J. Dairy Sci. 87:23752387 American Dairy Science Association, 2004.

Deconstructing Milk Yield and Composition During LactationUsing Biologically Based Lactation ModelsG. E. PollottDepartment of Agricultural Sciences, Imperial College London,Wye, Ashford, Kent, TN25 5AH, UK

ABSTRACT

A recently developed biological model of lactation de-scribed changes in dailymilk yield throughout lactationas the result of 3 processes, secretory cell differentia-tion, cell death, and secretion rate per cell. This paperextends the model to describe the production of milkcomponents (fat, protein, lactose, and water) through-out lactation by replacing milk secretion rate of theoriginal model with the secretion rates of the four com-ponents. Themilk componentmodel approach was usedto examine the relationship between milk yield and themajor determinants of its production, using the secre-tion of milk components throughout lactation. Newlyderived models were tested on 461 lactations from asingle Holstein herd and used to estimate variabilityof secretion rates throughout lactation. Because thepattern of cell numbers throughout lactation is not pre-cisely known, an alternative pattern of cell numberswas modeled and the concomitant change in secretionrates outlined. Fat secretion rate was themost variable,as measured by its weekly coefficient of variationthroughout lactation. Secretion rates of lactose and wa-ter were nearly constant throughout lactation andhighly correlated (0.94). Fat and protein secretion ratesalso were well correlated (0.53). The known biochemis-try of milk component production related well to thesecretion rate observations derived from the model.Lactose secretion rate and numbers of active secretorycells primarily determined daily milk yield.(Key words: lactation curve, milk composition, milkyield)

INTRODUCTION

Milk is a mixture of fat, protein, lactose, vitamins,and minerals, either dissolved or suspended in water.Each component is either produced by the active secre-tory cells or is filtered through the epithelial cells in

Received February 25, 2004.Accepted March 29, 2004.E-mail: g.pollott@imperial.ac.uk.

2375

the mammary gland during lactation, and the weightof each component produced per animal each day hasbeen described by numerous authors (see Olori et al.,1997). The secretion rate of milk components is com-monly described in relation to the whole udder. Theactual secretion pattern of milk and its components percell, throughout lactation, has not been clearly defined.In their review of mammary development and function,Knight et al. (1998) concluded that the decline in milkyield in late lactation paralleled the decline in secretorycell numbers and that secretory cell activity remainedhigh throughout declining lactation. They also statedthat secretory cells continued to be produced untilaround the peak of lactation. Patterns of secretion rateof milk components per cell are largely unknown dueto the difficulty of measuring the number of active cellson any given day of lactation, or individual cell metabo-lism throughout lactation.Milk production during lactation has been described

in biological terms using amodel comprising 3 functionsto represent cell proliferation and differentiation, celldeath (apoptosis), and secretion rate (Pollott, 2000).This biological approach to describe milk yield hasbeen shown to accommodate the effects of changes innutrition onmilk yield during lactation and the effect ofpregnancy on milk yield. It can also be used to calculatevarious characteristics of lactation curves, such as theday and extent of peak yield, rates of increase, anddecline in milk yield at specific points in the lactationand total milk yield. Use of this approach has beencompared with a range of empirical and biological mod-els in cattle (Pollott, 2000) and sheep (Pollott and Goot-wine, 2000). Using dairy sheep as an example, factorsaffecting both the parameters of the model and calcu-lated values have been investigated by Gootwine andPollott (2000). The genetics of these parameters, andcalculated values, and the relationships between themhave been described by Pollott and Gootwine (2001).The objective of this paper was to derive suitable

expressions to describe the production of milk constit-uents throughout lactation using the model of Pollott(2000), and to investigate some factors that influencecharacteristics of the lactation curves for these vari-ous components.

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MATERIALS AND METHODSDeveloping Lactation Curve Functionsfor Milk ComponentsThe multiplicative model, for describing changes in

milk yield (M; kg/d) during lactation using the biologi-cally-based approach of Pollott (2000), was M = (NDPC PR) SM. In this model, NDPC was the number ofdifferentiated parenchyma cells and PRwas the propor-tion of differentiated cells still active on any day oflactation. The term SM was the milk offtake rate onany given day of lactation, more loosely called the secre-tion rate of the active parenchyma cells, in kilogramsper cell, per day. The secretion rate term was based oncomments by Knight et al. (1998) and also on the factthat milk secretion starts from zero at or around d 1 oflactation (Pollott, 2000), reaches a maximal value, andthen remains at that value throughout lactation. How-ever, this is not a clearly defined relationship (Knight etal., 1998). Modeling work by Vetharaniam et al. (2003)suggested that secretion rate may be highly dependenton energy availability throughout lactation, and thusfollows a different pattern for each animal in relationto the range of factors that cause energy supply to theudder to vary. No reports exist from extensivemeasure-ments of cell numbers or secretion rate throughout lac-tation and the exact pattern of these key determinantsof milk production. Thus, the work reported herein isbased on assumptions about the pattern of cell numbersand secretion rates throughout lactation. Although thisis somewhat unsatisfactory, the approach outlinedherein will have wide application once such measure-ments become possible.Summarizing the description of daily milk yield in

full gives:

Mt = [N/(1 +1 P0P0

eGt)] [1 + Q0 {1/(1

+1 Q0Q0

eDt)}] [Sa(1 eSbt)] [1]

[NDPC] [PR] [SM]

whereMt is milk production in kg/d on day t of lactation,N = total number of parenchyma cells that become ac-tive during lactation, P0 = proportion of N present atthe start of lactation, G = the relative growth rate inactive cell numbers (G > 0), t = day of lactation, Q0 =proportion of N dead at start of lactation, D = the rela-tive death rate of cells (D > 0), Sa = the maximumsecretion rate of milk (kg/cell per day), and Sb = therelative rate of change in milk secretion rate (Sb > 0).This form of the model sets the rate of apoptosis at

parturition to zero by subtractingQ0 from the apoptosis

Journal of Dairy Science Vol. 87, No. 8, 2004

term, PR (Pollott, 2000; appendix). This is an empiricalmodification to the original model that may be aban-doned when levels of apoptosis throughout lactationhave been better characterized.This model can be used for any milk constituent by

using the weight of the component produced per dayas the dependent variable. Commonly, the availableweights are fat, protein, lactose, water, and total solids,all measured in kilograms per day, but any measurablemilk constituent, including ash, minerals, and vita-mins, could also be used. For any given lactation, theonly term in model 1 to differ among the different milkconstituents (and milk yield) is the last expression formilk offtake rate (SM), because the first 2 terms relateto cell differentiation and apoptosis of the same cellsthat are secreting all the components. If the same celldifferentiation and death curves are fitted to all 5 milkconstituent lactation curves, then the difference amongthem and the actual component weights per day is thesecretion rate of that component on the test-day. Forexample, weight of fat (F; kg/day)

F = (NDPC PR) SF [2]

where SF (kg/cell per day) is the weight of fat secretedby the average cell on any given day. Thus SF = F/(NDPC PR), and most of the parameters on the right-hand side of this relationship, using the full model(model 1), can be derived from an analysis of any suit-able lactation curve. Exceptions to this are the totalnumber of parenchyma cells that become active duringlactation (N), which cannot be derived from the com-monly collected farm data, and fat weight, which is ameasured value.Because commonly available farm records comprise

test-day milk yields and component proportions (sayFP for fat proportion), it may be better to think of SFusing these values. Thus by rewriting model 2 usingFP then:

F = (NDPC PR) SM FP [3]

Model 3 ismodel 1, describing the curve formilk produc-tion, multiplied by the fat proportion measured on agiven day.Substituting model 3 in model 2 and rearranging the

terms gives:

SF = FP SM [4]

This expression describes the daily quantity of fatproduced by the average cell. The pattern of fat secre-tion per cell throughout lactation differs from the pat-tern of fat secretion of the whole udder because of possi-

LACTATION CURVES FOR MILK COMPONENTS 2377

ble changes in the number of active cells during lacta-tion. Similar expressions for protein, lactose, totalsolids, and water may be derived by substituting theproportions of these components for fat in models 2, 3,and 4.Alternatively, it is possible to describe milk yield in

terms of the number of productive cells and the sum ofthe secretions of the components:

M = (NDPC PR) (SF + SP + SL + SW) [5]

where the subscripts P, L, and W refer to the secretionrate of protein, lactose, and water (plus vitamins, min-erals, and ash), respectively. This form of the modeldescribes milk yield as a product of several biologicalprocesses, cell production, differentiation, and death,and the factors that determine fat, protein, lactose, andwater secretion. If these different biological processesare controlled independently of each other, then milkyield results from the sum of a variable set of indepen-dent factors. If it were possible to characterize a lacta-tion in this way, then a more complete description ofmilk yield would be obtained and a more meaningfulway of considering its modification may be achievable.

Records Used to Test the BiologicalModel on Milk Component Weights

The lactation records used in the current study weredescribed previously by Olori et al. (1997, 1999a,b).Briefly, the cows were part of the multiple-ovulationand embryo-transfer herd owned by Genus Ltd. Thesecows were kept at Genus Ltd.s Bays Leap Farm, North-umberland,UK. Establishment andmanagement of theherd were described by Strathie and McGuirk (1995).Cows were housed indoors throughout the year andmilked thrice daily. They were fed a TMR all year. Milkproduction from individual cows was recorded at eachmilking and weekly test-day yields were recorded alongwith percentages of fat, protein, and lactose.First-lactation records, used in this analysis, com-

prised daily milk yields and weekly composition mea-surements collected from 488 cows from wk 2 to 44 oflactation. The original dataset comprised 20,095 test-day records, representing calvings between July 1990andDecember 1994.A number of factors thatmay affectmilk yield were recorded, including age at first calving,day of each test-day record, pregnancy status, time ofconception, calving date, and week of lactation. Monthof calving and month of production were grouped by 2-mo increments for these analyses, resulting in 6 groupsper calendar year (Olori et al., 1997).Weekly test-day records were used in these analyses.

Records available comprised test-day milk yield (kg)

Journal of Dairy Science Vol. 87, No. 8, 2004

and proportions of fat, protein, and lactose in test-daymilk samples. Weights (kg) of fat, protein, and lactoseproduced per day were calculated using the appropriateproportion value and test-day milk yield. Total solidswere calculated as the sum of the three componentweights (fat + protein + lactose); the weight of waterproduced daily was calculated as milk weight minustotal solids; and the proportion of water was calculatedas 1 minus the sum of fat, protein, and lactose propor-tions. Thus, water estimates included the small amountof ash, minerals, and vitamins produced in milk, typi-cally about 0.009 by weight (see Tow, 1984). In theseanalyses, the week of lactation was used as the timevariable (t in model 1).

Overall Production of Milk Components

To derive overall production curves for the 6 weighttraits (milk, fat, protein, lactose, total solids, and wa-ter), a least squares analyses was carried out to derivethe weekly test-day means free of the effects of cow,year,month of production, and pregnancy status.Model6 was fitted to 17,862 test-day records using the GLMprocedure of SAS (SAS Inst., Inc., Cary, NC):

yijklm = + wi + mj + pk + cl + wimj [6]+ wipk + mjpk + eijklm

where yijklm was the appropriate test-day weight (milk,fat, protein, lactose, water, or total solids) from the lthcow (c; l = 1 to 488) in the ith wk of lactation (w; i = 2to 44) and jth yr/mo of production (m; j = 1 to 27) duringthe kth month of pregnancy (p; k = 0 to 8), and eijklmwas the randomly distributed error term. The termswimj, wipk, and mipk were the interactions involvingweek of lactation, month of production, and month ofpregnancy. Only records with complete sets of informa-tion for all traits and class variables were used in theseanalyses. The proportion of the overall variance of atrait accounted for by each effect in model 6 was calcu-lated by fitting a reduced model, omitting the effect ofinterest. The proportional reduction in the residual sumof squares between the full and reduced models usedwas then calculated.Least square means for week of lactation (plus an

extrapolated estimate for wk 1) were used to describethe production of each of the 6 traits throughout lacta-tion. This produced 6 lactation curves, one each formilk,fat, protein, lactose, total solids, and water weight. Themultiplicative model of Pollott (2000) was then fittedto these 6 curves using an iterative, nonlinear, curve-fitting procedure (NLIN in SAS). This modified model1 took the form:

POLLOTT2378

Mt = [MS/(1 +1 P0P0

eGt)] [1 + Q0 [7]

{1/(1 +1 Q0Q0

eDt)}] [1 eSbt]

where MS was the maximal secretion potential of theudder. The termMS (kg/d) is a composite of the numberof parenchyma cells that become active during lactation(N), and the maximal secretion rate of cells (Sa) (i.e.,MS = N Sa). As Pollott (2000) pointed out, it is notpossible to separate these 2 components using com-monly collected farm records.Whenapplying thismodelto the 6 weight traits, the only difference among themwas in the secretion rate term, which was calculatedin its composite form MS, and Sb. The term for milkyield (M) was replaced by fat, protein, lactose, water,and total solids weight in model 7 to derive the other5 lactation curves. This approach was used to investi-gate the differences in the lactation curves of milk andits 5 components.The parameters of each curve were estimated using

an iterative least square method, and the computa-tional strategy of Marquardt (SAS, 1989) was used tofind the best fit solution. The best fit solution wasobtained for each curve when there was a

LACTATION CURVES FOR MILK COMPONENTS 2379

Figure 1. Two patterns of active cell numbers used to derivesecretion rates in this study, one based on Pollott (2000) and theother based on Vetharaniam et al. (2003).

changes in active secretory cell numbers, a balance be-tween cell production and death by apoptosis. If newphysiological research shows milk secretion to have adifferent pattern, then the relativities between the dif-ferent secretion rates will remain the same, but theactual secretion rate pattern of each component willdiffer in line with the new secretion rate pattern. Theseweekly component values were then analyzed in a GLManalysis using model 6 to see which factors had a sig-nificant influence on secretion rate of each component.Least squaremeans forweek of lactation andpregnancywere investigated to determine patterns of componentsecretion during lactation and effects of pregnancy oncomponent secretion.Weekly secretion rates for each milk component, for

each cow, were investigated to explore the relationshipsamong them. Correlations between secretion rate traitswere calculated using residual variances and covari-ances derived from multiple analyses of variance (MA-NOVA option inGLMprocedure of SAS) of all the traits,using model 6. Overall correlations were derived byfitting no fixed effects in the model. Within-cow correla-

Table 1. Means, standard deviations, coefficients of variation, and proportions of variation accounted forby fixed effects used in a least square analysis of the 20,095 test-day records for the 6 milk weight traits.

Milk Fat Protein Lactose Water Total solidsweight weight weight weight weight weight

Mean, kg/d 29.2 1.12 0.994 1.43 25.7 3.54SD, kg 5.98 0.216 0.181 0.304 5.35 0.653CV, % 20.5 19.3 18.2 21.3 20.8 18.4R2 0.76 0.58 0.67 0.76 0.76 0.70Proportion of variance explained by the effectCow 0.45*** 0.36*** 0.42*** 0.42*** 0.46*** 0.41***Month of production 0.02*** 0.02*** 0.05*** 0.03*** 0.02*** 0.03***Lactation week 0.07*** 0.01*** 0.04*** 0.09*** 0.07*** 0.05***Gestation 0.001*** 0.001*** 0.002*** 0.001*** 0.001*** 0.001***

***P < 0.001.

Journal of Dairy Science Vol. 87, No. 8, 2004

tions were derived by fitting cow as the only effect inthe model. A further analysis was carried out fittingweek of lactation and cow to detect the effect of remov-ing week of lactation on correlations between varioussecretion rate values.Weekly coefficients of variation for the 4 secretion

rate traits (fat, protein, lactose, and water) were calcu-lated to see how their variability changed throughoutlactation.

Investigating Alternative Patterns of Milk Secretion

The pattern of milk secretion (or active cell numbers)used so far is largely unknown, but was based on com-ments by Knight et al. (1998). If alternative patternsof milk secretion (or active cell numbers) become appar-ent, then relative secretion rate patterns derived hereinwill change. For example, Vetharaniam et al. (2003)suggested a different rate of development of cell num-bers during early lactation, usingmammary glandmod-eling studies. The difference between the 2 patterns ofcell numbers is shown in Figure 1. This new patternwas used to calculate secretion rates for cells from theoverall weeklymilk component production curve valuesderived from the least square analysis described aboveusing model 2.

RESULTS

Factors Affecting Test-Day Yields

Results of fitting the effects of cow, month of produc-tion, week of lactation, and month of gestation to the20,095 test-day records of the 6 milk component weighttraits are summarized in Table 1, with the associatedmeans, standard deviations, and coefficients of varia-tion.Nofirst-order interactionswere detected (P < 0.05).Milk weight was 29.2 kg/d on average, and comprised1.12 kg/d of fat, 0.99 kg/d of protein, 1.43 kg/d of lactose,and 25.7 kg/d of water (plus ash, minerals, vitamins,

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Figure 2. Lactation curves for milk and constituent weights de-rived from least square mean values for each week of lactation (kg).

etc.). Total solids produced per day was 3.54 kg. Themodel fitted to all traits accounted for between 58 (fat)and 76% (milk, lactose, and water) of the variance inthe traits, as shown by the coefficient of determination(R2). All 4 effects influenced the 6 weight traits (P