Development of a mechanistic model of liver metabolism in the lactating cow

Agricultural Systems 41 (1993) 157--195

Development of a Mechanistic Model of Liver Metabolism in the Lactating Cow H. C. F r e e t l y * , J. R. K n a p p , C. C. C a l v e r t & R. L B a l d w i n Department of Animal Science, University of California, Davis, California 95616, USA (Received 13 January 1992; accepted 18 March 1992)

A BSTRA CT In order to accurately predict nutrient requirements and utilization in ruminants, an assessment of hepatic nutrient utilization must be made. A mechanistic, dynamic, deterministic model of liver metabolism was constructed to integrate available data toward this end. The objective of the modeling exercise was to develop a mechanistic model of ruminant liver metabolism, based on known metabolic pathways, which was capable of predicting whole-organ metabolism and simulating 14C-radiotracer kinetics. The program was written in the Advanced Continuous Simulation Language and run on a VAX computer. ,4 fourth-order Runge-Kutta procedure was used for numerical integration. Development of kinetic arguments and subsequent parameterization relied heavily on in-vivo input:output data and underlying biochemistry. The model was evaluated using a combination of behavioral analyses, sensitivity analyses, and challenges from independent data sets.

INTRODUCTION The liver of the ruminant animal is actively involved in nutrient partitioning, as evidenced by its involvement with such important biological functions as gluconeogenesis and plasma protein, fatty acid, and urea synthesis. Its contribution to daily energy expenditures is high, 24% in * Present address: US Department of Agriculture, Agricultural Research Service, Roman L. Hruska US-Meat Animal Research Center, Clay Center, Nebraska 68933, USA. 157 Agricultural Systems 0308-521X/92/$05.00 © 1992 Elsevier SciencePublishers Ltd, England. Printed in Great Britain

158

H. C Freetly, J. R. Knapp, C C. Calvert & R. L. Baldw&

the lactating cow (Smith, 1967). A complete knowledge of liver metabolism in changing nutrient and physiological states is required to understand whole animal metabolism and subsequent nutrient requirements. Both in-vivo and in-vitro experimentation have been performed to investigate nutrient partitioning, but the data have not been integrated to allow for quantitative and dynamic evaluations of the adequacy of current concepts and data regarding liver metabolism. Interpretations of input: output data so far have focused on measuring differences in products over changing physiological states. The majority of these data have been analyzed solely with traditional statistical methods. Many of the in-vitro studies that have used 14C-radiotracers were analyzed in a similar manner. While these analyses have been useful in increasing our knowledge of liver function, they have not improved our understanding of the underlying biology that effects these changes. In-vivo and in-vitro 14C-radiotracer studies have also been conducted to determine the relative contributions of gluconeogenic compounds to glucose synthesis. Interpretation of these data, however, has been confounded by the randomization of label in the tricarboxylic acid (TCA) cycle and withdrawal of label from the TCA cycle to form alternate products. In an attempt to account for randomization of label, Goebel et al. (1982) proposed a mathematical model to describe the distribution of isotopic carbon atoms through the TCA cycle. Similar models were proposed by Weinman et al. (1957), Hetenyi (1982), Wiltrout and Satter (1972), and Thompson (1971). Steinhour and Bauman (1988) later improved on these models by accounting for alternate paths by which label is removed from the TCA cycle. While these models have had varying degrees of success in describing radiotracer flow and distribution, they do not describe metabolism of liver. Waghorn (1982) developed mathematical models of bovine mammary and liver metabolism to describe metabolism in these tissues. These models were developed to support whole-animal models and have been partially successful in accommodating patterns of nutrient metabolism. However, these models were based upon mass action kinetic arguments and therefore can yield spurious results. The following study uses a mechanistic modeling approach that incorporates saturation kinetic arguments and both input:output and laC-radiotracer data. The model is dynamic and deterministic in nature. There are five basic steps in the development of a mechanistic model. The first step is to set the objective. The objective of the modeling exercise was to develop a mechanistic model of ruminant liver metabolism, based on current knowledge of liver function, which would be capable of simulating whole-organ metabolism including the metabolism of

159

A mechanistic model of liver metabolism

=4C-radiotracers. The reason for including the simulation of 14Cradiotracers was to provide a vehicle for extending interpretations of radiotracer data to estimation of key kinetic parameter values not directly measurable. The second step in modeling is to identify the elements of the biological system that are to be modeled. This step usually results in the formulation of a block diagram. The block diagram developed in the current study is presented in Fig. 1. Metabolites involved in regulatory processes or which have multiple inputs and outputs were chosen to be state variables. The third step in modeling is to develop mathematical equations which describe relationships among the elements to be modeled, and the fourth is to collect numerical inputs required to implement the equations based upon literature surveys or laboratory work. Descriptions of the equations used and their numerical inputs are in the following text. The fifth step is evaluation of the model ............

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with respect to the objective. Methods of model evaluation include behavioral analyses, sensitivity analyses, and challenges with independent data sets. A combination of these three was used in the evaluation phase of this study. Sample evaluations follow the description of the model.

OVERALL STRUCTURE A N D NOTATION The model was developed to describe liver metabolism in a 638-kg Holstein cow producing 30 kg of milk per day. Liver weight was set at 10.6 kg based on regression equations developed by Smith and Baldwin (1974). The model is represented schematically in Fig. 1. The model was programmed in the ACSL (Advanced Continuous Simulation Language; Mitchell & Gauthier Associates, 1986). ACSL was designed to model systems described by time-dependent, nonlinear differential equations and thus is well suited for use in modeling metabolic processes. The model is also capable of running with SimuSolv ~ (The Dow Chemical Company, Midland, MI), a combination of ACSL and Run-Time Command Language used in sensitivity analyses, for the purpose of estimating parameters and comparing models. Metabolites in the model are expressed in moles, mass in kg, volume in liters, and time in days. Broken line boxes in Fig. 1 indicate input metabolites. Solid boxes represent intermediate metabolites. Each of these boxes is described by a differential equation with respect to time. In the reference state defined for model development and represented in Fig. 1, the difference equations equal zero. Arrows that connect boxes and the numbers on the arrows are reaction rates or fluxes of substrate going to product in mol/day in the reference state. In the model, rates of conversion of substrate to product are denoted as substrate going to product (UA.B), such that UGP,Fp is the flux of glucose-6-phosphate (GP) going to fructose-6-phosphate (FP) in mol/day. Tables 1 and 2 list, respectively, metabolite pool and substrate concentrations in the reference state. Seventeen equations were used to describe fluxes between state variables (Table 3). Equations (1)-(5) are mass action equations or modifications thereof. KA, 13 (day l) describes the coefficient for the mass action argument. Mass action equations can be used to describe hyperbolic functions when substrate concentrations are low with respect to the affinity constant. Equations (6)-(17) describe saturation kinetic arguments. Most of these are Michaelis-Menten in form. When multiple substrates are used, interaction terms were omitted as available data are not normally adequate (due to variance) to define these. Equations (10) and (16) are

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TABLE 3 Equation Forms of Metabolic Transactions Equation number (1) (2) (3) (4) (5)

(6) (7) (8) (9) (10) (11) (12) (13) (14)

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sigmoidal in form. This is achieved by raising the affinity c o n s t a n t : s u b strate ratio ( k / [ S ] ) to a power function (e). F o r a given metabolite exchange (UA,B), VA,B (mol/day) describes the m a x i m u m capacity o f the flux, a n d k n (mol/liter) describes an affinity c o n s t a n t for substrate S n (mol/liter). The value Jn is s y n o n y m o u s to kn, in t h a t it represents a competitive inhibitor o f a reaction rather t h a n a substrate or activator. T r a n s a c t i o n s a m o n g metabolites are given in Table 4. In cases where in-vivo values were n o t available for affinity or inhibition constants (kn a n d Jn), these were set equal to steady-state c o n c e n t r a t i o n s o f m e t a b o lites, M a x i m u m capacities (VA,B) were calculated based on flux rates a n d affinity constants. R e a c t i o n s representing single enzymes are described by e q u a t i o n s t h a t represent their behavior. In transactions where multiple enzymes are aggregated, the e q u a t i o n usually represents behavior o f the rate-limiting e n z y m e in t h a t effectors o f that reaction are represented explicitly. State variables

Each pool is described by a differential e q u a t i o n with respect to time. State variables are schematically represented in Fig. 1 by closed boxes.

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The differential or difference equations take the general form d(pool) = inputs - outputs dt where inputs are metabolites and effectors that determine fluxes, outputs are metabolites formed, and t denotes time in days. The differential equations are solved using the fourth order Runge-Kutta numerical integration method. In order to simulate ~4C-radiotracer pools, a separate differential equation is written for each carbon atom in the metabolite. Inputs and outputs of radiotracers are a function of the specific activity ( S A ) of the precursor pool multiplied by the flux (UA,B)- An example of this type equation is the differential equation for 14C-l-triose phosphate (TPI). dTP1 = (SAGp4 X 0.222 X Ucp,Ru -- SATp~ x 0.222 x dt UGp,RU) + SAAGPI X UAGP,Tp + SApEl X UpE,TP + (SAFp 3 x UFP,TP + SAFp 4 x UvP, Tp) -- SATp I X UTP, FP - - S A T p I x UTP,A G P - SATp I X UTp,pE

The specific activity of a given carbon is defined as the moles of ~4C in a given position divided by the total moles of substrate. The specific activity of TP1 is calculated as SATp ! -- TP1/TP

Numerical coefficients described stoichiometric flows of the carbon atoms and are not modifiers of specific activity. By using the above representation, the model allows for coordination between input:output relationships and radiotracer kinetics. Representing the flux of radiotracers as a function of specific activity assumes that there is no discrimination between labeled and unlabelled metabolites. The above strategy has proven to be superior to those used in earlier tissue models where independent equations for labeled and unlabelled metabolites were used (Baldwin et al., 1989). Earlier radiotracer models and unstable solutions due to the small pool sizes of radiolabeled species.

Model inputs and outputs The model defines 15 different entry points for carbon compounds: glucose, glycerol, lactate, propionate, isobutyrate, valerate and 2-methylbutyrate, butyrate, isovalerate, acetate, free fatty acids, carbon dioxide, alanine, aspartate, glutamate, and other amino acids. These are identified in Fig. 1 by dashed boxes. The model also allows 14C to be used as a tracer of each of the carbons of each substrate except other amino

A mechanistic model of liver metabolism

167

acids, long chain fatty acids, valerate and 2-methylbutyrate, and isobutyrate. Long chain fatty acids can be labeled only in the carbon one position. The model accommodates nine products: glucose, lactate, acetate, free fatty acids, carbon dioxide, ketone bodies, triacylglyceride, glutamate and protein. All of these except ketone bodies, triacylglyceride and protein act as both substrates and products. Ketone bodies, triacylglyceride and export protein are terminal products that require metabolism in peripheral tissues before re-entering the liver. These products are represented schematically in Fig. 1 by boxes made with lines and dots.

EQUATIONS FOR METABOLITE POOLS Steady-state concentrations for metabolite pools were calculated based on literature values (Tables 1 and 2) and distributions of metabolites in intracellular or extracellular fluids. Hexose and triose metabolism

Glucose-6-phosphate metabolism, GP The differential equation for GP is dGP dt - UeGL,GP + UFp,GP- UGP,RU -- UGpxGL- UGP,Fp

(54a)

The differential equations for '4C-GP are dGPn = S A G L n dt

X UeGL,GP "4- SAeGLi X UeGL, GP q- S A F p i x UFP,G P -- S A G p i X UGP, RU - - S A G p i )< UGP, eGL - S A G p i )< UGP, FP

(54b)

where i = 1, 2, 3, 4, 5 or 6. The differential equations that describe the '4C pools of GP allow uniformly labeled '4C-glucose (eGLU) to be used as a tracer: SAGL n =

0" 167 X eGLU / eGL

Respective inputs to GP from glucose and fructose-6-phosphate are defined by eqns (20) and (33) (Table 4). Outputs defined by eqns (35), (36) and (37) (Table 4) are to ribulose-5-phosphate (via 6-P-gluconolactone), glucose, and fructose-6-phosphate. The flux UG1,,Ru is a function of GP and NP. The ratio of NP/initial NP is raised to a power e (= 2) to increase sensitivity to NP. NP is a product of fatty acid synthesis (Ucc, FA). NADPH required for fatty acid

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H. C, Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

synthesis is generated by UGP, RU and UIC, AKG. In the reference state, 60% o f N A D P H is from UGP, RU and 40% comes from UIC,AI~. Fructose-6-phosphate metabolism, FP

The differential equation for FP is d F P = UGP,FP + UTP,FP X 0'5 + UGP,RU X 0"667 -- UFp,O p d t

UFP, TP

(55a)

The differential equations for 14C-FP are dFP1 = S A o p 1 x UGP, FP d- S A T p 3 x UTP, FP X 0 . 5 -F UFP, GP - -

(55b)

dFP2 = SAGp2 x UGp,FP + SATp2 X UTP, Fa X 0"5 qdt S A G p 3 x 0 . 4 4 4 x UGP, RU - - S A F p 2 x UFP, GP - SAFp2 X UFp,Tp

(55C)

dt

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dFP3 =SAGe 3 x UGp, FP + SATp 1 X UTP,FP X 0"5 + dt SAGp2 X 0.444 X UGP,RU + SATp3 X 0.222 X UGP, RU - - S A F p 3 x

UFP, GP - - S A F p 3 x

(55d)

UFP, TP

dFP4 = SAGp 4 X UGP, FP q" S A z p 1 X UTP, FP X 0"5 + dt

SAGp 3 X 0 " 4 4 4 X UGP, RU + SATp 1 X 0 . 2 2 2 X

(55e)

UGP, RU -- SAFp 4 X UFP, Ga -- SAFp 4 X UFP, TP

dFP5 = SAGp 5 X UGP, FP -}- SATp 2 X UTP, FP X 0"5 + dt SAGp4 x 0.444 x UGr, Rv + SATp2 X 0.222 X UGP, R U -- S A F p 5 x

UFP, GP - - S A F p 5 X UFP, TP

dFP6 = SAGp 6 X UGP, F P -I- SATp 3 X UTP, F P X 0"5 ddt SAGp 6 x 0.444 X UGP,RU + SATp 3 x 0.222 X UGP, RU - - S A F p 6 x

(55f)

(55g)

UFP, GP -- S A F p 6 X UFP, TP

Respective inputs to FP from glucose-6-phosphate, triose phosphate, and ribulose-5-phosphate are defined by eqns (36), (37) and (52). FP is converted to glucose-6-phosphate and triose phosphate. The UTp,Fp flux describes moles o f TP going to FP. By convention, fluxes are represented as moles of substrate going to product. Since 2 mol o f TP are required to make 1 mol o f FP, the flux was multiplied by the stoichiometric coefficient 0-5 to correct for moles o f F P formed from TP. The a m o u n t of FP formed from ribulose-5-phosphate is calculated by multiplying a stoichiometric coefficient (0.667) by UGI,,Ru flux (eqn (37)). For simplicity, ribulose-5-phosphate is not a state variable. Rather, carbon that enters the pentose cycle is converted to the ultimate

A mechanistic model of liver metabolism

169

products of the pathway, fructose-6-phosphate, triose phosphate and carbon di-oxide. There is a net production of TP in the pentose cycle. The terms SATpi × 0"222 X UGp,Ru provide the necessary stoichiometric coefficients required for incorporation of TP tracer into FP resulting from pentose cycle reactions. Outputs from FP are defined by eqns (33) and (34).

Triose phosphate metabolism, TP The differential equation for TP is dTP = UGP,RU X 0"333 + UAGP,Tp 4- UpE,TP 4- UFP, TP X dt

2"0-

UTP, F P -- UTP, A G P -

(56a)

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Thc differentialequations for ~4C-TP are

dTP1 = SAop4 x 0-222 X UGP, RU - - S A T p I X 0.222 X dt UCp,RU + SAAopl X UAOP,Tp + SApEI X UpE,TP + S A F p 3 x UFP, TP 4- S A F p 4 x UFP, TP - - S A T p I X UTP, FP - - S A T p I x UTP, A G P - S A T p I X UTp, p E

dTP2 = S A G p 5 X 0.222 x UCp,RU - S A T p 2 x 0.222 x dt Ucp,gu + SAAGP2 X UAGP, Tp 4- S A p E 2 X UpE, Tp 4S A F p 2 X UFP, TP 4- S A F p 5 X UFP, TP - - S A T p 2 x UTP, FP - - S A T p 2 X UTP, AGP -- S A T p 2 X UTp, pE

dTP3 = S A G p 6 X 0.222 x UGP, RU -- S A T p 3 x 0.222 x dt UGp,Ru + SAAGP3 X UAGP, Tp 4- S A p E 3 x UpE,T P 4S A F p I X UFP,T P 4- S A F p 6 X UFP, TP - - S A T p 3 x

(56b)

(56c)

(568)

UTP, Fp - - S g T p 3 x UTP, AGP - - S g T p 3 x UTp, pE

Triose phosphate includes the glyceraldehyde-3-phosphate and dihydroxyacetone phosphate pools. Combining these two pools implies that the two pools are in isotopic equilibrium due to the very high activity of triose phosphate isomerase. Inputs to TP from ribulose-5-phosphate (a function of the UGP, RU flux), a-glycerol phosphate, phospho-enol-pyruvate, and fructose-6-phosphate are described by eqns (1), (34), (37) and (46). Triose phosphates are converted to fructose-6-phosphate, a-glycerol phosphate, and phosphor-enol-pyruvate. The flux UpE,TP is an aggregated step that incorporates the reactions catalyzed by glyceraldehyde-3-phosphate dehydrogenase, 3-phosphoglycerate kinase, 2-phosphoglycerate phosphatase, and enolase. Equation (46) incorporates effects of substrates required by these enzymes, including PE, NDH and AT. As described earlier for the contribution of ribulose-5-phosphate to FP, the contribution of RU to TP is 0-333 ×

H . C . Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

170

UGP, RU . UFP,T P is multiplied by 2.0 to account for the formation of 2 mol of TP for every mole of FP converted. Outputs are described by eqns (51), (52) and (53). As described for UpE,TP, UTp, p E is an aggregation of reactions.

Alpha-glycerol phosphate metabolism, A GP The differential equation for AGP is dAGP dt

-

UGY, AGP -t- UTP,AGP -- UAGP, TP - - UAGP, TG

(57a)

The differential equations for 14C-AGP are dAGPi dt

= SAeGYi X UeGY, AGP -1- S A T p i x UTP, AGP -SAAGPi X U A G P , T p - SAAGPi X UAGP,TG

(57b)

where i = 1, 2 or 3. Respective inputs to AGP from glycerol and triose phosphate are described by eqns (23) and (51). Alpha-glycerol phosphate is converted to triose phosphate and is used in triacylglyceride synthesis. Outputs are described by eqns (1) and (57c): UAGP, TG =

0.333

x

UFA,TG

(57c)

The stoichiometric coefficient, 0.333, is derived from the assumption that liver makes only triacylglyceride and for every 3 mol of fatty acids incorporated, 1 mol of a-glycerol phosphate is used. This assumption implies that availability of fatty acids rather than a-glycerol phosphate is rate-limiting.

Phospho-enol-pyruvate metabolism, PE The differential equation for PE is dPE dt

-

UTp'PE q- U D C ' P E - U p E ' P Y - UpE'TP

(58a)

The differential equations that describe the laC transactions for PE are dPEi dt

= SATp i X UTp, p z + S A D c i X UDC, PE - - SApE i X UpE, pY - - SApE i X UpE, TP

(58b)

where i = 1, 2 or 3. Inputs to PE from triose phosphate and dicarboxylic acid are described by eqns (13) and (53), Phospho-enol-pyruvate conversion to pyruvate and triose phosphates are defined by eqns (45) and (46).

A mechanistic model of liver metabolism

171

Pyruvate metabolism, P Y The differential equation for PY is

dPY dt

= UpE, pY + UeLA,Py + UAL,PY + UCT,PY + UAA,PY -UpV,eLA- Upy, A L - Upy, M C - Upy, DC

(59a)

The differential equations for 14C-pY are d P Y 1 = l A p E l X UpE, pY + SAeLAI X UeLA,aY + SAAL 1 X dt UAL, PY + S A c T 6 x UCT,PY -- S A p y I )< Upy,eLA -S A p y I X Upy, AL - - S A p y 1 X Upy,M C - - S A p y I X

(59b}

Upy, DC d P Y 2 = S A p E 2 x UpE, pY + SAeLA2 )< UeLA,Py + SAAL 2 X dt UAL, Pv + S A c T 3 >( UCT,Pv -- S A p y 2 X Upy,eLA -S A p y 2 X Upy, g L - - S A p y 2 >( Upy, MC - - S A a y 2 X

(59c)

UPY, DC d P Y 3 = S A p E 3 x UpE, pv + SAeLA3 X UeLA,Py + SAAL 3 )< dt UAL, PY + S A c T 2 X UCT,PY -- S A p y 3 )< Upy,eLA -S A p y 3 X Upy,A L - - S A p y 3 X Upy, MC - - S A p y 3 X Upy, DC

(59d)

Inputs to PY from phospho-enol-pyruvate, lactate, alanine, amino acids and citrate are defined by eqns (4), (10), (26), (45) and (68b). Pyruvate conversions to lactate, alanine, mitochrondrial acetyl-CoA and dicarboxylic acid are described by eqns (47), (48), (49) and (50).

Tricarboxylic acid cycle

Dicarboxylic acid metabolism, DC The differential equation for DC is

dDC dt

= Upy, DC + UAS,DC + UAK,DC + UpC,DC + UAA,DC --

UDC,AS-UDC,PE- UDC,CT

(60a)

The differential equations for 14C-DC are dDC1 =0.5 x SApy I x Upy,Dc + 0-5 X S A c D U X Upy, DC + dt 0"5 X SAAs I X UAS,DC 4- 0"5 X SAAs 4 X UAS,DC +

(60b) 0"5 X SAAK 2 X UAK, DC -'l- 0 ' 5 X SAAK 5 X UAK, DC + 0.5 x SApc 1 × UpC, DC 4- 0"5 X S A c D U X UpC, DC -S A D c I X UDC, AS - - S A D c 1 X UDC, PE - - S A D c I x UDC,Cw

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H. C Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

dDC2 = 0 - 5 x SApy 2 )K Upy,DC + 0"5 X S A p y 3 X Upy, DC + dt

0.5 x SAAs 2 x UAS.DC + 0"5 X SAAs 3 x UAS,DC + 0"5 X SAAK 3 X UAK,DC Jr 0"5 X SAAK 4 X UAK,DC + 0"5 X S A p c 2 X UpC,DC + 0"5 )K S A p c 3 x

(60c)

Upc, DC --

S A D c 2 x UDC,AS -- SADc 2 X UDC.PE -- S A D c 2 X UDC,CT

dDC3 =0-5 × SApy2 x Upy,DC + 0"5 × SApy 3 X Upy,Dc + dt 0.5 X SAAs 2 x UAS,DC q- 0 ' 5 X SAAs 3 ;K UAS,DC q-

(60d)

0 ' 5 X SAAK 3 X UAK,DC + 0"5 X SAAK 4 X UAK,DC + 0"5 X S A p c 2 X UpC.DC + 0 ' 5 X S A p c 3 X UpC,DC -SADc 3 x UDC,AS - - S A D c 3 X UDC.PE - - S A D c 3 x UDC,CT

dDC4 =0.5 × SAp¥~ × Upy.DC + 0"5 × SAcD U X Upy,DC + dt 0'5 × SAAs I × UAS,DC + 0"5 × SAAs4 × UAS.DC+ (60e)

0"5 X SAAK 2 X UAK,DC q'- 0"5 X SAAK 5 X UAK,DC "40"5 X S A p c I X UpC,Dc + 0"5 X S A c D U X UpC,DC -SADc 4 x

UDC,AS -- S A D c 4 x UDC,PE -- S A D c 4 x UDC,CT

Succinyl-CoA, succinate, fumarate, malate and oxaloacetate are represented in aggregate as the dicarboxylic acid pool. It is generally considered that these metabolites interchange and equilibrate rapidly. The differential equations for J4C reflect randomization of radiotracer in the symmetrical intermediate succinate. These simplifications will not compromise the radiotracer model as long as isotopic equilibria among these metabolites are achieved. Redox units and energy charge are accommodated in the stoichiometric coefficients in the model (see below). The size of the dicarboxylic acid pool was set to the sum of oxaloacetate and malate. Inputs to DC from pyruvate, aspartate, propionyl-CoA, amino acids and a-ketoglutarate are defined by eqns (2), (5), (44), (48) and (68c). Dicarboxylic acid conversions to aspartate, phospho-enol-pyruvate and citrate are defined by eqns (11), (12) and (13). Citrate metabolism, C T The differential equation for CT is

dCT d t - UDC'CT + U I C ' C T - UCT'AK -- UCT,IC -- UCT,PY

(61a)

The differential equations for 14C-CT are dCT1 = S A D c 4 X UDC.CT + S A i c 1 × UIC,C T - SAcTJ X dt

dCT2 dt

UCT,AK

S A c T 1 X UCT,IC - - S A c T 1 X UCT,PY

= S A D c 3 x UDC,CT + S A I c 2 X Vlc, c T -- S A c T 2 X UCT.AK -- S A c T 2 x UCT,IC -- S A c T 2 x VcT, PY

(61b)

(61c)

A mechanistic model of liver metabolism

dCT3

= S A D c 2 )< VDc, c T + S A i c 3 X V i c , c T UCT, AK - - S A c T 3 X UCT,1C -

dt

d C T 4 = SAMc 2 X VDc,c T + SAIc 4 X dt UCT, A K - - SAcT 4 X UCT, IC dCT5 = SAMc I dt UCT, A K

>( --

Vlc, c T - - S A c T 4 x

(61d) (61e)

S A c T 4 X VcT, Pv

UDC,CT + SAIc 5 )< VIc, c T SAcT 5 X UCT, IC - - S A c T 5 X

d C T 6 = SADc I × UDC,CT + SAIc6 dt UCT, A K - - SAcT 6 X UCT, IC

SAcT 3 X

S A c T 3 X VcT, Pv

173

SAcT 5 X VcT, PY

X UIC, CT - - S A c T 6 X - - S A c T 6 X UCT, PY

(61f) (61g)

Inputs to C T from isocitrate and mitochondrial acetyl-CoA plus dicarboxylic acid are described in eqns (12) and (40). Outputs of C T to a-ketoglutarate, cytoplasmic isocitrate and pyruvate plus cytoplasmic acetyl-CoA are described by eqns (8), (9) and (10).

Cytoplasmic isocitrate metabolism, IC The differential equation for IC is dlC dt

-

U C T ' I C - UIC'CT- UIC'AK

(62a)

The differential equations for ~4C-IC are dlCi d t - SAcxi × Ucx, ic - SAlci × UIC,C T - SAici × UXC,AK

(62b)

where i -- 1, 2, 3, 4, 5 or 6 for C T and IC. The cytoplasmic isocitrate pool allows for generation of N A D P H . Since data on isocitrate concentration are not available, a concentration was assigned. Input to IC from citrate is described by eqn (9). Equations (40) and (41) describe IC conversion to citrate and a-ketoglutarate.

Alpha-ketoglutarate metabolism, AK The differential equation for A K is dAK dt

= UCT, AK q- UIC, AK -4- UGU,AK d- UAA, AK - - UAK, OC - UAK, G U

(63a)

The differential equations for 14C-AK are dAKi dt

= S A c T i X UCT, AK + S A t c i X U1C,AK -]- S A G u i X UGU, AK - - S A A K i X UAK, DC - - S A A K i X UAK, G U

where i = l, 2, 3 4 or 5 for CTi, ICi, G U i and AKi.

(63b)

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H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldw&

Alpha-ketoglutarate formation from amino acids are described by eqns (8), glutarate forms dicarboxylic acid and from AK are described by eqns (2) and

citrate, isocitrate, glutamate and (38), (41) and (68d). Alpha-ketointracellular glutamate. Outputs (3).

PropionyI-CoA metabolism, PC The differential equation for propionyl-CoA is dPC dt - UePR'PC + UeVA'PC + UelB'PC + U A A ' P C - UpC'DC

(64a)

The differential equations for 14C-PC are dPCi dt - SAePRi X VpR,p C -- S A p c i >( Vpc, o C

(64b)

where i = 1, 2 or 3 for PR and PC. Inputs to PC from propionate, isobutyrate, amino acids and volatile acids are defined in eqns (24), (27), (29) and (68e). Since data are not available to determine propionyl-CoA concentration, a concentration was assigned. Equation (44) describes propionyl-CoA conversion to dicarboxylic acid. The radiotracer equations only provide for input of labeled propionate.

Mitochondrial acetyl-CoA metabolism, M C The differential equation for MC is dMC = Upy,MC -~- UeAC,MC + 8 ' 0 X UFA, MC + 2.0 × dt UeBU,MC+ UelV.MC q- UeVA,PC q- UAA,MC -- UDC,CT -UMC, KB

(65a)

UMC,eAC

The differential equations for ~4C-MC are dMC1 = SApyz × Upy,Mc + SAcAcl × UeAC,MC + SAFAI × dt 8-0 × UFA,MC+ SAeuu1 × UeBU,MC + SA~Bu3 × UeBU,MC S A M c I X UDC,CT - - S A M c I x UMC, KB - - S A M c I >4 UMC,eAC

dMC2 = SApy3 × Upy,MC + SAcAc2 × UeAC,MC + SAeBu2 × dt UeBU,MC+ SAeuu4 )< UeBU,MC -- S A M c 2 )< UDC,CT - - S A M c 2 X UMC,KB

(65b)

(65c)

S A M c 2 X UMC,eAC

Inputs to MC from pyruvate, acetate, volatile acids, butyrate, isovalerate, amino acids and fatty acids are described by eqns (15), (18), (25), (29), (31), (50) and (68f). The outputs citrate, ketone bodies, and acetate are described by eqns (12), (42) and (43).

A mechanistic model of liver metabolism

175

Lipid metabolism Cytoplasmic acetyl-CoA metabolism, CC The differential equation for CC is

dCC _ UCT,CC + dt

UeAC, CC __ UCC, FA

(66a)

The differential equations for 14C-CC are dCCI = SAcT5 × UCT,Pv + SA~Acl × U~Ac,cc -- SAccl × dt UCC,FA dCC2 = S A c T 4 × UCT,PY q- SAeAc2 × UeAC,CC -- S A c c 2 × dt Ucc, FA

(66b)

(66C)

Inputs to CC from citrate and acetate are described by eqns (10) and (14). Cytoplasmic acetyl-CoA conversion to fatty acids is described by eqn (7). Fatty acid metabolism, FA The differential equation for FA is

dFA = UeFV,VA + 0"125 × UCC,FA + UeUL,V A - UVA~MCdt UFA,e F F - UFA,TG

(67a)

The differential equation for 14C-FA is dFA1 dt

=SAeFFI × UeFF,FA -4- SAcc I × 0"125 ×

Ucc, vn SAFA1× UFA,~C- SARA1 × UVAxVF- SAFA1 ×

(67b)

UFA,TG

Inputs to the fatty acid pool from free fatty acids, undefined lipids and cytoplasmic acetyl-CoA are described by eqns (7), (19) and (28). Outputs to free fatty acids, mitochondrial acetyl-CoA and triacylglyceride are described by eqns (30), (31) and (32). Estimates of free fatty acid uptake A-V difference data and estimates of the capacity of liver for de novo synthesis of fatty acids are not sufficient to account for free fatty acid oxidation and triacylglyceride synthesis. In order to balance the model, free fatty acids are provided from an undefined source. These undefined lipids (eUL) may be cholesterol esters of phospholipids. A concentration equal to free fatty acids was assigned to undefined lipids.

Nitrogen metabolism Protein metabolism Currently data are not available to rigorously parameterize the model for

176

H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

protein synthesis and metabolism. However, omission of such provisions would clearly lead to erroneous solutions in the tracer portions of the model. Thus, these components were included even though a number of assumptions were required. Synthesized proteins fall into two categories: liver and export proteins. Estimates of protein turnover rates in liver and of synthetic rates of export proteins were made to account for ~4C incorporation into proteins and to allow calculation of the energy expenditures in protein synthesis. Hibbitt and Baird (1967) reported nitrogen contents of 26-9-32-3 mg/g wet weight in the livers of normal lactating cows. Based upon these estimates, a protein content of 20% for liver was assumed. Lobley et al. (1978) reported liver protein fractional synthesis rates (FSR) of 0.24/day and 0-38/day in growing heifers and 0.144/day in a mature non-lactating cow. Lobley et al. (1980) reported respective FSR in growing heifers and a mature non-lactating cow of 0.10/day and 0-108/day for the heifers and 0.074/day for the cow. These values were lower than earlier measurements, leading Lobley and colleagues to suggest that the actual FSR is between values reported in these studies. A likely, conservative FSR of 0.15/day was adopted for use in model development. This FSR yields a turnover of liver protein of 318 g/day in the reference animal. Export protein synthesis rates for albumin, globulins, and fibrinogen were adopted from Waghorn's (1982) model. Few experiments have been conducted on bovine albumin synthesis in cows. Saad et al. (1984) reported albumin synthetic rates of 134 mg/kg BW in Zebu calves. Waghorn (1982) established a relationship between albumin synthesis and body weight using data from multiple species of 0.42 g/kg BW °75. Based upon the observations, albumin synthesis was set at 54 g/day for model development. Plasma albumin concentrations vary with physiological state, season and age, and range between 2.95 and 3.63 g/dl (Payne et al., 1973, 1974; Rowlands et al., 1975; Swanson, 1977). The estimate of 54 g/day indicates an FSR of 6-5%/day for albumin based on a plasma albumin concentration of 3.3 g/dl and a plasma volume of 25 liters. Fibrinogen and globulins are synthesized and exported from liver-like albumin. Fibrinogen is synthesized solely in the liver. Approximately 59% of the circulating globulins are synthesized in liver. Bovine plasma concentrations of fibrinogen and globulins are 0-66 and 3.75 g/dl, respectively (Mylrea & Healy, 1968; Payne et al., 1973, 1974; Rowlands et al., 1975; Swanson, 1977). With an assumed FSR similar to that set for albumin for globulins and fibrinogen, total globulin-fibrinogen synthesis would be 47 g/day. Thus, total export protein synthesis was set at 101 g/day in the reference state. The above considerations led to the estimate that total hepatic protein

A mechanistic model of liver metabolism

177

synthesis in the reference state is 419 g/day or 3.81 mol of amino acid incorporation into protein/day. Of this, 2.89 mol/day are incorporated into intracellular liver protein (UAA,PT) and 0.92 mol/day are converted to export proteins (UAA,evr). Amino acid compositions of synthesized and degraded proteins were assumed to be the same (Table 5). A m i n o acid metabolism

Amino acids taken up by the liver are used for protein synthesis or are catabolized. Reynolds et al. (1986b, 1988a) reported net uptakes of 4.0 and 4.2 mol/day of a-amino nitrogen by liver. Currently, the model provides explicitly for entry of alanine, aspartate and glutamate since these have been used as ~4C-radiotracers. Limited data are available on whole blood amino acid profiles in portal blood and intake by the livers or lactating cows (Baird et al., 1975; Reynolds et al., 1988a). For the purposes of model construction, an arbitrary amino acid profile was set for portal blood (Table 5). It was assumed that amino acids were not preferentially removed by the liver and that entry of a given amino acid into the liver was proportional to its concentration. In the reference state, 5.328 mol/day (UAA, UP) of amino acids are removed from blood. Amino acids available for catabolism come from amino acids removed from blood plus those arising from protein degradation minus amino acids used for protein synthesis: UAA,CAT = UAA, UP -Jr UpT, AA - - UAA, P T -

UAA, ePT

(68a)

Products of amino acid catabolism enter the PY, DC, AK, PC, MC and KB pools. Conversions to PY, DC and AK were adjusted to accommodate explicit representations of AL, AS and GU metabolism, UAA, PY ---- (0"133 X UAA,CAT ) - - UAL, CAT UAA,DC =

(0-133 X UAA, CAT) - - UAS, C A T - VAN,CAT

UAA,AK = (0"327 X UAA,CAT) -- UGN,CAT- UGU,CAT

(68b)

(68c) (68d)

while conversions to PC, MC and KB were set as simple functions of stoichiometric relationships for catabolism of the mixed amino acids taken up by liver (Table 5; Krebs, 1964). UAA, PC ---- 0 ' 2 0 6

X UAA,CAT

(68e)

UAA, MC ---- 0 ' 2 8 3

X UAA,CAT

(68f)

UAA, KB : 0"012

X UAA.CAT

(68g)

H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

178

N A D H , ATP, carbon dioxide and ammonia produced during amino acid catabolism and prior to entry into metabolite pools were also calculated: UAA, NDH ---- (1"428 ×

UAA,CAT ) - - UAS, C A T -

UAN,CAT-

(68h)

UoN,cA- UGu,cA X UAA,CAT

(68i)

0-528 × UAA,CAT

(68j)

UAA, AT

= 0"818

UAA,C o

=

UAA,NH3 = (1"2105 × UAA,CAT)- Uau, eau

(68k)

Alanine metabolism, AL The differential equation for A L is dAL dt

= UeAL, AL -I- Upy, A L + UpT, A L - - UAL, PY - - UAL, PT - -

(69a)

UAL,ePT

The differential equations for ~4C-AL are dALi dt

= SAeALi × UeAL, AL + S A p y i × Vpy, A L - - S A A L i X UAL,Py - - S A A L i )< UAL, PT - - S A A L i )< UAL, ePT

(69b)

where i = 1, 2 or 3 for A L and PY. Inputs to A L from extracellular alanine, pyruvate and protein are described by eqns (16), (47) and (69c):

(6%)

UpT, AL ---- SpT, AL X UpT, AA

where SpT,AL is the proportion of alanine in the reference protein (Table 5). Alanine conversion to pyruvate and protein are described by eqns (4), (69d) and (69e): UAL, PT = S A L P T X UAA, PT

(69d)

UAL, ePT ~- SAL, PT X UAA,ePT

(69e)

where SAL,Vr is the proportion of alanine in the reference protein (Table 5).

Aspartate metabolism, AS The differential equation for AS is dAS dt

= UeAS,AS q- UeAN,AS -t- UDC, A S - - UAS, DC - - UAS, PT - -

(70a)

UAS,ePT

and the differential equations for 14C AS are dASi dt

= SAeAsi ×

UeAS, AS -{- S A D c i x

UDC, A S - - S A A s i X

UAS,OC -- SAAsi × UAS,PT -- SAAsi × UASxPT

where i = 1, 2, 3 or 4 for AS and DC.

(70b)

A mechanistic model of liver metabolism

179

Inputs to AS from extracellular aspartate, dicarboxylic acid, and protein are described in eqns (11), (17), (70c) and (70d): UeAN,AS ---- SeAN,AS X UeAN,AS

(70C)

UpT, As

(70d)

---- SpT, AS X UpT, AA

SpT,AS is the proportion of aspartate in the reference protein (Table 5). Aspartate conversions to dicarboxylic acid and protein are described in eqns (5), (70e) and (70f): UAS, PT = SAS, PT X UAA. PT

(70e)

UAS,ePT = SAS,PT × UAA,ePT

(70f)

Glutamate metabolism, GU The differential equation for GU is

dGU = UeGtJ,GU+ UeGN,GU + UAK,GU -'t- UpT, G U - - UGU,AKUGU,eGU -dt UGU,PT- UGU,ePT (71a)

TABLE 5

Estimated Amino Acid Composition of Blood and Protein

Amino acid

Proportion of amino acids

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

0-039 6 0.037 9 0.058 7 0.020 9 0.082 7 0.082 7 0.029 3 0.022 0 0.051 2 0.077 3 0.061 8 0.022 9 0-036 0 0.101 4 0.066 0 0-039 7 0.007 0 0.038 3 0.069 6

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H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

The differential equations that describe 14C-GU are d G U i = SAeGui X UeGU,GU + SAAK i X UAK,GU - - S A G u i X dt

U G U , AK

S A G u i y UGU,eGU--SAGu i x UGu, PT --

(71b)

SAGu i X UGU,ePT where i -- 1, 2, 3, 4 or 5 for GU and AK. Inputs to GU from extracellular glutamate, extracellular glutamine, o~ketoglutarate and protein are described by eqns (3), (21), (22) and (71c): UpT, GU = SpT, G U X Up.LG U

(71c)

where SpT,GU is the proportion of glutamate in the reference protein (Table 5). Glutamate is converted to c~-ketoglutarate, extracellular glutamate, and protein according to eqns (38), (39), (71d) and (71e): UGU.PT = SGu,P T X UAA,PT

(71d)

UGUxPT = SGU,PT X UAA,ePT

(71e)

where SGU,P T is the proportion of glutamate in the reference protein (Table 5). Urea production, UR

The differential equation for UR is dUREA dt - (UBNH3"UR +- UAA'NH3) / 2 + UARG, UR

(72a)

Urea nitrogen is generated from blood ammonia and amino acid catabolism. Reynolds et al. (1986b, 1988a) reported net ammonia uptakes from blood of 531 and 554 mmol/h in 639-kg and 645-kg lactating cows. The reference ammonia uptake for model development was set at 13.3 mol/day (eqn (6)). Urea nitrogen results from amino acid catabolism via ammonia (eqns (6) and (68k)) or from arginine metabolism: UARG, UR z SAR,UR X UAA,CAT

(72b)

where SAR.UR is the proportion of arginine in the reference amino acid profile (Table 5). Reynolds et al. (1986b, 1988a) reported net hepatic urea release rates of 8.80 and 9.40 tool/day. In the reference state 8-9 tool/day of urea are released. Since model estimates of urea production account for only amino acid and blood ammonia nitrogen, urea production is underestimated to the extent that other nitrogen sources contribute approximately 0.4 mol/day.

A mechanistic model of liver metabolism

181

E N E R G Y METABOLISM

Oxygen consumption Limited studies of oxygen consumption by liver have been performed in dairy cattle. Baird et al. (1975) reported liver oxygen consumption to be 48-0 mol/day in 400-kg cows yielding 12 kg of milk/day. Reynolds et al. (1988b) reported rates of oxygen consumption of 73.5 mol/day in livers of 645-kg cows yielding 32.2 kg of milk/day in agreement with earlier estimates of 74.0 mol/day in 660-kg cows (Reynolds et al., 1986a; Huntington & Reynolds, 1987). Given the relationships developed by Smith and Baldwin (1974), liver weights of the Baird et al. (1975) and Reynolds et al. (1988b) cattle would be 7.72 and 10-68 kg, indicating respective oxygen consumption rates of 9-66 ml (100 g rain) -1 and 10.47 ml (100 g min) -l. For model construction, a value of 10-0 ml (100 g min) -~ was used.

Carbon dioxide production Baird et al. (1975) reported liver carbon dioxide production to be 43.1 tool/day with a respiratory quotient (RQ) of 0.90, while Reynolds et al. (1988b) reported carbon dioxide production to be 51.5 mol/day with a RQ of 0.70. A RQ of 0-80 was used to estimate carbon dioxide production rates in the course of model development. Carbon dioxide can act as both a substrate and a product. In the model, carbon dioxide is not represented as an explicit effector of reaction rates. However, provision for ~4C-carbon dioxide (CDU) use as a radiotracer was incorporated. CDU can enter the model in the Upy,DC and Upc,oc reactions as defined in differential equations for 14C-DC (60b) and 60(e). Carbon dioxide as a product is expressed as carbon dioxide from carbon dioxide releasing reactions (GRCD) and as carbon dioxide released into the media/blood (eCD). The latter expression depends on an estimate of the size of the cellular carbon dioxide pool (nC) and assumes that there is no accumulation of carbon dioxide in nC. In the model's reference state, cellular carbon dioxide is set to the dissolved fraction of carbon dioxide in blood (31 mM; Airman & Dittmer, 1974). The differential equations for G R C D and eCD are dGRCD dt

= UIC, A K -~- UGP, R U -t-- Upy, M C 4- UCT, AK -{-- UC T pv _i_ UAK,DC -I- UDC, PE -1- UAA, C D '

deCD dt - Unc,eCO

(73a)

(73b)

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H. C Freetly, ,I. R. Knapp, C C Calvert & R. L. Baldwin

where U,C,eCD is the flux from the intracellular carbon dioxide pool to the extracellular carbon dioxide pool (eCD) and is described by eqn (73c): UnC, eCD ~-

(73c)

K n C e C D × nC

The differential equation for intracellular carbon dioxide is dnC = UIC,AK + UGP,RU + Upy,MC 4- UCT,AK 4- UCT.PY 4dt UAK,D C 4- UDC. P E 4- UAA.C D - - Upy,D C Upc, v C - -

(73d)

Unc,eCD Similar expressions described C D U formation via releasing reactions ( G R C D U ) and extracellular carbon dioxide (eCDU). d G R C D U = SAcT6 x UIC,AK + SAGp I x UGP,RU + SApy I X dt Upy,MC + S A c T 6 x UCT,AK 4- S A c T I X UCT,PY 4-

(73e)

SAAK 1 X UAK, DC 4- S A D c 4 X UDC,PE

deCDU dt where

(73f)

-- UnC, eCDU

UnC, eCD U ---- SA~c U ×

(73g)

UnC, eCD

The differential equation for nCU is d n C U = S A c T 6 X UIC, A K 4- SAGp I X UGP, RU + S A p y I X dt Upy,MC+ SAcT6 × UC+,AK + SAcTI × Ucr, pY + SAAK 1 X UAK,D ( + S A D c 4 x UDC,P E S A n e U X Upy.D c - - S A n e U × UpC,DC - -

SAncu

×

(73h)

UnC,eCD

Coenzyme metabolism Coenzyme pool sizes are very small and their rates o f turnover are high with respect to other metabolites. If physiological concentrations of the coenzymes are used in the model, very short integration steps are required to attain stable solutions. In order to increase solution speed, the pool sizes specified for N A D +, N A D H , N A D P +, N A D P H , F A D ÷, F A D H , A D P and ATP were artificially increased. This strategy decreases the computer time required to solve the model without compromising solutions.

A mechanistic model o f liver metabolism

183

N A D H and N A D ÷ metabolism, N D H and ND Oxidation of N A D H to N A D + resulting in the production of A T P is given by eqn (74a): UNDH, AT = K N D H A T

× NDH

(74a)

The differential equation for N A D H is d N D H = UTp,pE + UTP,AGP + Upv,Mc + UCT,AK + UAK,DC + dt U~LA,PY+ UAK,DC + Upc~Dc + 7"0 × UVA,MC+ UAL, Pv -t- UAS, DC -1- UGU,A K -t- 2"0 × UelB.PC "k-

(74b)

UeVA,P C q'- UeBU,M C -t'- UelV, M C q- UAA, ND H q'- Upy,eL A - UpE, T p - - UAGP, T p -

UMC, K B - - Vpy,A L - - V p y , e L A -

UDC, A S - - UAK,G U -- UNDH,AT

(74c)

ND=TNAD-NDH

N A D P H and N A D P + metabolism, N P H and N P

The differential equation for N A D P H is dNPH - 2.0 × UGP, R U -.b UIC,AK - - 1-75 X UCC,FA dt

(75a)

NP = TNDP - NPH

(75b)

N A D P + availability as a substrate determines the rates of the pentose cycle (UGP,RU) and the N A D P - l i n k e d isocitrate dehydrogenase (U~c,AK) reactions. As a result, the rate of N A D P + formation during fatty acid synthesis determines UGP,RU and U~C,AK fluxes. F A D H and FAD + metabolism, F D H and FD

The differential equation for F A D H is d F D H = UAK,DC + UpC,DC + 7"0 × UFA,MC + U~m,pc + dt UeVA,PC d- UelV, PC q- UeBU,PC q'- UAA.FDH - -

(76a)

UFDH. AT

FD = TFAD - FDH

(76b)

The oxidation of F A D H resulting in the formation of A T P is given by eqn (76c): UFDH,AT ~ K F D H A T

× FDH

(76c)

184

H. C Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

A T P and A D P metabolism, A T and A D

The differential equation for ATP is dAT = UVp,pE + UpE,py + UpC,DC + UAK,OC + 3'0 × dt UNDH,AT+ 2'0 × UFDH,AT+ UAA,AT- U~y, ACpUDC, PE - - 2 . 0 × UpR, pC - - UpC, DC -- UCT, CC - - 2 . 0 X UeAC,CC

2 ' 0 X UeAC, M C -

UeFF, F A -

0"875 ×

UCC,F A - U~m.pc UeVA,P C - UeW,MC- UoBU,MCUpy, DC -- U e G L , G P - UGP, FP - - UpE, TP - - 2 - 0 ×

(77a)

(UAA, NH4 + UBNH4,UP) -- SAA, PT X (UAA,Pv + UAA,ePT ) -SpT, AA X UpT, AA -- UAT, MP

ATP is generated from substrate level and oxidative phosphorylation. ATP expenditures in the differential equation include phosphorylation, carboxylation and thiokinase reactions, urea synthesis, protein synthesis, protein degradation, and ATP required for basal metabolic processes. Stoichiometric coefficients for most of these processes are well established. ATP use for protein synthesis is specified by the stoichiometric coefficient (SAA, PT) which was set to 5.0 (Buttery & Annison, 1973; Millward et al., 1976). The stoichiometric coefficient (Sr,r, AA) for the breakage of a peptide bond was set at 1.0 (Rapoport et al., 1985). A T P used for basal metabolic processes (UAT, MP) includes expenditures such as ion transport, nucleic acid synthesis, and macromolecule transport. C o A S H metabolism, Cs

Cytoplasmic and mitochondrial CoASH are treated as one pool. The differential equation for CoASH is dCs = UpC,DC + 0"875 × UCC,FA + UMC,CT q- UMC,eAChdt UFA, eFF "J- UFA,TG q- UMC, KB -- VePR, PC - - 2"0 X UeVA,P C - UelB, P C - 2"0 X UoBU.MC- UelV, M C Ac, M c - VeAc.cc UoFF,FA- Uo L, A- Upy,M c -

(78a)

7"0 X UFA,M C -- UCT, PY -- UAA, PC - - UAA, MC

MODEL EVALUATION A combination of behavioral analyses, sensitivity analyses, and challenges from independent data sets were used to evaluate the model. Examples of each are presented in the following text.

Behavioral analyses Behavioral and sensitivity analyses were conducted using a 90-min simu-

A mechanistic model of liver metabolism

185

lation time. This guaranteed that the model reached steady-state characteristic of the new nutritional and physiological states specified as inputs. The specific activities of all substrate radiotracers were set to one to allow for convenient interpretations of results obtained (Figs 2 and 3). The reference condition used for model formulation was steady-state. This strategy allows differential equations to be set to zero thereby simplifying computation of numerical inputs to the model. Behavioral analyses test whether the aggregate equations and parameter values developed on the basis of the reference steady-state condition behave in a reasonable manner when perturbed with inputs other than those used in model development. Since the equations upon which the model is based reflect our current knowledge of underlying biology and data available for formulation of numerical inputs, failure of the model to mimic the behaviour of the whole-organ indicates that one or more of the equation forms are incorrect or that data used to formulate parameter inputs was inadequate. The most common behavioral analyses to perform are to vary substrate concentrations and monitor the effects upon formation of products. An example of this type of analysis is Fig. 2(a). Most of the equations in the model represent saturation kinetics. Thus, overall behavior of the model tends towards non-linear responses with increasing substrate. This behavior is demonstrated in Fig. 2(a), where glucose production with increasing propionate concentrations approaches saturation. This behavioral relationship is in agreement with literature values for isolated hepatocytes (Looney, 1985) and liver slices (Mesbah & Baldwin, 1983). Once substrate product relationships were established, behavioral analyses concentrated on 14C-radiotracer behavior with varying substrates. Rates of 1-14C-propionate oxidation to carbon dioxide with increasing propionate concentrations are presented in Fig, 2(b). The limited data of Looney (1985) suggest a curvilinear relationship as well. Figure 2(c) illustrates relative differences in the specific activities of glucose, propionylCoA and citrate as the concentration of 1J4C-propionate increases. As its concentration increases, the contribution of propionate to each pool increases. The fractions of carbon originating from 1J4C-propionate in the citrate and propionyl-CoA pools is greater than in the glucose produced. Relative rates of incorporation of 1- and 2-14C-acetate into products are a function of rates of reactions in the TCA cycle and rates of removal of TCA intermediates from the cycle. Randomization of 14C-label at succinate and subsequent re-entry of oxaloacetate into the TCA cycle results in the loss of 1J4C- acetate as carbon dioxide in the second turn of the cycle. Oxidation of 2J4C-acetate to carbon dioxide requires three

186

1t. C Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin 0.04-

1"0"

-6 E ~ 0-9-

1:1 ¢d ¢,

8.6 o.o3~L E

II

.E ~ 0-02"

~0 0-8

~

u

°

1

~

0

Propionate,

:~

- - 1- 14 C - A c e t a t e --- 2-14C - A c e t a t e

!

O.Ol

_./

0.7

// /

~

°o

Labeled

(a)

1.0 o-8

"u'6 8 E 0.6.

0.03

x 0 0~

/

~, ~ 0.02 13

rll lID

u ~ 04"

~

acetate, mM

(b)

"o

_~ ~

I0

g

m M

U m

~ g o.ol.

~ 1 - 1 4 C-Acetate 2-14C-Acetat e -

-

-

o.2.

d

o

d

,i

o

oo

1-14 C - p r o p i o n a t e , mM

Labeled acetate, m M

(d)

(c)

1.o-

0.12 ' ........................................................

010"

0-8 ."d .>

0-6

u

0"4

~ u'l

0'2

> 008" u ',,7.

0,06-

14C-Acetate ~Citrate-1C itrate-2-14C-Acetate ..... Dicarb. - 1 J 4 C - A c e t a t e - - - Dica rb,- 2-14C- A c e t a t e

~ 004-

cose

..... Propionyl - CoA ---

o

~

Citrate

~

~ 0,02"

~

1-14 C - p r o p i o n a t e , m M (e)

~

o

/..~._--.--..........................................

o

g Labeled

lb acetate,

1~ mM

(f)

Fig. 2. Behavioral analyses: substrate product relationships. Simulations were conducted for 90 min and ~4C-substrates were set with a specific activity of one. (a) Glucose released from U~p,eGL with increasing concentrations of extracellular propionate. (b) Labeled carbon dioxide released from carbon dioxide producing reactions with increasing concentrations of IJaC-propionate. (c) Specific activity of glucose, propionyl-CoA, and citrate with increasing concentrations of lJ4C-propionate. (d) Glucose released from UGP, eGL with increasing concentrations of 1- or 2J4C-acetate. (e) Labeled carbon dioxide released from carbon dioxide producing reactions with increasing concentrations of 1- or 2J4C-acetate. (f) Specific activity of citrate and dicarboxylic acids with increasing concentrations of 1- or 2J4C-acetate.

A mechanistic model of liver metabolism

187

0"4-

3.23 3-22

0"3"

--~

3.21

~ ~

3-20

~5 E c

' 3"18 "

O t~ tJ

3"17 2

0.2"

c -- 0.1"

3.16' 8O

go

100

110

120

~

o-

20

0

40

(a) 0.8-

-0~'~~o2~Eo4. .°6 ~~o,,~.~

0.04

003 "

~

o-o2

e

ooi

100

80

(b)

0"05

L

60

Time, min

S t e a d y s t a t e basal ATP, *1.

0

0

10

20

30

40

50

0

60

, 20

•

UeAC,MC as a p e r c e n t of UMC.eAC

, 40

-

, 60

•

, 80

•

, 100

Time, min

(c)

(d)

0-5,

0.3,

F...........................

T, ~, o.4.

----

2 -14 C - l a c t a t e 1 14 C- propionate

i;- 0.2

f --~-ketogluterate

•~ ~

~- 0.1

0.3"

---phospho-enoi-pyruvate

X= 0.2

•

0

0

0~5

"

~'0

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1.~5

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270

Proportion of s t e a d y s t a t e kp(.D C (e)

.

0

,

20

.

,

.

40

,

60

.

.

.

80

.

100

Time, rain

(f)

Fig. 3. Sensitivity analyses. Simulations were conducted for 90 min at steady-state substrate concentrations and specific activity of 14C-tracers were set at one. (a) Carbon dioxide released from carbon dioxide producing reactions as basal ATP expenditures vary as a percentage of steady-state basal ATP expenditures. (b) Labeled carbon dioxide released from carbon dioxide producing reactions when 1- or 2-14C-acetate is used as a tracer and UeAC,MCincreases as a percentage of UMC,oACwhile the net acetate flux is held constant. (c) Specific activity of glucose released from UCP,eOLas the affinity constant for pyruvate (kpv,oc) in the UpV,Dc flux (eqn (48)) changes as a proportion of the steady-state value when 2-14C-lactate and 1J4C-propionate are used as tracers. (d) Moles of laC-carbon released by the Uop.mLflux with increasing simulation time and 1J4C-propionate used as a tracer. (e) Labeled carbon dioxide released from carbon dioxide producing reactions with increasing simulation time and 1J4C-propionate used as a tracer. (f) Specific activity of a-ketoglutarate and phospho-enol-pyruvate with increasing simulation time and 1JaC-propionate used as a tracer.

188

H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

or more turns of the TCA cycle. Model simulations produce a behavioral pattern consistent with our knowledge of the biology. Figure 2(d) depicts the appearance of acetate carbons in glucose as the acetate concentration increases. Figures 2(d) and 2(e) indicate that incorporation of 2-14C-acetate into glucose is greater than that of 1-~4C-acetate, while the converse is true for incorporation into carbon dioxide. Figure 2(f) further demonstrates the mechanism of carbon dioxide production in the TCA cycle. The specific activities of citrate from 1- and 2-~4C-acetate are similar to each other. After citrate is converted to dicarboxylic acids, a difference in specific activities of intermediates formed from 1- and 2-~4C-acetate becomes evident as the biology of the system dictates. These and many more behavioral analyses agreed with literature values. Where literature values were not available, the behavioral analyses were consistent with the biology of liver metabolism.

Sensitivity analyses Sensitivity analyses are an important part of model construction and evaluation. Following the initial construction of a model, sensitivity analysis can be used to test the sensitivity to a given parameter. Sensitivity analyses can be divided into two major categories. The first is to test sensitivity of model outputs to alternative equation forms, degrees of aggregation, etc. These analyses are generally undertaken during model formulation to determine what elements of the biological system must be represented and at what level of complexity. Results of these types of analyses are reflected in the equations adopted (Table 4). The second type is to evaluate effects of varying parameters, such as affinity constants on model behavior. Based on this type of sensitivity analysis, a decision can be made on a given parameter. If the model is insensitive to changes in the parameter value, that parameter can be eliminated from the model if it is not required to represent a biological function essential to representation of the whole model. In this way, simplification of the model can occur. Conversely, if the model is sensitive to a parameter, it shows the importance of accurate estimation of that parameter. If such a parameter is not firmly defined by experimental data, collection of data to accurately define the parameter is the next logical step. Results of some of the sensitivity analyses conducted during evaluation of the model are presented in Fig. 3. ATP acts as a substrate and regulator of many key reactions in the model. Figure 3(a) illustrates sensitivity to changes in the amount of ATP required for basal metabolic processes (UAT,Mp) on carbon dioxide production (eqn (77)). Varying UAT,MP, mimics variation in energy expen-

A mechanistic model of liver metabolism

1189

ditures in the tissue. The results indicate that carbon dioxide production increases as ATP use increases until a maximum is reached. Over this portion of the curve, the model indicates that reaction rates in the TCA cycle increase and while there is a minor decline in carbon dioxide production from the pentose cycle due to a decrease in long chain fatty acid synthesis. The simulated decline in carbon dioxide production after reaching this maximum results from a sharp decrease in amounts of dicarboxylic acid entering gluconeogenesis and a lesser rate of increase in TCA cycle flux. This analysis clearly demonstrates that an accurate estimation of ATP use is critical to model behavior. Input: output data indicate a net release of acetate from liver while isotope studies show significant rates of ~4C-acetate incorporation into liver metabolites and products. A sensitivity analysis was conducted to determine sensitivity of the ~4C-radiotracer portion of the model to estimates of U~AC,MC (eqn (15)) and UMC,eAC (eqn (42)) while net flux to eAC was held constant (Fig. 3(b)). It is clear that accurate estimates of these flux rates are required when acetate is used as radiotracer and that apparent rates of 1-~4C-acetate oxidation are more sensitive than those of 2 - N C acetate. An inverse relationship was evident when 14C-label incorporation into glucose was estimated (Figure not shown). The conversion of pyruvate to dicarboxylic acid represents a potential regulatory step of gluconeogenesis (Looney, 1985). Sensitivity to changes in affinity for pyruvate (kpy, DC) in this reaction (Upy,DC) was tested (eqn (48)). As the affinity constant increased, there was a small decline in glucose production. The specific activities of glucose produced using 2-14C lactate and 1-~4C-propionate as substrates are presented in Fig. 3(c). When 2-~4C-lactate was used as a radiotracer, the specific activity of glucose declined, but when 1-~4C-propionate was used as a radiotracer, there was a slight increase in specific activity. These results suggest that as kaY, DC increases, the proportion of glucose derived from propionate increases while the proportion derived from pyruvate decreases. The data in Fig. 3(c) also shows how the model can be used to select a tracer to enable estimation of a given parameter in the model. The relative sensitivity of glucose production to changes in kev, Dc is greater for 2-~4C-lactate than 1-~4C-propionate, suggesting that 2-~4C-lactate can be used to estimate kpy, o C when specific activity in glucose is measured. Most radiotracer studies assume rapid mixing of the tracer in the tissue and stability of the specific activities of all metabolites. Figures 3(d) and 3(e) are the simulated production of 14C-labeled carbon dioxide and glucose from 1-14C-propionate over time. Both curves lag briefly before they become linear. This lag is more exaggerated for glucose than for carbon dioxide. Neither of these lags can be examined experimentally,

190

11. c. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

because their deviation from linearity is small with respect to experimental error. They do suggest, however, that specific activities of intermediate pools are not stable over early times. Simulated specific activities of the cr-ketoglutarate and phosphoenol-pyruvate over time using 1 - 1 4 C propionate as tracer are depicted in Fig. 3(f). The time required for the specific activities of metabolites to reach a steady-state appears to be unique for each pool and varies with different radiotracers (data not shown). The time required for ,-ketoglutarate to reach a constant specific activity was greater than that for phosphoenol-pyruvate. While the appearance of 14C-label into carbon dioxide over time seems nearly linear, the relative contribution of each of the carbon dioxide-producing reactions changes until all metabolites come to stable specific activities. This simulation suggests that calculations based on assumptions that stable specific activities are instantaneously obtained may be incorrect, and the shorter the incubation time, the greater the potential error. This sensitivity analysis demonstrates how the model can suggest modifications in experimental design.

Independent data challenge Data sets used to evaluate a model must not have been used in model construction. Limited data were available for model parameterization and, as a result, all available in-vivo data were used. However, in-vitro ~4C-radiotracer data were not used in model construction. These data sets were used to challenge the model. Some comparisons of experimental data and model outputs for radiotracers were presented in the behavioral analysis section. Experimental and simulated substrate interactions were tested. As noted in the behavioral analyses section, glucose production with varying concentrations of lactate or propionate agree with literature values. Invitro data have shown that additions of amino acids stimulate glucose production. Simulation runs indicated that additions of amino acids to limited incubation media increased 2-laC-propionate conversion to glucose and decreased proportionate oxidation. Aiello and Armentano (1987) observed an increase in propionate incorporation in glucose but no difference in its oxidation. The simulated difference in carbon dioxide production is within the error term reported by Aiello and Armentano (1987). Looney (1975) added aspartate to a limited medium containing propionate and demonstrated a subsequent increase in glucose production. The model predicted similar behavior when aspartate was added to a medium containing propionate. Potential interactions of propionate with other volatile fatty acids that

A mechanistic model of liver metabolism

191

form CoASH esters were tested. Addition of valerate to limited media with 2-14C-propionate has been shown to increase label incorporation into glucose and decrease its appearance in carbon dioxide (Aiello & Armentano, 1987). The same result was simulated when valerate was added to the limited media. Previous studies have shown that butyrate added to limited media containing propionate decreased glucose synthesis (Looney, 1985; Aiello et al., 1989). Aiello et al. (1989) also observed no increase in carbon dioxide from 2-14C-propionate with butyrate addition. Aiello et al. (1989) argued that the effect is not the result of competition for substrates like CoASH and ATP, but rather an inhibition of one of the enzymes required for propionate conversion to succinyl-CoA. A sensitivity analysis was undertaken to test this hypothesis. Affinity constants and maximum capacities for uptake of butyrate were varied to test the hypothesis that inhibition was the result of substrate competition. The maximum depression of 2-~4C-propionate incorporation into glucose achieved with rational parameter values was 5%. Reported decreases in 2-14C-propionate incorporation into glucose by butyrate were 32-42% (Aiello & Armentano, 1987; Aiello et al., 1989). This decrease suggests that butyrate or one of its metabolites may inhibit propionate entry. Until the mechanism of this inhibition is elucidated, equations required to simulate this response cannot be formulated. The model predicts a seven-fold higher incorporation of 1-~4C-acetate than 2-14C-acetate into carbon dioxide at maximum capacity (Fig. 2(d)). The model also predicts a two-fold greater rate of 2-~4C-acetate incorporation into glucose than for 1-14C-acetate at maximum capacity (Fig. 2(d)). This is consistent with the work of Knapp et al. (1992). Simulation results compare favorably with data presented by Knapp et al. (1992) for oxidation of 1-14C-lactate and 2-14C-lactate. More 14C-carbon dioxide is derived from 1-~4C-lactate than 2-~4C-lactate at maximum capacity. Increasing concentrations of lactate from Ks to capacity result in increased oxidation of 1-14C-lactate to ~4C-carbon dioxide but not an increase in the oxidation of 2-~4C-lactate to ~4C-carbon dioxide.

SUMMARY This modeling study resulted in the formulation of a mechanistic model of ruminant liver metabolism based on current knowledge of underlying biochemical functions. The model integrates input:output and ~4C-radiotracer data. Behavioral analysis and challenges with independent data sets suggest that model behavior is correct. Sensitivity analyses have identified additional research that is required to further our knowledge of

192

H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

liver metabolism and in doing so has illustrated the usefulness of models of this type in the research process. Integration of i n p u t : o u t p u t and ~4Cradiotracer elements has resulted in a new tool for analysis of radiotracer data. Parameterization was based primarily on in-vivo data, so its use in evaluations of in-vitro data requires appropriate scaling to account for differences in activities observed between whole-organ, isolated hepatocytes, and liver slices. From this analysis of ruminant liver metabolism, it is clear that additional research must be conducted to fully understand nutrient partitioning. Additional information is required to accurately assess the contribution of amino acids to gluconeogenesis. More in-depth studies that measure the composition of amino acid uptake and release are needed. Similarly, rates and composition of lipid uptake and release are required. Sensitivity analyses have shown that data on flux rates that provide potential points of exit for radiotracer are required. Examples of these include acetate and lactate exchange. Integration of the existing data provides insight into liver function and experimental design. This model serves primarily to guide research in ruminant liver metabolism in an organized fashion while allowing integration of information to insure improved understanding of the organ as a whole.

REFERENCES Aiello, R. J. & Armentano, L. E. (1987). Gluconeogenesis in goat hepatocytes is affected by calcium, ammonia and other key metabolites but not primarily through cytosolic redox state. Comp. Biochem. Physiol., 88B, 193-201. Aiello, R. J., Armentano, L. E., Bertics, S. J. & Murphy A. T. (1989). Volatile fatty acid uptake and propionate metabolism in ruminant hepatocytes. J. Dairy Sci., 72, 942-9. Altman, P. L. & Dittmer, D. S. (1974). Biology Data Book, Vol. 1II. Federation of American Societies for Experimental Biology, Bethesda, MD. Baird, G. D. & Heitzman, R. J. (1970). Gluconeogenesis in the cow. The effect of a glucocorticoid on hepatic intermediary metabolism. Biochem, J., 116, 865-74. Baird, G. D., Heitzman, R. J. & Hibbitt, K. G. (1972). Effects of starvation on intermediary metabolism in the lactating cow. A comparison with metabolic changes occurring during bovine ketosis. Biochem. J., 128, 1311-18. Baird, G. D., Symonds, H. W. & Ash, R. (1975). Some observations on metabolite production and utilization in vivo by the gut and liver of adult dairy cows. J. Agri. Sci. Camb., 85, 281-96. Baird, G. D., Heitzman, R. J., Reid, I. M., Symonds, H. W. & Lomax, M. A. (1979). Effect of food deprivation on ketonaemia, ketogenesis and hepatic intermediary metabolism in the non-lactating dairy cow. Biochem. J., 178, 35-44.

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Baldwin, R. L., Miller, P. S., Freetly, H. C., Hanigan, M. D., Fadel, J., Bowers, M. K. & Calvert, C. C. (1989). Future of tissue level models. Proceedings Third International Workshop on Modelling Digestion and Metabolism in Farm Animals, eds B. Robinson & D. Poppi. Lincoln University Press, Canterbury, New Zealand, pp. 345-57 Brumby, P. E., Anderson, M., Tackley, B., Storry, J. E. & Hibbitt, K. G. (1975). Lipid metabolism in the cow during starvation induced ketosis. Biochem. J., 146, 609-15. Buttery, P. J. & Annison, E. F. (1973). Considerations of the efficiency of amino acid and protein metabolism in animals. In The Biological Efficiency of Protein Production, ed. J. W. G. Jones. Cambridge University Press, London, pp. 141-71. Goebel, R., Berman, M. & Foster, D. (1982). Mathematical model for the distribution of isotopic carbon atoms through the tricarboxylic acid cycle. Fed. Proc., 41, 96--103. Hetenyi, G., Jr. (1982). Correction for the metabolism exchange of t2C and 14C atoms in the pathway of gluconeogensis in vivo. Fd. Proc., 41, 104-9. Hibbitt, K. G. & Baird, G. D. (1967). An induced ketosis and its role in the study of primary spontaneous bovine acetonaemia. Vet. Rec., 81, 511-17. Huntington, G. B. & Reynolds, C. K. (1987). Oxygen consumption and metabolite flux of bovine portal-drained viscera and liver. J. Nutr., 117, 1167-73. Knapp, J. R., Freetly, H. C., Reis, B. L., Calvert, C. C. & Baldwin, R. L. (1992). Effects of somatotropin substrates on patterns of liver metabolism in lactating dairy cattle. J. Dairy Sci., 75, 1025-35. Krebs, H. A. (1964). The metabolic fate of amino acids. Mammalian Protein Metabolism, ed. H. N. Munro & J. B. Allison, Academic Press, New York, pp. 125-76. Lobley, G. E., Reeds, P. J. & Pennie, K. (1978). Protein synthesis in cattle. Nutr. Soc. Proc., 37, 96A. Lobley, G. E., Milne, V., Lovie, J. M., Reeds, P. J. & Pennie, K. (1980). Whole body tissue protein synthesis in cattle. Br. J. Nutr., 43, 491-502. Lomax, M. A. & Baird, G. D. (1983). Blood flow and nutrient exchange across the liver and gut of the dairy cow. Effects of lactation and fasting. Br. J. Nutr., 49, 481-96. Looney, M. E. C. (1985). Studies of gluconeogenesis in isolated sheep hepatocytes. PhD Dissertation, University of California, Davis. Mesbah, M. M. & Baldwin, R. L. (1983) Effects of diet, pregnancy, and lactation on enzyme activities and gluconeogenesis in ruminant liver. J. Dairy Sci., 66, 783-8. Millward, D. J., Garlick, P. J. & Reeds, P. J. (1976). The energy cost of growth. Proc. Nutr. Soc., 35, 339-49. Mitchell & Gauthier Associates (1986). Advanced Continuous Simulation Language Reference Manual. Concord, MA. Mylrea, P. J. & Healy, P. J. (1968). Concentrations of some components in the blood and serum of apparently healthy dairy cattle. 2. Serum proteins, enzymes, bilirubin, and creatinine. Aust. Veter. J., 44, 570-3. Payne, J. M., Rowlands, G. J., Manston, R. & Dew, S. M. (1973). A statistical appraisal of the results and metabolic profiles test on 75 dairy herds. Br. Vet. J., 129, 370-85.

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H. C. Freetly, J. R. Knapp, C. C. Calvert & R. L. Baldwin

Payne, J. M., Rowlands, G. J., Manston, R., Dew, S. M. & Parker, W. H. (1974). A statistical appraisal of the results of 191 herds in the B.V.A./A.D.S.A. joint exercise in animal health and production. Br. Pet. J., 130, 34-43. Rapoport, S., Dubiel, W. & Muller, M. (1985). Proteolysis of mitochondria in reticulocytes during maturation is ubiquitin-dependent and is accompanied by a high rate of ATP hydrolysis. Europ. J. Bioehem., 290, 249-52. Reid, I. M., Baird, G. D. & Heitzman, R. J. (1977). Effects of fasting in nonlactating cows. A correlated mechanical and sterological study of fastinginduced fatty liver. J. Agric. Sci., 89, 319-25. Reynolds, C. K., Huntington, G. B., Tyrrell, H. F. & Reynolds, P. J. (1988a). Net portal-drained visceral and hepatic metabolism of glucose, L-lactate, and nitrogenous compounds in lactating Holstein cows. J. Dairy Sei., 71, 1803-12. Reynolds, C. K., Huntington, G. B., Tyrrell, H. F. & Reynolds, P. J. (1988b). Net metabolism of volatile fatty acids, fl-hydroxybutyrate, non-esterified fatty acids, blood gasses by portal-drained viscera and liver of lactating Holstein cows. J. Dairy Sci., 71, 2395-405. Reynolds, C. K., Huntington, G. B., Tyrrell, H. F. & Reynolds, P. J. (1986a). Splanchnic tissue and whole animal oxygen consumption by lactating Holstein cows. J. Dairy Sci., 69(Supple 1), 193, (abstract). Reynolds, C. K., Huntington, G. B., Tyrrell, H. F., Reynolds, P. J. & Elsasser, T. H. (1986b). Net portal-drained visceral and hepatic flux of nutrients in lactating cows. Fed. Proc., 45, 240 (abstract). Rowlands, G. J., Manston, R., Pocock, R. M. & Dew, S. M. (1975). Relationships between stage of lactation and pregnancy and blood composition in a herd of dairy cows and the influences of seasonal changes in management on these relationships. J. Dairy Res., 42, 349-62. Saad, A. M., Hussein M. F. Bushara, H. O., Dargie, J. D. & Taylor, M. G. (1984). Erythrokinetics and albumin metabolism in primary experimental Schistosoma bovis infection in Zebu calves. J. Comp Path., 94, 249-62. Smith, N. E. (1967). Quantitative simulation analyses of ruminant metabolic functions: basal; lactation; milk fat depression. PhD Dissertation, University of California, Davis. Smith, N. E. & Baldwin, R. L. (1974). Effects of breed, pregnancy and lactation on weight of organs and tissues in dairy cattle. J. Dairy Sci., 57, 1055-60. Snoswell, A. M., Costa, N. D., McLean, J. G., Baird, G. D., Lomax, M. A. & Symonds, H. W. (1978). Interrelationships between acylation and the disposal of acetyl groups in the livers of dairy cows. J. Dairy Res., 45, 331-8. Steinhour, W. D. & Bauman, D. E. (1988). Propionate metabolism: a new interpretation. In Aspects of Digestive Physiology in Ruminants, eds. A. Dobson & M. J. Dobson. Cornell University Press, Ithaca, New York, pp. 238-56. Swanson, E. W. (1977). Factors for computing requirements for protein for maintenance of cattle. J. Dairy Sci., 60, 1583-93. Thompson, J. R. (1971). Gluconeogenesis from propionate in the lactating cow. PhD Dissertation, University of California, Davis. Waghorn, G. (1982). Modelling analyses of bovine mammary and liver metabolism. PhD Dissertation, University of California, Davis. Weinman, E. O., Strisower, E. H. & Chaikoff, I. L. (1957). Conversion of fatty

A mechanistic model of liver metabol&m

195

acids to carbohydrate: Application of isotopes to this problem and role of the Krebs cycle as a synthetic pathway. Physiological Reviews, 37, 252-72 Wiltrout, D. W. & Satter, L. D. (1972). Contribution of propionate to glucose synthesis in the lactating and non-lactating cow. J. Dairy Sci., 55, 307-17.