DEB parameters estimation for Mytilus edulis

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DEBparametersestimationforMytilusedulis

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Journal of Sea Research 66 (2011) 289–296

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DEB parameters estimation for Mytilus edulis

S. Saraiva a,b,c,⁎, J. van der Meer a,b, S.A.L.M. Kooijman b, T. Sousa c

a Royal Netherlands Institute for Sea Research (NIOZ), P.O. Box 59, 1790 AB Den Burg, Texel, The Netherlandsb Vrije Universiteit, Department of Theoretical Biology, de Boelelaan 1087, 1081 HV Amsterdam, The Netherlandsc Instituto Superior T'ecnico, Environment and Energy Section, Av. Rovisco Pais, 1, 1049–001 Lisboa, Portugal

⁎ Corresponding author at: Royal Netherlands InstitutBox 59, 1790 AB Den Burg, Texel, The Netherlands.

E-mail addresses: sofia.saraiva@nioz.nl, sofia.marete

a b s t r a c t

Article history:Received 28 December 2010Received in revised form 7 June 2011Accepted 7 June 2011Available online 2 July 2011

Keywords:ModelingBivalvesParameters estimationDEB theory

The potential of DEB theory to simulate an organism life-cycle has been demonstrated at numerous occasions.However, its applicability requires parameter estimates that are not easily obtained by direct observations.During the last years various attempts were made to estimate the main DEB parameters for bivalve species.The estimation procedure was by then, however, rather ad-hoc and based on additional assumptions thatwere not always consistent with the DEB theory principles. A new approach has now been developed – thecovariation method – based on simultaneous minimization of the weighted sum of squared deviationsbetween data sets and model predictions in one single procedure. This paper presents the implementation ofthis method to estimate the DEB parameters for the bluemusselMytilus edulis, using several data sets from theliterature. After comparison with previous trials we conclude that the parameter set obtained by thecovariation method leads to a better fit between model and observations, with potentially more consistencyand robustness.

e for Sea Research (NIOZ), P.O.

c@ist.utl.pt (S. Saraiva).

l rights reserved.

1. Introduction

The Dynamic Energy Budget (DEB) theory for metabolic organi-zation has all the essential components to deal with energy and massbalances, including stoichiometry (Kooijman, 2010; Sousa et al.,2008). However, DEB models use state variables that cannot directlybe measured, such as ‘reserves’ and ‘structure’ both contributing tobody mass. Similarly, many biological quantities that are relativelyeasy to measure have contributions from different basic processes.Respiration rate, as measured by oxygen consumption, for example,does not represent maintenance costs only, but also overhead costs ofgrowth and reproduction. In addition, most DEB parameters, such asthe fraction of energy spent on growth and maintenance (κ), themaintenance rate per unit of volume ([pM]), or the maximum energydensity ([Em]), cannot be measured directly. This implies thatparameter estimation procedures are by necessity complex.

For bivalves, van Haren and Kooijman (1993) presented firstestimates of DEB parameters. Data from the literature were used toestimate several parameters, but rather different values wereobtained from various data sets. This was attributed to differencesin experimental methods, temperature assumptions, salinities, waterdepths and food conditions and, to some extent, genetic variation(van Haren and Kooijman, 1993). After that, van der Veer et al. (2006)

developed a protocol to estimate a complete set of DEB parameters forvarious bivalve species. The authors advocated the use of data fromfactorially designed experiments. But such data sets were, and stillare, lacking and several compromises had to be made. For example, toestimate the shape coefficient parameter (δM, relation betweenphysical and structural length), data on physical length vs. weightwas used, thus disregarding the contribution of reserves to the totalbiomass. Rosland et al. (2009) tried to overcome this problem byestimating this parameter using data from starved organisms,assuming that their body mass would be only structure, but theywere confronted with another problem, which is that their set ofparameter estimates resulted in physiological inconsistencies. Aparticularly important quantity is the yield of reserves on structure(yVE), a measure of the growth efficiency, which is the number ofC-moles of structure built with 1 C-mol of reserves. A value higherthan 1 for this quantity is impossible due to mass conservation. Theparameter set previously obtained by Rosland et al. (2009) and vander Veer et al. (2006) resulted in a yVE value higher than one. Thus, theparameter estimation method should be updated in order to avoidsuch inconsistencies.

As suggested by van der Meer (2006), the standard procedure forparameter estimation should combine all available data sets andestimate parameters by means of simultaneous regression. The authorpresents an exercise using this approach, where part of the estimationprocedure of van Haren and Kooijman (1993) was repeated using twostatistical approaches (simultaneous regression by means of weightednon-linear regression, and repeated measurements or time-seriesregressions). The underlying idea was that if two or more functions

290 S. Saraiva et al. / Journal of Sea Research 66 (2011) 289–296

contain commonparameters it is possible to arrive at a single parameterset estimate using a weighted least-squares algorithm. This proceduremay at the same time result in lower standard errors of the parameters.

Kooijman et al. (2008) provided further guidance by structuringthe estimation of parameters in 10 steps with a minimum set of dataand the use of the regression routines included in the softwarepackage DEBtool (http://www.bio.vu.nl/thb/deb/deblab/debtool/).These routines used several generic algorithms to obtain the best fit,from slow algorithms with a large domain of attraction (geneticalgorithms, Nelder–Mead method), to fast algorithms with a smalldomain of attraction (Newton–Raphson method).

Finally, the contribution by Lika et al. (2011), in this special issue,provides a method based on the simultaneous minimization of theweighted sum of squared deviations between data sets and modelpredictions in one single-step procedure, including physiologicalconstraints on the estimated parameter set (covariation method). Theintended physiological consistency, apart from optimizing thegoodness of fit for all available data, can be obtained using theconcept of sloppy constraints, where ‘pseudo-observations’ are fittedfor particular parameters, simultaneously with real observations(Kooijman et al., 2008). By choosing the weight coefficients in theregression procedure that minimize the weighted sum of squareddeviations, the observations can be obtained (high weight coefficientfor observations) without high deviation of the standard parameters(slightly lower weight coefficient). This study presents the imple-mentation of this approach to estimate DEB parameters for Mytilusedulis, using several collections of literature data. The results arecompared with previous attempts from van der Veer et al. (2006) andRosland et al. (2009).

Table 1Data used in parameter estimation procedure and the weight coefficient βi. The generalized

Data type Symbol Description Value/specific c

Zero-variate ab Age at birth (18 C) 0.2as Age at first spawning 365am Life span 24×365Lb Length at birth (18 C) 0.009Lp Length at puberty 1.2Li Maximum length observed 15WW

b Wet weight at birth 0.037Lb3

WWp Wet weight at puberty 0.037Lp3

WWi Ultimate wet weight 0.037Li3

GSI Gonado-somatic index 0.2Generalized animalparameters

υ Energy conductance 0.02κ Allocation fraction 0.8pM� �

Volume specific somaticmaintenance

κG Growth efficiency 0.8Uni-variate Data01 TPM vs. pseudofeces

production rate7 cm, T=15 °C

Data02 TPM vs. ingestion rate 3.7 cm, T=15Data03 TPM vs. pseudofeces

production rate6 cm, T=6:5 °C

Data04 TPM vs. ingestion rateData05 TPM vs. algae feces

contributionData06, Data07,Data08

Time vs. shell length 0, 13 and 33% oexposure

Data09 Shell length vs. oxygenconsumption

T=15 °C

Data10 Shell length vs. wet weight field observatio

Data11 Time vs. growth starvation, T=L=1:7 cm

Data12, Data13 Ingestion vs. oxygenconsumption

T=15 °C, L=2and 4:5 cm

2. Material and methods

2.1. Covariation method

The covariation method for the parameters estimation is based ona collection of observations (single data points and/or time series) anda set of pseudo-data (used to restrict the possible parametercombinations, see below). The general idea behind the covariationmethod is to let all available information compete, or interact, toproduce the end result, implying the estimation of all parametersfrom all data sets simultaneously (Lika et al., 2011). Thus, all theparameter's values are estimated in one single step for all the availabledata, by minimizing the weighted sum of squared deviations betweendata and model predictions:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n∑n

i=1βi

Yi− Y i

!2vuut

where n is the total number of data points (real and ‘pseudo-data’), βi

theweight coefficient and Yi and Yˆi are the observations and themodelpredictions respectively. The concept of ‘pseudo-data’ is used to avoidan unrealistic combination of parameters and tomaintain the rules forthe covariation of parameter values implied by the physical laws, onwhich DEB theory is built. ‘Pseudo-data’ is in fact a set of values ofprimary or compound parameters for a generalized animal obtainedfrom a large collection of estimated parameters from various data setsfor a wide variety of species (Kooijman, 2010). Estimates should notresult in large deviations from these values, since animals share most

animal parameters refer to T=20 °C.

onditions Dimensions β value References

d 102 Newell (1989)d 102 Newell (1989)d 104 Sukhotin et al. (2007)cm 102 Newell (1989)cm 102 van der Veer et al. (2006)cm 104 van der Veer et al. (2006)gWW 102 van Haren and Kooijman

(1993)gWW 102 van Haren and Kooijman

(1993)gWW 104 van Haren and Kooijman

(1993)mol mol−1 102 Cardoso et al. (2007)cm 10−1 Kooijman (2010)adim 10−1

Jd−1 cm−3 10−1

adim 102

mgdw l−1 vs. mgdwh−1 102 Widdows et al. (1979)

°C mgdw l−1 vs. mgdwh−1 103 Kiørboe et al. (1980)mgdw l−1 vs. mgdwh−1 102

mgdw l−1 vs. μg Chlah−1 102 Prins et al. (1991)mgdw l−1 vs. μg Chlah−1 102

f aerial y vs. mm 102 Rodhouse et al. (1984)

cm vs. cm3O2 h−1 102 Kruger (1960)

ns cm vs. g 10 Borchardt (1985);Pieters et al. (1979)

21:8 °C, d vs. mm d−1 10 Strömgren and Cary (1984)

:5 mg POMh−1 vs. cm3O2 h−1 10 Bayne et al. (1987, 1989)

Table 2Model formulations and main assumptions used to estimate parameters and compute data found in literature. Equations and sections refer to Kooijman (2010). Parametersdescription and dimensions can be found in Table 3.

Type Data Model formulations Model main assumptions

Zero-variate (step 1) ab Eq. 2.39ap Eq. 2.53am Section 6.1.1Lb Eq. 2.46Lp Eq. 2.54Li Li= fLm f=1

Lm¼ υkMg k Tð Þ = k1e

TAT1− TA

� �kM¼ pM½ �

EG½ � kJ=kM

g ¼ ½EG �κ½Em �

½Em� ¼ fpAm gυ

pAmf g = z pM½ �κ

WW¼V 1 + f Em½ �wEμEdV

� �WW

GSI eq. 4.89 f=0.8, T=12 °CUni-variate

Data01 JPI = ∑n

iJXiF− JXi I

� �i=1, inorganic material

JXi I =ρXi I

1 + ∑n

iρXi I

JXi Imf gJXiF = CRXi i=0,algae

CR =CRm� �

1 + ∑n

Xi CRmf gJXi Fmf g

V2 =3 Xi particle concentration

Data02 ∑n

iJXi I ρX1I=ρX0I

Data03 JPI ; similar to Data01 Assumptions described in Saraiva et al. (2011)

Data04 ∑ JXi I ; similar to Data02

Data05 JPA = ∑n

iJXi I− JEiA� �

Data06/07/08 ddt l = kM

e−le + g finitial=1(assumed),T=12°C

f=0.60, 0% aerial exposure(estimated)ddt e = f−eð ÞgkM

l ; scaled reserves density f=0.52, 13% aerial exposuref=0.40, 33% aerial exposure

Data09 pD = κ l3 + 1−κð Þel2 g + lg + e

e= f=1,aclimated organisms(estimated)pG = κ l2 e−l1 + e

JO,oxygen mass balance

Data10 WW = V 1 + f Em½ �wEμEdV

� �e= f=1,aclimated organisms(estimated)

Data11 UEinitial =L3

υfinitial finitial=0.25(estimated)

ddt l;

ddt e f=0, starvation

Data12/13 f = JXyEX

JEAmf gL2 e= f, aclimated organisms

ddt l; pD; pG; JO y1EX=0.13,2.5 cm(estimated)

Similar to Data09 y2EX=0.53,4.5 cm(estimated)

291S. Saraiva et al. / Journal of Sea Research 66 (2011) 289–296

metabolic properties and machinery (Kooijman, 2010; Lika et al.,2011). Different weight coefficients are associated to both the trueobservations and the pseudo-data: a high weight coefficient implieshigh confidence in the data and/or pseudo-data. Pseudo-data typicallyget lower weight coefficients relative to real data. Among real data,the weight coefficients mainly aim to reflect the certainty of the useddata. The comparison between the different parameter sets (van derVeer et al., 2006, Rosland et al., 2009 and this study) is made by anoverall error (E) and a goodness-of-fit mark computed as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑n

i=1βi

Yi− Y iYi

i=1βi

vuuuuuuut

Goodness�of�fit = 10 1−Eð Þ

where n is the number of data points, βi the weight coefficient and Yiand Yˆi are the observations and the model predictions respectively.The inclusion of the weight coefficients in the computation of theoverall data will reflect the uncertainty of the data in line with theassumptions made in the estimation procedure. An overall error of0 (goodness-of-fit mark of 10) represents a perfect model predictionand a high value (low goodness-of-fit mark) represents highdiscrepancies between model and observations.

2.2. Observations

Several observations (zero and uni-variate data) were used in theestimation procedure.

Zero-variate data (single data points) include general physicalcharacteristics (a, age; L length and WW wet weight) on particularstages of M. edulis development: i) birth, the moment when they areable to feed; ii) puberty, the moment when they are able to reproduceand iii) adult stage; and also information about the maximumgonado-somatic index (gonadal mass fraction relative to other tissue)found in field.

The uni-variate data include: (i) detailed feeding observations ontotal pseudofeces production (Prins et al., 1991; Widdows et al.,1979), ingestion rate (Kiørboe et al., 1980; Prins et al., 1991) and algaefeces contribution (Kiørboe et al., 1980; Prins et al., 1991) for differenttotal particulate matter concentrations (TPM) in the water; (ii) fieldobservations on length versus age (Rodhouse et al., 1984) and on wetweight versus length (Borchardt, 1985; Pieters et al., 1979) presentedby van Haren and Kooijman (1993); (iii) oxygen consumption rate atdifferent length from experiments where food level was kept constant(Kruger, 1960); (iv) growth rate during starvation (Strömgren andCary, 1984) and (v) oxygen consumption at different ingestion rates,for two different length groups (Bayne et al., 1987; Bayne et al., 1989).Table 1 lists the data and their relative weights used in the parameterestimation. The relative weight coefficients were arbitrary assignedfor each observation although following a logical reasoning. Relativelymore weight is given to information on ultimate length and weightbecause it is considered reliable and it is crucial for estimating thefeeding parameters, pM

� �and κ. Information on the growth versus age

also has a high relative weight, because it is important that the model

Table 3DEB parameters for Mytilus edulis and other parameters and conversion factors at the refstandard deviation of the estimated parameters.

Symbol Description Units

υ Energy conductance cm d

κ Allocation fraction to growth and somatic Maintenance –

pM½ � Volume specific somatic maintenance Jd−1

[EG] Specific cost for structure J cm−

δM Shape coefficient –

ρX1I Algae binding probability –

ρX0I Inorganic material binding probability –

JX1 Im Algae maximum ingestion rate molCmolC

JX0 Im Inorganic material maximum ingestion rate g d−

g d−

EHb Maturity at birth JEHp Maturity at puberty Jha Weibull aging acceleration d−1

{ĊRm} Maximum surface area-specific clearance rate m3 dJX1Fm

� �Algae maximum surface area-specific filtration rate molC

� �Inorganic material maximum surface area-specific filtration rate g d−

sG Gompertz Stress coefficient –

κR Reproduction efficiency –

Tref Reference temperature KTA Arrhenius temperature KdV=dE Structure and reserves specific density g(dw)

wE=wV Reserves and structure relative molecular mass g(dw)

μE Reserves chemical potential J moVmol Oxygen molar volume l molψdw/C Algae dry weight to carbon conversion factor mg(d

captures the observed main pattern of the organism growth. Theweight coefficient of the pseudo-data is set relatively low mainlybecause their function is to avoid unrealistic parameter combinationwithout forcing toomuch the estimation. An exception is made for thegrowth efficiency, κG, because it is mandatory that this quantity islower than 1.

2.3. Model description

To predict observations presented as Data01 to Data05, thestandard DEB model was coupled with a detailed model for feedingprocesses, as suggested by Saraiva et al. (2011). Total assimilation rateis now related to both food quantity (concentration) and quality(composition). Filtration, ingestion and assimilation are threedifferent processes, using the concept of synthesizing units to describemathematically the fluxes between these processes. The use of a fooddensity parameter (f) and the maximum specific-assimilation rateparameter ( pAmf g) as used in the standard DEB model, is no longerneeded. pAmf g is derived from some additional parameters, alsoestimated in this study. Data sets Data06 to Data13 lacked detailedinformation on feeding conditions. In these cases the standard DEBmodel was applied using the extra parameter food density, f (rangingfrom 0, a situation without food, to 1, a situation with optimalconditions). Formulations and specific assumptions used in this studyare listed in Table 2. Parameters and conversion factors are listed inTable 3. Due to the lack of data on the length of birth and puberty at

erence temperature of T=20 °C. (dw) represents dry weight and (sd) represents the

Value (sd) References

−1 0.056 (0.0077) This study0.094 Rosland et al. (2009)0.067 van der Veer et al. (2006)0.67 (0.13) This study0.45 Rosland et al. (2009)0.7 van der Veer et al. (2006)

cm−3 11.6 (3.52) This study27.8 Rosland et al. (2009)24 van der Veer et al. (2006)

3 5993 (1744) This study1900 Rosland et al. (2009)1900 van der Veer et al. (2006)0.297 (0.0058) this study0.231 Rosland et al. (2009)0.287 van der Veer et al. (2006)0.37 (0.06) This study0.99 Saraiva et al. (2011)0.37 (0.06) This study0.45 Saraiva et al. (2011)

d−1 1.3×104(7×1010) This studyd−1 0.65×104 Saraiva et al. (2011)

1 0.11 (0.078) This study1 0.23 Saraiva et al. (2011)

2.95×10−5 (1.4×10−4) This study1.58×102 (2.5×102) This study5.23×10−10 (2.23×10−9) This study

−1 cm−2 0.096 Saraiva et al. (2011)d−1 cm−2 4.8×10−4 Thomas et al. 2011–(this issue)

1 cm−2 3.5 Saraiva et al. (2011)0.0001 Kooijman (2010)0.95 Kooijman (2010)293 This study7022 van der Veer et al. (2006)

cm−3 0.2 Rosland et al. (2009); Brey (2001)mol−1 25.22 C1 H1.8O0.53 Kooijman (2010)

N0.21 Smaal and Vonck (1997)l−1 697,000 van der Veer et al. (2006)−1 22.4 CODATAw)mgC−1 C 2.5 Slobodkin and Richman (1961)

Table 4Model predictions by data type for each parameter set: overall error and goodness-of-fit mark. Er =

Yi− Y iYi

van der Veer et al. (2006) Rosland et al. (2009) Saraiva et al. (2011) This study

Zero-variate data Value Prediction Er Prediction Er Prediction Er Prediction Er

ab 0.2 0.48 1.40 0.39 0.93 0.23 – 0.23 0.13ap 365 17.26 0.95 34.5 0.97 151.3 – 151.3 0.58am 8760 8078 0.078 8760 0.0011 8777 – 8777 0.003Lb 0.009 0.026 1.89 0.034 2.79 0.0073 – 0.0073 0.200Lp 1.2 0.61 0.49 1.53 0.55 1.27 – 1.27 0.057Li 15 15 0 14.3 0.047 15.7 – 15.7 0.043WW

b 2.7×10−8 5.88×10−7 20.83 6.89×10−7 24.40 1.26×10−8 – 1:26×10−8 0.53WW

p 0.064 0.0077 0.88 0.062 0.96 0.067 – 0.067 0.057WW

i 124.6 111.6 0.10 50.1 0.60 126.6 – 126.6 0.016GSI 0.2 0.33 0.64 1.09 4.44 0.20 – 0.20 0.0043Uni-variate dataData01 – – 0.32 0.46Data02 – – 0.26 0.13Data03 – – 0.16 0.24Data04 – – 4.3 0.74Data05 – – 1.96 0.31Data06 0.32 0.36 – 0.012Data07 0.32 0.34 – 0.014Data08 0.53 0.56 – 0.014Data09 0.034 0.026 – 0.011Data10 0.12 0.45 – 0.12Data11 2.19 3.85 – 0.34Data12 0.0085 0.0047 – 0.0032Data13 0.036 0.027 – 0.0094Overall E 0.01 0.02 0.005 0.001Goodness-of-fit 9.92 9.831 9.94 9.99

different food levels, the maturity maintenance rate coefficient (kJ) isassumed equal to the somatic maintenance rate coefficient (kM), withthe implication that these lengths do not depend on food level. Inorder to compare model predictions and observations, severaladditional assumptions were made concerning the environmentalconditions, including the f parameter estimates (when needed),and/or initial conditions of the organisms in the beginning of theexperiment, also listed in Table 2. More details about the feedingprocess' model extension and the standard DEB model and assump-tions can be found in Saraiva et al. (2011) and Kooijman (2010).

3. Results

The parameter's estimates obtained in this and previous studiesare presented in Table 3. To enable a better evaluation of the results,

a) Windows et al. (1979) b

Fig. 1. Observations vs. model predictions: pseudofeces production fromWiddows et al. (197different parameters set: (i) Saraiva et al. (2011) and (ii) this study.

the observations were predicted using the three different DEBparameter sets: van der Veer et al. (2006), Rosland et al. (2009) andthis study. Besides, two ingestion parameter sets are used: Saraiva etal. (2011) and this study. Table 4 presents the results for the zero-variate data and Figs. 1–4 for the uni-variate data. Additionally,Table 5 presents some quantities useful to verify the consistency andrealism of the parameters combination and species comparison. Itshould be noted that additional information was often needed toperform a model simulation, mainly concerning the conditions of theorganisms in the beginning of the experiments, the food availabilityand also the parameters defining the maturity level at birth andpuberty (controlling the transition stages) and the Weibull agingacceleration (which controls the life span of the organism). In thisstudy these additional parameters were included in the estimationprocedure. Because the parameter sets found in literature do not

) Kiørboe et al.(1980)

9) and ingestion rate from Kiørboe et al. (1980). Simulations were performed using two

Fig. 2. Observations vs. model predictions: pseudofeces production, ingestion rate and algae feces production from Prins et al. (1991). Simulations were performed using twodifferent parameters set: (i) Saraiva et al. (2011) and (ii) this study.

include these quantities, they were separately estimated whilekeeping the main parameters fixed.

The volume specific somatic maintenance, pM� �

and also themaximumspecific assimilation rate, pAm

� �(a parameter in the standard

DEB model but in this study a quantity computed by the feedingprocessesmodel, Table 5) obtained in this study aremuch lower than inthe other studies. This means that the organism is not able to obtain somuch food as previously thought, but it also spends less energy onsomatic maintenance. On the other hand, the obtained specific cost forstructure, [EG], is much higher and closer to 4600 J cm−3, which is theestimated energy content of bivalve structure assuming 23 kJ/g(AFDW)

(van der Veer et al., 2006), 0.2 g(AFDW)/g(WW) (Brey, 2001; Rosland et al.,2009) and 1 g cm−3. The total overhead costs for growth representsthen about 20%, which is a plausible value. The estimates for the energyconductance, v, are close between studies. The allocation fraction togrowth and somatic maintenance, κ, obtained here (κ=0.67) is veryclose to the value obtained by van der Veer et al. (2006) (κ=0.7) andhigher than obtained by Rosland et al. (2009) (κ=0.45). Thosedifferences imply differences on the reproduction strategy of theorganism: a higher value of κ represents higher investment on growthand lower gonado-somatic index as a direct consequence. In Table 4besides the overall error and the goodness-of-fitmark, the relative error

by data type is presented, computed as Er = ∑��Yi− Y i

��. As expected,lower relative errors, Er, are obtained for the data with higher weightcoefficients: life span, am and ultimate length and weight (Li and WW

a) 0% aereal exposure, =0:6 b) 13% aereal expo

Fig. 3. Observations vs. model predictions: shell length over time from Rodhouse et al. (1984al. (2006); (ii) Rosland et al. (2009) and (iii) this study.

High values are found for the weight of birth,Wb, and age at puberty ap,for all the data sets.Wb is very difficult tomeasure and ap is a very roughestimate. Although it is commonly assumed that bivalves can spawn1 year after birth, not much information is available for the exactmoment of puberty. Overall, predictions from this study imply muchlower Er formost dataused comparedwithpreviousestimates.am, Li andData10 are exceptions but the differences are not significant. Figs. 1 and2 concern the detailed feeding processes using Saraiva et al. (2011) andparameters from this study. Both predictions approximate the obser-vations, with a slightly better result from previous estimations. Yet,decrease in goodness of fit for these particular data sets are balancedwith a better goodness offit for the other data sets. The newly estimatedparameter set yielded much better fits for the uni-variate data (Figs. 3and 4), with the oxygen consumption rates as an exception. The oxygenconsumption rate depends very much on the biochemical compositionof the organic compounds (mussel structure and reserves, food andfeces), which can be different from the values assumed in this study dueto lack of data. The overall error, E, obtained in this study is muchlower than before and the obtained goodness-of-fitmark is very close to10.

4. Discussion

The parameters related with the filtration process (maximumclearance rate, CRm

� �and the maximum filtration rates, JX1Fm

� �) were

sure, =0:52 c) 33% aereal exposure, =0:4

). Simulations were performed using three different parameters set: (i) van der Veer et

a) Kruger (1960), =1 b) Borchardt (1985); Pietersetal. (1979), =1

c) Stromgren and Cary(1984), =0 d) Bayne et al. (1987,1989), variable

Fig. 4. Observations vs. model predictions: several authors. Simulations were performed using three different parameters set: (i) van der Veer et al. (2006); (ii) Rosland et al. (2009)and (iii) this study.

estimated and validated by Saraiva et al. (2011) using severalobservations from several authors. Those parameter values were notfurther estimated in this study. In contrary, the ingestion parameters(binding probability, ρX1I, andmaximum ingestion rate JX1 Im) were notkept fixed because the previous estimates were less reliable due tolack of detailed data and other types of relevant data are nowavailable.

The results indicate that the parameter set obtained using thecovariationmethod is a considerable improvement compared to thoseobtained before, using ad-hoc procedures. However, the implemen-tation of the method is not trivial and requires a consistent choice ofthe data to be used and a careful establishment of assumptions. Thechoice of different weight coefficients for the different data is animportant step in the estimation procedure, not only to normalize theerror to a dimensionless number, but also to account for theuncertainty and relative importance of the data, though it should beadmitted that there is subjectivity in the assignment of weightcoefficients. Data available in the literature are not always suitablemainly because: i) information is lacking and too many assumptionshave to be made to run the model and/or ii) the extended standardDEBmodel could still be too simple to cope with the observations. Theuse of the standard DEB model in the covariation method is notimperative. The model can be as complex as the user wants, but it isimportant to keep in mind that technical problems, namely on theconvergence of the estimation method, increase with the complexityof themodel and complexmodels require more detailed input data. Inthis study, the use of an additional parameter to quantify the food

density, f, to predict some of the experiments/field measurements isan example of a model simplification, disregarding for instance theseasonal pattern of food density and quality. However, this parameter,when used, was estimated or adjusted for each data type and it isbelieved that this approximation is consistent with the type of dataused in the procedure, particularly for the growth prediction at thescale of years. This study uses also observations directly linked withthe feeding processes (particularly ingestion) in order to improveprevious estimations of the respective parameters. The use ofobservations on oxygen consumption during starvation was avoided.In order to correctly simulate extreme starvation, the model wouldhave to account with the possible utilization of the reproductionbuffer contents (gametes) and the possible shrinking of the organism,which would imply more complexity and, evenmore important, moreaccurate information about the reproduction processes, which are inthis study summarized by the gonado-somatic index observation.More information on the eggs production and/or energy contentwould improve the κ parameter estimation as well as the reproduc-tion efficiency, kR, assumed constant in this study.

5. Conclusion

The optimal parameter set for a model is the one that best predictsthe observations but without losing physiological realism. The para-meter's estimation procedure implemented in this study – thecovariation method – is based not only on minimization of theweighted sum of squared deviations for all data sets simultaneously,

Table 5Other quantities computed by the model (at T=20 °C).

Symbol Description Units van der Veer et al. (2006) Rosland et al. (2009) This study

ME0 Initial reserve mass at optimal food conditions mol 2.97×10−9 4.50×10−9 1.49×10−10

W0 Initial weight at optimal food conditions g 7.48×10−8 1.14×10−7 3.75×10−9

UEb/UE0 Fraction of reserve left at birth – 0.45 0.34 0.14MV

b Structural mass at birth mol 3.33×10−9 3.89×10−9 7.92×10−11

Wb/Wm Birth weight as fraction of maximum – 5.23×10−9 1.36×10−8 9.94×10−11

MVp Structural mass at puberty mol 4.36×10−5 3.5×10−4 4.24×10−4

WP /Wm Puberty weight as fraction of maximum – 6.89×10−5 1.24×10−3 5.32×10−4

MVi Ultimate structural mass mol 0.63 0.28 0.80

delV Fraction of weight that is structure – 0.72 0.72 0.79fGb f for growth ceasing at birth – 1.74×10−3 2.39×10−3 4.63×10−4

fJb f for maturation ceasing at birth – 1.74×10−3 2.39×10−3 4.63×10−4

fJp f for maturation and growth ceasing at puberty – 0.041 0.11 0.081[Em] / [pM] Maximum survival time when starved d 91.2 78.1 124z Zoom factor relative to reference Lm=1 cm – 4.30 3.30 4.65pAm� �

Maximum specific assimilation rate J d−1 cm−2 147.6 204 80.5JEAm� �

maximum surface-specific assimilation rate mold−1 cm−2 2.11×10−4 2.93×10−4 1.15×10−4

tE Maximum reserve residence time d 64.2 35.1 83.03[Em] Reserve capacity J cm−3 2203 2170 1438kM Somatic maintenance rate coefficient at T d−1 0.0126 0.0146 0.0019kJ Maturity maintenance rate coefficient at T d−1 0.0126 0.0146 0.0019k maintenance ratio – 1 1 1JEM Volume-specific somatic maintenance costs mol d−1 cm−3 3.44×10−5 3.99×10−5 1.66×10−5

RQ Respiration quotient at maximum length molC molO−1 0.97 0.97 0.97UQ Urination quotient at maximum length molN molO−1 0.20 0.20 0.20WQi Watering quotient at maximum length molH molO−1 0.57 0.57 0.57JO Dioxygen use per gram at maximum length lg−1 h−1 1.21×10−4 1.46×10−4 6.49×10−5

GSI Gonado-somatic index at optimal food mol mol−1 0.49 1.6 0.30Rm Ultimate reproduction rate d−1 3.77×105 3.70×105 5.26×106yVE Yield of structure on reserve mol mol−1 2.91 2.91 0.92g Energy investment ratio – 1.23 1.95 6.2rB von Bertalanffy growth rate d−1 2:32×10−3 3.22×10−3 5.6×10−4

[MV] Volume-specific structural mass mol cm−3 7.93×10−3 7.93×10−3 7.93×10−3

[EV] Volume-specific structural energy J cm−3 5527 5527 5527ψEW Energy density of whole body J g−1 2.76×104 2.76×104 2.76×104

but also on the inclusion of physiological constraints by introducing theconcept of pseudo-data. The parameter set obtained reveals not onlymore consistency and realism but also better predictions than previousestimates. For that reason, the use of the new parameter set should beused in future implementations of a DEB model onM. edulis energetics.

Acknowledgments

Thisworkwas supported by Fundação para a Ciência e a Tecnologia,Portugal (SFRH/BD/44448/2008) and by Life Engine — Development ofbiological realism in computer games (QREN 3111). The authors wouldalso like to thank the contribution of two anonymous reviewers.

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