference concentration of glucagon was varied from 0.25 to 4. Carbon required to support Gl synthesis was derived primarily from

increased rates of La removal and decreased rates of Asp and Glu release with lesser contributions from Ala, Asn, Pro, and Gln. It was surprising that Ala contributed so much less than La given the common intermediate, Py. This occurred as a result of a greater than expected removal of La, due to a decrease in Ndh concentration as glucagon concentrations increased. Release of Kb (Fig. 4a) and Ac (data not shown) increased as glucagon concentration increased due to a decrease in Oa and an increase in As. The increased rates of carbon release in the form of Ac and Kb, caused a decrease in O2 removal and CO2 release (data not shown). Release of Ur was increased in response to changes in deamination rates of AA used for gluconeogenesis as glucagon was increased. As hormonal regulation of EAA flows was not included in the model, removal of these AA in support of Gl synthesis was not observed. Consequently, the pattern of AA released in the hepatic vein would deviate from portal and arterial concentrations, but those deviations would result from changes in non-essential AA (NEAA) (Fig. 5) and other metabolite concentrations. Hepatic release of total AA declined due to the increased removal of AA in support of Gl synthesis. The effects of glucagon and portal blood flow on predicted patterns of AA in the hepatic vein are

ARTICLE IN PRESS

5.0E-03

0.12

Oa Asp Glu

1.5E-01 1.0E-01

0.0E+00

0.0E+00 0.08

0.1

3.6E-03

0.14

3.1E-03

0.12

2.6E-03

0.1

2.1E-03

0. 0.08

1.6E-03

0.06

1.1E-03

0.04

6.0E-04

0.02

1.0E-04

Py Ak Ndh At Glu

0 .2 0.2

. 0.4

0.6

. 0.8

1

Time (d)

Ak Oa Ndh Glu Asp

Ak Arg Glu Asp Am

0.02 0 0.2

(b)

Time (d)

0

0.04

0

Arg and Ak (mol)

Pool Size (mol)

(a)

0.06

0.06

2.0E-03 1.0E-03

0.04

0.08

3.0E-03

5.0E-02

0.02

0.1

4.0E-03

Asp and Glu (mol)

2.0E-01

Pool o Size (mol) o

0.14

Pool Size (mol)

6.0E-03

0

(c)

281

2.5E-01

Glu (mol)

Pool Size (mol)


0.4

0.6

0.8

1

Time (d)

2.0E-03 1.8E-03 1.6E-03 1.4E-03 1.2E-03 1.0E-03 8.0E-04 6.0E-04 4.0E-04 2.0E-04 0.0E+00

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

(d)

0.2

0.4

0.6

0.8

1

Time (d)

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.

presented in Fig. 5. As noted above, EAA were not generally responsive to glucagon changes, however they are responsive to changes in portal flow. The predicted pattern of EAA changes, whereby Met and Phe are affected to a greater degree than are Ile, Leu, and Val with intermediate changes in His, Thr, Arg, and Lys. Hepatic vein concentrations were predicted to change more dramatically for NEAA and these changes were responsive to both glucagon and portal flow. Relatively large changes were observed in Asp and Glu concentrations with very marginal responses for Cys and Gln. The sensitivity of hepatic vein release and hepatic tissue outflows to independent variations in portal concentrations of metabolites was evaluated, and the results are presented in Table 8. Most outflows were not very responsive to changes in portal concentrations. However, those AA involved in transammination were changed significantly with changes in other nitrogen donors, 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 intermediates were affected by changes in either donor, i.e. La and Ala release each responded to changes in the other metabolite. Gl was most responsive to Pr and Tg concentration changes, with little response to other metabolites. However, this response was small relative to that elicited by changes in glucagon. Changes in Cit concentration decreased the release of Asn, Asp, Gln, Glu, Pro, Ur and La.

5. Discussion The work described herein provides an integrative and quantitative representation of AA metabolism in the liver of a lactating cow. As the data used to parameterize the model were derived from mid-lactation cows, it seems prudent to restrict use of the dynamic model to the same physiological state. However, the model was found to be stable over a broad range of inputs, suggesting that it could be applied to cattle in similar physiological states after appropriate evaluation. The large deficit in carbon (about 13 mol/d), also observed previously by Freetly et al. (1993), is troubling as it represents either an unknown substrate or is indicative of a deficiency in the measurements commonly made. It seems unlikely that it could be caused by bias in any of the measurements currently undertaken. For example, it represents almost a third of the observed carbon outflow in Gl and a quarter of the observed CO2 release. Such large errors in measurement would have been identified. It appears more likely that the deficit is due to failure to measure all the inputs to the organ. Freetly et al. (1993) hypothesized that the deficit in carbon 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. Although Reynolds et al. (2003) observed a positive removal of Tg, the flow was not adequate to cover the apparent deficit.


Hepatic Removal (mol/d)

282 10 5

Ala

0

Gl

Hepatic Removal (mol/d)

Kb

-10

La

-15

Ur

-20

(a)

0

0.5

1

1.5

2

2.5

3

3.5

4

1.5 1

Asn

0.5

Asp Cit

0

Gln

-0.5

Orn

-1

Pro

-1.5

(b) Relative Concentration

Glu

-5

0

0.5

1

1.5

2

2.5

3

3.5

4

3 Arg

2.5

As

2

At

1.5

Ndh

1

Oa

0.5

Py

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.

0 0

(c)

0.5

1

1.5

2

2.5

3

3.5

4

Relative Glucagon Concentration

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.

Hepatic nitrogen balance would be affected if a portion of the carbon deficit comprises nitrogen containing compounds such as peptides (Koeln et al., 1993). Given that there is slightly more nitrogen being removed by liver than could be accounted for in Ur output (Table 7), additional nitrogen removal associated with uptake and catabolism of peptides seems unlikely unless catabolized nitrogen is being lost from the liver in forms other than those considered herein. The branched chain AA may be an exception to this conclusion, as their removal is nearly zero and can be negative (Reynolds et al., 1995, 1999, 2000). Removal of the corresponding keto-acids is not adequate to provide the missing carbon, suggesting that peptides may represent the deficit. 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; defined as CO2/O2). The normal RQ range for an animal is from 0.7 to 1.2 (Kleiber, 1975). The RQ derived for the data used herein was 0.56. However, it was reduced to 0.42 when glucagon concentrations were increased.

Gl synthesis was responsive to hormonal regulation, as is observed in vivo, with the potential for increasing Gl production by at least 5 mol/d when arterial and portal inputs were held constant. Changes in substrate supply in the absence of a change in hormonal signals elicited very little change in Gl production (Table 8). In response to a glucagon signal, La was the largest contributor to increases in gluconeogenesis. As most La is derived from Gl oxidation in peripheral tissues, it is not clear if a 5 mol/d increase is sustainable when the driving factor behind the glucagon signal is an irreversible 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 increases in blood Gl concentration during 14 d infusions of glucagon in lactating cows. The predicted decline in Ndh concentrations associated with increasing glucagon is not supported by previous observations (Ayuso et al., 1986). The predicted reduction appears to be due to a decline in Oa concentrations (Fig. 4), which in turn leads to a decrease in Ndh generation from the tri-carboxylic acid cycle. This inconsistency with observed data suggests that the description of glucogenic carbon entry into the tricarboxylic acid cycle was not adequate. Regulation of pyruvate carboxylase (UPy,As) by As was included in the model as represented by fC, however, the available data were not adequate to define fC accurately and therefore an assumed value of two was adopted. Increasing the


283

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 Ala Am Arg Asn Asp Bu Cit Cys Fa Gln Glu Gly Gy His Ile La Leu Lys Met Phe Pr Pro Ser Tg Thr Trp Tyr Va Val

98.99 69.75 27.43 92.02 79.35 29.24 45.64 95.07 98.34 92.77 91.73 82.82 86.91 45.47 93.04 99.42 72.17 99.41 96.22 88.33 88.91 27.95 89.97 78.44 89.41 93.54 94.43 87.65 26.78 98.73

0.01 8.42 0.04 0.02 0.01 0.01 0.01 0.05 0.02 0.04 0.01 0.01 0.00 0.19 0.00 0.00 1.06 0.00 0.00 0.00 0.00 0.18 0.01 0.19 0.04 0.01 0.03 0.00 0.03 0.00

0.00 0.14 0.04 0.07 9.50 0.08 0.03 0.19 0.00 0.03 0.02 0.02 0.01 0.01 0.00 0.00 0.06 0.00 0.01 0.00 0.01 0.18 0.01 0.01 0.06 0.01 0.00 0.01 0.03 0.01

0.08 1.74 0.51 0.83 2.17 2.52 0.34 2.39 0.03 0.46 0.26 0.34 0.15 0.15 0.06 0.02 0.72 0.02 0.11 0.03 0.15 2.29 0.20 0.16 0.73 0.11 0.06 0.15 0.39 0.07

0.01 0.16 0.04 0.06 0.04 0.02 0.04 0.18 0.00 0.05 9.42 0.06 0.01 0.02 0.01 0.00 0.08 0.00 0.01 0.00 0.01 0.26 0.04 0.01 0.08 0.01 0.01 0.01 0.04 0.01

0.05 1.08 0.26 0.41 0.25 0.10 0.25 1.22 0.02 0.35 0.30 6.24 0.08 0.11 0.07 0.01 0.50 0.01 0.06 0.02 0.10 1.73 0.24 0.10 0.56 0.07 0.04 0.10 0.29 0.05

0.00 0.09 0.02 0.04 0.02 0.01 0.02 0.11 0.00 0.03 0.03 0.03 0.01 0.01 0.01 0.00 0.04 0.00 0.01 0.00 0.01 0.15 9.66 0.01 0.05 0.01 0.00 0.01 0.02 0.00

0.00 0.19 0.46 0.09 0.11 0.03 0.01 0.18 0.00 0.01 0.05 0.03 0.09 0.00 0.01 0.00 0.01 0.00 0.02 0.00 0.01 0.07 0.01 0.05 0.02 0.02 0.01 0.01 0.01 0.01

0.01 0.02 0.06 0.01 0.01 0.01 0.00 0.04 0.00 0.02 0.00 0.01 0.04 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.10 0.00 0.00 0.02 0.00 0.00 0.00 0.02 0.00

0.03 0.13 0.04 0.11 0.08 0.04 0.12 0.32 0.00 0.49 0.06 0.09 0.04 0.06 0.01 0.00 0.26 0.00 0.02 0.01 0.13 1.63 0.05 0.02 0.74 0.05 0.05 0.09 0.27 0.03

0.01 0.04 0.03 0.03 0.04 0.02 0.07 0.08 0.00 0.10 0.02 0.02 0.00 0.02 0.01 0.01 0.09 0.00 0.00 0.01 0.02 0.44 0.02 0.02 0.15 0.01 0.01 0.02 0.07 0.01

0.02 0.76 0.07 0.08 0.08 0.03 0.09 0.21 0.01 0.08 0.04 0.05 0.06 0.22 0.01 0.00 9.27 0.00 0.01 0.01 0.02 0.63 0.04 0.16 0.21 0.03 0.03 0.02 0.11 0.01

0.03 0.03 0.01 0.00 0.01 0.01 0.14 0.01 0.00 0.20 0.00 0.00 0.00 0.01 0.00 0.00 0.04 0.01 0.04 0.00 0.07 0.16 0.00 0.00 0.30 0.00 0.03 0.07 0.35 0.00

9.37 0.02 0.01 0.01 0.01 0.01 0.13 0.01 0.00 0.19 0.01 0.01 0.00 0.01 0.00 0.00 0.04 0.00 0.00 0.00 0.01 0.15 0.00 0.00 0.28 0.01 0.00 0.01 0.04 0.00

a

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

sensitivity to As (via increasing the numerical value of fC, from two) should prevent the decline in Oa concentration and the assocatiated decline in Ndh concentrations. The decrease in Oa concentrations, associated with increasing glucagon concentrations, also results in an increase in predicted As concentrations, which leads to an increase in Kb generation from As via mass action (Fig. 4). Increasing blood propionate concentrations had the opposite effects, causing an increase in Oa (Fig. 3) and a reduction in Kb release (Table 8). A reduction in ketogenesis in association with the provision of additional propionate is consistent with both in vivo (Grohn, 1985) and in vitro (Lomax et al., 1983) observations. Glucagon infusions in lactating cows have been shown to result in a decrease in the blood concentration of both b-hydroxybutyrate and NEFA. The decease in NEFA concentrations also results in a decline in their rate of removal by the liver, which reduces 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 by allowing more As to enter the tri-carboxylic acid cycle. Therefore, the combination of decreasing blood NEFA concentrations and increasing sensitivity of pyruvate carboxylase to As should result in better predictions of Kb release when simulating glucagon infusions. Gluconeogenesis appears to place significant demands on AA supply (Fig. 5). However, the largest impact is on NEAA, which are not thought to be critical in terms of maintaining productive capacity in the lactating cow. The possibility that EAA removal by liver is regulated by endocrines such as insulin or glucagon cannot be ruled out. A number of EAA including Met, Thr and His are glucogenic. Removal of Phe, Arg and His has been observed to increase during glucagon infusion using a perfused rat liver model (Mallette et al., 1969). If transport activity was stimulated by glucagon in the ruminant liver, there could be a significant impact on peripheral availability of these AA. This is of particular interest in early lactation, when intake is not increasing as rapidly as milk production. During this time, EAA



supply may be compromised if removal of EAA by the liver is enhanced by gluconeogenic signals. However, such a mechanism was not included in the current model, as there is no firm evidence to support it at this time. Although EAA removal for gluconeogenic purposes could impact EAA supply significantly, it seems likely that it would have little impact on total Gl flow as the mass of carbon available from the EAA is small relative 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 the composition of AA in blood flowing from the liver (Fig. 5). Concentrations of Met and Phe were affected to the greatest degree. Met is of particular concern, as it is believed to often limit production (NRC, 2001). It is not clear whether the increased flow of absorbed EAA that would normally be associated with a meal contains a pattern of AA that is complimentary to the predicted changes associated with changes in portal flow. Perhaps, hepatic removal of Met and Phe are offset by an absorption pattern that contains greater proportions of Met and Phe than are needed. Metabolite removal by the liver in response to increases in input concentrations ranges from about 1% for Ac, Ile, Leu and Val to about 70% for Asp, Am, Pr and Va. Although Gl release is predicted to respond to substrate supply, the response to a 10% change in portal propionate was slightly less than 0.5%, and responses to similar changes in other substrates resulted in changes in Gl release of less than 0.1% (Table 8). Consequently, it would appear that Gl release is driven primarily by endocrine status. The balance model is useful for interpreting experimental results. Measurement of arterial, portal and hepatic vein concentrations of AA, and energetic substrates provides enough information to allow calculation of a number of intracellular flows with a minimum of assumptions. This model might be used with individual animal data and the estimates derived subjected to similar statistical analyses as the observed values. In this manner, the effects of treatments such as increasing glucagon concentrations could be evaluated with respect to extracellular and intracellular flows and rate parameters. Changes in flow patterns, associated with a particular treatment, might be used to distinguish between regulatory effects and metabolic responses. The dynamic model appears to be fairly robust with respect to input variation and can be defined largely by arterio-venous difference data. Once tested thoroughly against independent data, it could be used to predict changes in metabolite removal and release. Additionally, it provides insight as to simplifications that could be tolerated when incorporating metabolic knowledge into a more aggregated whole animal model.

Acknowledgements This work was funded by a consortium of governmental and industrial organizations, including the Department for Environment, Food and Rural Affairs, the Biotechnology and Biological Science Research Council, Purina Mills Inc., the Milk Development Council of England and Wales, and NUTRECO Inc. Additional aspects of the work were funded through DEFRA contract LS3608. The comments and support of Dr N. E. Smith are greatly appreciated.

Appendix A A.1. Differential and associated equations The following are the differential equations of the model. Flows used are defined in Table 3. Numerical coefficients preceding a flow variable represent standard stoichiometries for conversion of moles of substrate to moles of product (Freetly et al., 1993; Newsholme and Leech, 1986; Salway, 1994). Stoichiometries for eVa metabolism were derived by assuming a 1.0:0.8:1.8 mixture of valerate, isobutyrate, and isovalerate, respectively, based on standard stoichiometries and the observations of Reynolds et al. (1988a) dQPy ¼ UeAla;Py þ UeCys;Py þ UeGy;Py dt 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 ¼ UAk;Oa þ UnAsp;Oa þ UeIle;Oa dt 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 ¼ UeAc;As þ 2UeBu;As þ 8UeFa;As dt 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


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

dQnAsp ¼ UeAsn;nAsp þ UeAsp;nAsp þ UOa;nAsp dt UnAsp;eAsp UnAsp;xPrt UnAsp;eAsn UnAsp;Oa ;

ðA:7Þ

dQAk ¼ UnCit;nArg þ UnGlu;Ak þ UOa;Ak þ UOa;nAsp dt 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 ¼ UAk;nGlu þ 2UnArg;nGlu þ UeAla;Py þ UeCys;Py dt 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 and UeLys,eKb in Eq. (A.9) reflect the stoichiometries associated with conversion of nArg and eLys to nGlu and As, respectively. dQnCit ¼ UeCit;nCit þ UnArg;nCit dt UnCit;eCit UnCit;nArg ;

ðA:10Þ

dQnArg ¼ UeArg;nArg þ UnCit;nArg UnArg;eOrn dt UnArg;xPrt UnArg;nCit UnArg;nGlu ;

ðA:11Þ

dQnAm ¼ UeAm;nAm þ UeAsn;nAsp þ UeGln;nGlu dt þ UeGly;nCd þ UeHis;nGlu þ UeMet;Oa þ UeSer;Py þ UeThr;Oa þ UeTrp;Py

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 RNdh;TNd CNdh UAk;nGlu 0:5UAs;eKb iCNdh 0:67UeFa;eTg UNdh;Ox UnGlu;ePro UOa;eGl UPy;eLa ;

ðA:14Þ

dQNd dQNdh ¼ : ðA:15Þ dt dt Multipliers in Eq. (A.14) reflect the stoichiometries of Ndh production associated with metabolism. The factor 0.67 associated with UeFa,eTg represents the cost of converting Py to phosphorylated glycerol for use in eTg synthesis from eFa. The term RNdh,TNdCNdh/ iCNdh is used to calculate the mixture of b-hydroxybutyrate and acetoacetate that is released from liver as predicted from Ndh concentrations, where RNdh,TNd represents the observed Ndh:(Ndh+Nd) ratio for hepatic tissue in vivo (Baird et al., 1975), and CNdh/ iCNdh is used to reference RNdh,TNd to model concentrations of Ndh. dQAt ¼ UAk;Oa þ UeGy;Py þ UeTrp;Py þ 2UFdh;Ox dt þ 3UNdh;Ox þ 3UnpH;Ox 5UAA;xPrt 0:67UeFa;eTg UeLeu;As 2UeMet;Oa 2UePr;Oa UeTg;As UeVa;Oa

ðA:12Þ

2UnArg;nCit 2UnAsp;eAsn 2UnCit;nArg UnGlu;eGln 2UOa;eGl UPy;Oa ;

ðA:16Þ

dQAd dQAt ¼ ; ðA:17Þ dt dt where UnpH,Ox was used to correct a slight imbalance of intracell pH, with an assumed energetic equivalency of Ndh. UFdh,Ox was assumed to be equal to its rate of synthesis (France et al., 1992):

dQnCd ¼ UAk;Oa þ 2UeGly;nCd þ UeHis;nGlu dt 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 :

dQNdh ¼ 2UAk;Oa þ UeBu;As þ 7UeFa;As dt þ UeGly;nCd þ 2UeGy;Py þ 3UeIle;Oa þ UeLa;Py þ UeLeu;As þ 3UeLys;eKb

UAt;Und 2UeAc;As UeBu;As

þ UnGlu;Ak UAk;nGlu UnArg;nCit UnAsp;eAsn UnGlu;eGln ;

285

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) correspond with the stoichiometries associated with conver-

UFdh;Ox ¼ UAk;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



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

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 Fat Protein Lactose

27.8 1.052 0.862 1.357

29.6 1.124 0.944 1.431

32.1 1.242 0.963 1.556

0.84 0.025 0.032 0.057

0.02 0.004 0.07 0.06

Concentration (g/kg) Fat Protein Lactose

37.76 31.09 48.64

38.48 32.24 48.45

38.62 30.17 48.73

0.692 0.535 0.377

0.47 0.04 0.77

UeO2 ;H2 O 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 Ndh oxidation. 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 ; UnUr;eUr ¼ UnArg;eOrn þ UnArg;nCit þ UnArg;nGlu ;

ðA:20Þ ðA:21Þ

where nUr is a zero pool, since the model assumes that all intracellularly formed urea is instanteously pumped out of the cell into the extracellular pool. A.2. Methods for experiment C7 Four multiparous Holstein–Friesian cows were fitted with catheters in the hepatic-portal vein, hepatic vein, mesenteric vein, and an artery (mesenteric or intercostal) 10 to 14 days post-calving. This surgical preparation allowed measurement of net metabolite uptake and production across the portal-drained viscera (PDV), hepatic and splanchnic beds. The portal catheter of one cow did not give consistent results, therefore only three animals were used to determined PDV and hepatic uptakes; four were used for all other determinations. Beginning week 9 of lactation, cows were fed three diets of grass silage (170 g crude protein (CP)/kg DM):concentrates in a 40:60 ratio (DM basis). The three barleybased concentrates were formulated to contain 105(L), 154(M), or 204(H) g CP/kg; the additional protein was provided by a protected soya [Sopralin, BP Nutrition (UK), Ltd]. Silage and concentrates were given hourly via automated feeders. Diets were fed to meet the energy requirements of the cows at their level of milk production. A 3 3 Latin square design with 3-week periods 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 day of each period. Blood flow was determined using the dye

dilution technique (with para-amino hippuric acid) to enable measurement of net metabolite flow across the PDV and liver. A summary of the results is provided in Table 9.

A.3. Parameter estimation from the balance model Measurements of arterio-venous difference and blood flow were adequate to parameterize the majority of flows in the model although some inconsistencies were observed. The removal of eHis, eIle, eLeu and eVal was negative, which is inconsistent with known biology, and suggests the presence of alternative unmeasured sources for these AA. Therefore, an undefined source was included in the balance model. A number of internal flows could not be defined directly from the available data. There also was a shortage of carbon entering the tissue relative to measured outputs and slightly less Ur output than was required to dispose of the observed nitrogen inputs from AA and Am. In an attempt both to derive the remaining unknown flows and to balance the model as closely as possible to observed values so that intracellular parameters could be calculated, an optimization problem was formulated and solved. The optimization problem included the following equations, which were obtained from the balance model. The differential equations for Ak, As, nAm, nArg, nAsp, nCd, nCit, Ndh, nGlu, nUr, Oa and Py were set to zero and rearranged with known flows (denoted by U) on the # on the left to right and unknown flows (denoted by U) yield the following, respectively. U# Ak;nGlu þ U# Ak;Oa þ U# nArg;nGlu þ U# nAsp;Oa U# nCit;nArg U# Oa;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


U# Oa;Ak U# Py;As 24U# eTg;As ¼ UeAc;As þ 2UeBu;As þ 8UeFa;As þ UeIle;As þ UeLeu;As þ 0:778UeVa;As UAs;eAc UAs;eKb ;

U# Oa;Ak þ U# Oa;eGl U# Ak;Oa U# nAsp;Oa U# Py;Oa ¼ UeIle;Oa þ UeMet;Oa þ UePhe;Oa ðA:23Þ

U# Ak;nGlu þ U# nArg;nCit U# eAm;nAm ¼ UeAsn;nAsp þ UeGln;nGlu þ UeGly;nCd þ UeTrp;Py þ UnGlu;Ak UnAsp;eAsn UnGlu;eGln ; ðA:24Þ U# nArg;nCit þ U# nArg;nGlu U# nCit;nArg ðA:25Þ

U# nAsp;Oa ¼ UeAsn;nAsp þ UeAsp;nAsp þ UOa;nAsp UnAsp;eAsn UnAsp;eAsp UnAsp;xPrt ;

ðA:26Þ

U# nArg;nCit þ U# nCd;eCd þ U# Py;Oa U# Ak;Oa U# Oa;Ak U# Oa;eGl U# Py;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

U# nCit;nArg U# nArg;nCit ¼ UeCit;nCit UnCit;eCit ;

ðA:28Þ

U# Ak;nGlu þ U# Ndh;Ox þ U# Oa;eGl 2U# Ak;Oa U# nArg;nGlu U# nCit;nArg U# Oa;Ak U# Py;As 23U# eTg;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;eKb RNdh;TNd CNdh =iCNdh 0:67UeFa;eTg UnGlu;ePro UPy;eLa ;

þ UePr;Oa þ UeThr;Oa þ UeTyr;Oa þ 0:222UeVa;Oa þ UeVal;Oa UOa;nAsp ;

ðA:32Þ

U# Py;As þ U# Py;Oa U# eTg;As ¼ UeAla;Py þ UeCys;Py þ UeGy;Py

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

¼ UeArg;nArg UnArg;eOrn UnArg;xPrt ;

287

þ UeLa;Py þ UeSer;Py þ UeTrp;Py 0:33UeFa;eTg UPy;eAla UPy;eLa :

ðA:33Þ

UeTg;As was included as the unknown source of carbon, as it was not measured in the experiments used for parameter estimation, and previous work indicates that it may be used by the liver (Reid et al., 1979; Reynolds et al., 2003), though the rates of use would appear to be inadequate to supply all of the missing carbon. Although UeAm,nAm, UnCd,eCd, UnUr,eUr and UOa,eGl were measured, they were included in the above equations as unknowns to allow manipulation to achieve carbon and nitrogen balance, which could not be achieved when they were constrained to the observed values. As the above flows were manipulated, At and Ndh balance was maintained by automatic adjustments of the flow UAt,Und. To minimize substrate cycling, the problem was originally constrained to solutions where observed and predicted uptakes of O2 were equal. However, such a constraint prevented the problem from converging. The use of 3.0 and 2.0 instead of 2.5 and 1.5 (Nelson and Cox, 2000) for the stoichiometries of At synthesis from Ndh and Fdh would account for some of the error in predicting O2 consumption. Consequently, the constraint was converted to a residual error representation and added to the objective function yielding the following equation: 2 2 obj ¼ U# eAm;nAm UeAm;nAm þ U# eO2 ;H2 O UeO2 ;H2 O 2 2 þ U# eTg;As UeTg;As þ U# nCd;eCd UnCd;eCd 2 2 þ U# nUr;eUr UnUr;eUr þ U# Oa;eGl UOa;eGl ; ðA:34Þ

ðA:29Þ where U represented the experimentally observed flows and U# the predicted ones. Additional assumptions were that nVol was 10.4l (Freetly et al., 1993) and RNdh,TNd was 0.91. The problem was solved by minimizing Eq. (A.34) subject to Eqs. (A.22)–(A.33) and the restriction that all U# were nonnegative.

U# nCit;nArg U# Ak;nGlu 2U# nArg;nGlu U# nAsp;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 ; U# Ur;eUr U# nArg;nCit U# nArg;nGlu ¼ UnArg;eOrn ;

References ðA:30Þ ðA:31Þ

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