REGULATION OF REPRODUCTIVE PROCESSES WITH DYNAMIC ENERGY BUDGETS

Abstract

1. The simple bioenergetic models in the family of Dynamic Energy Budget (DEB) consist of a small number of state equations quantifying universal processes, such as feeding, maintenance, development, reproduction and growth. Linking these organismal level processes to underlying suborganismal mechanisms at the molecular, cellular and organ level constitutes a major challenge for predictive ecological risk assessments.

2. Motivated by the need for process-based models to evaluate the impact of endocrine disruptors on ecologically relevant endpoints, this paper develops and evaluates two general modeling modules describing demand-driven feedback mechanisms exerted by gonads on the allocation of resources to production of reproductive matter within the DEB modeling framework.

3. These modules describe iteroparous, semelparous and batch-mode reproductive strategies. The modules have a generic form with both positive and negative feedback components; species and sex specific attributes of endocrine regulation can be added without changing the core of the modules.

4. We demonstrate that these modules successfully describe time-resolved measurements of wet weight of body, ovaries and liver, egg diameter and plasma content of vitellogenin and estradiol in rainbow trout (Oncorynchus mykiss) by fitting these models to published and new data, which require the estimation of less than two parameters per data type.

5. We illustrate the general applicability of the concept of demand-driven allocation of resources to reproduction as worked out in this paper by evaluating one of the modules with data on growth and seed production of an annual plant, the common bean (Phaseolis vulgaris).

Introduction

Dynamic Energy Budget (DEB) theory offers a remarkably general mathematical and conceptual framework for physiological ecology. Originally formulated to describe growth and reproduction in animals, DEB theory now describes widespread empirical patterns in metabolic behavior of a steadily increasing number species (approaching 1,000 at the time of writing) from phyla from all three domains (Sousa, Domingos & Kooijman 2008; Kooijman 2010; AmP 2017; Jusup et al. 2017). Its core concepts are consistent with some general trends in evolutionary history (Kooijman 1986; Kooijman & Troost 2007) and with the principles of thermodynamics (Sousa et al. 2010; Jusup et al. 2017). In addition, the theory offers a powerful framework for modeling organismal response to environmental stress, notably in ecotoxicology (Kooijman & Bedaux 1996; Jager et al. 2014; Muller et al. 2014) and, more recently, in the context of ocean acidification (Muller & Nisbet 2014; Jager, Ravagnan & Dupont 2016), starvation (Gergs & Jager 2014) and crowding stress (Gergs, Preuss & Palmqvist 2014). The versatility of the theory is due to its modular structure, through which specific attributes or ‘details’ of a particular environment, stressor or species can be included without changing the core of the model. Here we follow a similar approach to accommodate life history strategies by which organisms allocate resources to reproduction. Since reproduction generally constitutes a major fraction of the total energy budget of an adult organism, the energetic implications of different reproductive strategies and their trade-offs play a fundamental role in life history theory (Stearns 1992).

An important feature of most DEB models is that resources are first assimilated into somatic reserves, which are then committed to support somatic, developmental and/or reproductive functions, depending on nutritional status and life stage. In the standard formulation of DEB (stDEB), applicable to animals, the rate at which reserves are allocated to reproduction depends only on the reserve density and the size of the animal (see Figure 1). Control mechanisms regulating the partitioning of reserves to favor growth over reproduction, or vice versa, are absent. Standard DEB ignores control mechanisms regulating the development of gonads, as the specifics of those mechanisms vary widely among taxa and sexes (but see Pecquerie, Petitgas and Kooijman (2009), Einarsson, Birnir and Sigurosson (2011) and Llandres et al. (2015) for species or group specific DEB gonad loading modeling modules for anchovy, capelin and parasitic wasps, respectively). This lack of feedback simplifies the dynamics of resource allocation, with obvious mathematical advantages as a result. Yet, Standard DEB quantifies reproductive output sufficiently accurately for many purposes, for instance those that require estimates of reproductive output over longer time spans or those involving species that release gametes in a nearly continuous manner. However, it is important to consider feedback, e.g. mediated by endocrine regulation mechanisms, in order to capture the dynamics of gamete production in iteroparous and semelparous organisms, in which gametes are primarily formed during the later part of the reproductive cycle or near the end of the life cycle, respectively. In addition, this kind of feedback could provide an entry to mechanistic modeling of the impact of endocrine disruptors on growth and reproduction in the DEB framework.

Figure 1.

Conceptual representations of the standard DEB (stDEB) model for healthy adults and of two types of modifications, dDEB and stDEB+. stDEB (Nisbet et al. 2000; Kooijman 2010; Jusup et al. 2017) describes the rates at which an adult animal acquires food, assimilates the energy and nutrients therein into general reserves, and allocates those reserves to somatic and maturity maintenance, growth and reproduction; this allocation is defined as catabolism. A fixed fraction κ of the catabolic flux is allocated to somatic maintenance and growth. Somatic and maturity maintenance are demand-driven processes and take priority over growth and reproduction; all other processes in stDEB are supply-driven. In dDEB, stDEB is modified to include positive and negative feedback of the reproductive buffer on the allocation of the catabolic flux. Thus, in dDEB, reproduction is a demand-driven process with a variable fraction λ of the catabolic flux allocated to maturity maintenance and reproduction. stDEB+ separates the reproductive buffer in two pools: reproductive reserves and actual reproductive matter (gonads). The rate at which reproductive reserves are converted into reproductive matter depends on the densities of reproductive reserve and reproductive matter, implying that gonad loading is a demand-driven process. Solid arrows represent energy and material fluxes; broken arrows represent feedback mechanisms; boxes represent state variables; modifications of dDEB and stDEB+ relative to stDEB are presented in black while communalities are shown in grey. Note that DEB processes and quantities are abstractions; auxiliary rules are required to relate them to experimental quantities – see Table 1.

To more accurately accommodate the alternative reproductive strategies of iteroparous and semelparous organisms, we develop and evaluate the performance of two extensions of the standard DEB model. These extensions include demand-driven feedback mechanisms on gonad development, guided by the premise that hormones produced in the reproductive organs, among other organs, commonly mediate those feedback mechanisms. We center our evaluation of model performance on a single fish species, the rainbow trout (Oncorynchus mykiss), due to the expansive data set on its growth and reproductive biology. However, we argue that the model extensions are based on general principles, and therefore applicable to other species. As an illustration, we discuss how simplified formalism from one of the model extensions can be applied to describe the growth and reproductive patterns in a species very different from trout, namely the common bean (Phaseolis vulgaris). Beans have a reproductive strategy typical for many annual plants, namely an allocation strategy that favors seed production over somatic growth during the later phases of the life cycle. In addition, we discuss how these extensions can be useful in exploring physiological mechanisms by which stressors, in particular endocrine disruptors, affect resource allocation, and ultimately adverse outcomes to reproduction and growth.

Materials and methods

DATA SOURCES

Three data sets about female rainbow trout (O. mykiss) were analyzed to evaluate model performance. The most expansive set, referred to as main data set, was from Nagler et al. (2012) with additional data from Gillies et al. (2016), and concerns a reproductively synchronized autumn-spawning population obtained from a commercial supplier (Troutlodge, Inc., Sumner, WA) and maintained in a temperature controlled flow-through system under a natural lighting regime at the Battelle Marine Science Facility (Sequim, WA). The main data set included time-resolved measurements of wet weight of body, ovaries and liver, egg diameter and plasma content of vitellogenin and estradiol of 58 individuals. The two supplementary data sets, SD1 and SD2, were more limited in scope. SD1 included time resolved measurements of body weight and egg mass of 12 and 9 individuals, respectively, of a spring spawning strain obtained from Troutlodge Inc. (Sumner, WA). SD2 included initial and final total body and egg weights as well as weights and diameters of individual eggs of 16 individuals of a fall-spawning strain obtained from Nisqually Trout Farm (Lacey, WA). Fish of SD1 and SD2 were kept in the same facility as those of the main set; see Nagler et al. (2012), Schultz et al. (2013) and the Supplemental Information for experimental detail. All sets span a single breeding cycle of approximately 11–14 months starting immediately after the time of first spawning.

The common bean, Phaseolis vulgaris, was used to evaluate the potential of the principle of demand driven resource allocation to reproduction (see next section) to capture the dynamics of growth and reproduction of a species wildly different from iteroparous rainbow trout; beans have a semelparous reproductive strategy typical for many annual plants, namely an allocation strategy that favors seed production over somatic growth during the later phases of the life cycle. Data are from Lima et al. (2005) and include time-resolved measurements of vegetative above ground biomass, leaf cover and pod biomass of 6 cultivars grown in a field setting in coastal Brazil from May to August (mean growing conditions: 21.2°C, 70% humidity, 6.9 h solar radiation per day; 12 seeds per row meter at 0.5 m row distance; plots fertilized with 2.5 g N, 4.0 g P and 4.0 g K per square meter).

DYNAMIC ENERGY BUDGET THEORY

This study uses the standard model of Dynamic Energy Budget (stDEB) theory as a reference. Since Kooijman (2010) has described this theory and its standard formulation in detail and several other publications provide extensive summaries (Nisbet et al. 2000; Sousa, Domingos & Kooijman 2008; Jusup et al. 2017), we only present features of the theory that are essential to evaluate the models developed in this study.

The stDEB formulation (see Fig. 1), describes the rates at which a ‘generalized’ animal acquires resources from its environment and uses the energy therein for somatic and maturity maintenance, growth, maturation (juveniles) and reproduction (adults). A ‘generalized’ animal is heterotrophic, grows isometrically (constant shape), does not encounter conditions of stress (including debilitating forms of starvation), and has three life stages: embryonic (during which it does not feed), juvenile (feeding but no reproduction) and adult. Since this study involves the adult stage only, from now on, all references to animals pertain to adults, unless other life stages are explicitly mentioned. stDEB distinguishes three pools of biomass: structure, general reserve and material in the reproductive buffer. Structure is defined as the biomass requiring maintenance in order to remain viable. The reproductive buffer contains resources tagged for reproduction (irreversibly, except potentially during starvation conditions). General reserve is functionally defined as all other metabolizable biomass; in practice, general reserve typically includes conventional storage materials as well as compounds that are traditionally not thought of as reserve, such as ribosomes in excess of the minimal amount needed to ensure vitality of an organism of a given size (Nisbet et al. 2000). The gross biochemical composition of each pool is considered to be invariant, implying that the costs to produce a unit of each type of biomass and the cost to maintain a unit of structure are constant. The general reserve density, i.e. the ratio of general reserve and structure, stabilizes in a constant food environment.

Environmental resources are first assimilated into general reserve, which is subsequently committed to somatic and developmental/ reproductive functions, with each set of functions receiving a constant fraction κ of committed general reserve (see Figure 1). In order to accommodate the changing rate of gamete development during a reproductive cycle in female rainbow trout, we studied two extensions to the standard model (see Figure 1). In the first variant, the proportion of committed general reserve allocated to reproduction is subject to feedback regulation of the reproductive buffer, implying that the allocation of general reserve to reproduction is driven by demand of the reproductive buffer. This variant is denoted dDEB, with the ‘d’ standing for ‘demand-driven’. The second variant, a modified version of a capelin model by Einarsson, Birnir and Sigurosson (2011), assumes stDEB but separates the reproductive buffer in pools of unspecified reproductive reserve and actual reproductive matter. A gonad loading modeling module describes the rate at which reproductive reserve are converted into actual reproductive matter. This variant will be denoted stDEB+, with the ‘+’ referring to the gonad loading module. Regulation of the allocation of reserves to the reproductive buffer in dDEB and of gonad loading in stDEB+ are subject to endocrine control.

The derivations of the dDEB and stDEB+ model equations in Table 1 are presented in full in the Supplementary Information. Here, only the assumptions that are not part of stDEB are presented and evaluated. The following list contains assumptions shared by and specific to both model variants, though it should be stressed that reproductive matter is defined differently in those variants. In dDEB, reproductive matter refers to all matter in the reproductive buffer regardless of location in the body, whereas reproductive matter roughly corresponds to gametes in stDEB+. The assumptions are:

Table 1.

Equations.

DEB Model Expressions
General reserve density (constant food), mE
All variants	fmEm	(1)
Fraction mobilized general reserves to reproduction and maturity maintenance, λ
stDEB+, stDEB	1−κ	(2)
dDEB	$4{\lambda }_{\mathrm{m}}{m}_F\left(m_{F\mathrm{m}}-m_F\right)m_{F\mathrm{m}}^{-2}$	(3)
Growth rate, dMV / dt = jVMV
All variants	$\left(\left(1-\lambda \right)k_{E}S{m}_E-j_M\right)M_V{\left(\left(1-\lambda \right)m_E+y_V^{-1}\right)}^{-1}$	(4)
Dynamics of the density of reproductive buffer in between spawning events, dmF / dt
dDEB, stDEB	$y_F\left(\lambda m_E\left(k_{E}S-j_V\right)-k_J{M}_{HD}{M}_V^{-1}\right)-j_V{m}_F$	(5)
Dynamics of the density of reproductive reserves, dmRE / dt
stDEB+	$y_{RE}\left(\left(1-\kappa \right)\left(k_{E}S-j_V\right)m_E-k_J{M}_{HD}{M}_V^{-1}\right)-m_{RE}\left(j_V+k_F{m}_G\left(m_{G\mathrm{m}}-m_G\right)\right)$	(6)
Dynamics of the density of reproductive matter in between spawning events, dmG / dt
stDEB+	yGkFmREmG(mGm−mG)−jVmG	(7)
Equations Linking Trout Data to DEB quantities
Total body wet weight, WB
dDEB	(1 + mE + mF)dMMV/dW	(8)
stDEB+	(1 + mE + mF + mG)dMMV/dW	(9)
Ovary wet weight, WO
dDEB	κOVmFdMMV/dW	(10)
stDEB+	κOVmGdMMV/dW	(11)
Liver wet weight, WL
dDEB	(p + mF)(1−κOV)dMMV/dW	(12)
stDEB+	(p + mG)(1−κOV)dMMV/dW	(13)
Mean follicle diameter, LF
dDEB	${\left(6{\kappa }_{OV}{d}_M{M}_V{m}_F/\pi n{d}_W\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	(14)
stDEB+	${\left(6{\kappa }_{OV}{d}_M{M}_V{m}_G/\pi n{d}_W\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$	(15)
Plasma estradiol concentration, E2
dDEB	q1λ	(16)
stDEB+	q2mG(mGm − mG)	(17)
Plasma vitellogenin concentration, VT
dDEB	$d_T{y}_F\left(\lambda m_E\left(k_{E}S-j_V\right)-k_J{M}_{HD}{M}_V^{-1}\right)-\left(k_T+j_V\right)V_T$	(18)
stDEB+	dTkREmREmG(mGm − mG)−(kT + jV)VT	(19)

1. At the onset of a reproductive cycle, a small fraction of somatic biomass is converted to reproductive matter, e.g. due to meiosis. General reserve and structure contribute proportionally to the initial formation of reproductive matter, and the costs of this conversion are negligible. The latter two assumptions are rather arbitrary but quantitatively insubstantial.

2. The initial density of reproductive matter is constant. This assumption maintains parameter parsimony and model simplicity.

3. An adult has a bounded capacity to carry reproductive matter. In non-starving adults, this capacity is proportional to the amount of structural biomass, i.e. the maximum density of reproductive matter is a constant. This assumption maintains parameter parsimony and model simplicity.

4. dDEB only: the fraction of mobilized general reserve allocated to reproduction and maturity maintenance in adults is proportional to (1) the density of reproductive matter, and (2) the difference between the maximum and actual density of reproductive matter. The first proportionality introduces positive feedback and is based on the general observation that the ovaries in fish produce estrogen, which stimulates the production of vitellogenin, the precursor of egg reserve material (Tyler & Sumpter 1996). The second proportionality provides a simple negative feedback (i.e. deceleration mechanism that causes the accumulation of gonadal material to slow down towards the end of a reproductive cycle.

5. stDEB+ only: the rate at which reproductive reserves are released is proportional to (1) the density of reproductive reserves, (2) the density of reproductive matter, (3) the difference between the maximum and actual density of reproductive matter, and (4) the amount of structural biomass. The first proportionality ensures the density of reproductive reserves cannot become negative; for arguments for the two subsequent proportionalities, see previous assumption.

6. The efficiency with which reproductive reserves are converted into reproductive matter is constant.

7. Spawning requires the density of reproductive matter to exceed a threshold and, additionally, may be under the control of a time trigger or environmental factor, depending on species.

LINK BETWEEN DEB QUANTITIES AND DATA

Variables in DEB models are abstract quantities and therefore do not correspond directly with measurable quantities. The mapping of DEB quantities onto the data analyzed in this study, including total body, ovary and liver wet weights, follicle diameter and plasma levels of estradiol and vitellogenin, is achieved through auxiliary assumptions stated in this section; the corresponding equations, summarized in Table 1, are derived in the Supplementary Information. The relationship between measurable quantities pertaining to the common bean and those of a DEB model of bean growth and fecundity can be found in the Supplementary Information.

In order to convert DEB mass quantities to wet weights, we use conversion factors from the trout entry in the DEB parameter database (AmP 2017). Considering that the ovaries mainly consist of storage materials in eggs, we assume the contributions of structure and general reserves to the wet weight of the ovaries are negligible (to avoid confusion, we will use ‘storage’ to refer to physical materials and ‘reserves’ as the conceptual abstraction in the context of DEB). We also assume that the fraction of reproductive matter that is in the ovaries is constant. Furthermore, we assume that reproductive matter is either in the ovaries or in the liver, which produces the precursors of egg storage materials. It seems natural also to include plasma vitellogenin, the precursor of egg storage materials. However, plasma vitellogenin levels are especially high just prior and after ovulation, indicating that not all plasma vitellogenin ends up in eggs. Furthermore, the amount of plasma vitellogenin is relatively small. Plasma contributes 2.5% to 5.5% to body wet weight in teleost fish (Brill et al. 1998, and references therein) and contains about 25 mg vitellogenin/ ml during the phase of accelerating ovary growth in a typical individual in this study (see figure 2F), which corresponds to only about 1.5–3.5 g vitellogenin in a 2.5 kg fish. Thus, it is reasonable to ignore the contribution of vitellogenin to reproductive matter, though its dynamics are informative and are modeled later. Furthermore, we assume that the fractions of structure and reserves that are part of the liver are constants for both model variants, and, for stDEB+, in order to retain simplicity, that the amount of reproductive reserves in the liver is negligible.

Figure 2.

Model fits of dDEB (solid line) and stDEB+ (dashed line) to main data set with rainbow trout (symbols), including (A) total body wet weight; (B) total body wet weight less wet weight of ovaries; (C) wet weight of ovaries; (D) wet weight of liver; (E) mean diameter of maturing follicles (mean per fish); (F) plasma vitellogenin content; and (G) plasma estradiol content. Measurements denoted ‘x’ in Panel A were used to calculate corresponding data in Panel B and were therefore omitted in the fitting procedure. Error bars denote standard deviations (n = 3 or 4). Parameter estimates are given in Table 1d and goodness-of-fit measures in Table 2. Data from Nagler et al. (2012) and Gillies et al. (2016).

This leaves the follicle diameter and estradiol and vitellogenin plasma levels as the experimental quantities that need to be related to DEB variables. In order to relate the mean diameter of a follicle to reproductive matter, we assume that follicles are perfect spheres and that the specific gravity of biomass equals unity. Estradiol is produced by the ovaries and regulates the flow of vitellogenin to the ovaries. Accordingly, we link the gonad loading module of stDEB+ and the reproduction flux in dDEB to the plasma estradiol concentration assuming simple proportionality.

To model the dynamics of plasma vitellogenin, we assume that the volume of plasma is proportional to the amount of structural biomass, and that the rate at which vitellogenin is cleared from plasma is proportional to the amount of structural biomass (e.g. by structural mass in the ovaries). Furthermore, for dDEB, we assume that the rate at which vitellogenin is released into the blood stream is proportional to the rate at which somatic reserves are allocated to reproduction. For stDEB+, we assume that the rate at which vitellogenin is released into the blood stream is proportional to the rate at which reproductive reserves are allocated to reproductive matter.

PARAMETERIZATION

In the evaluation of model performance with trout data, the values of some or all parameters in Table 2 were fixed, depending on the information content of the data and on the purpose of the analysis (see legend to Figure 4 for information about parameter values regarding the analysis of bean data). The main data set was used to parameterize the model variants; subsequently, this parameterization was used to predict the observations in the supplementary data sets SD1 and SD2 (with one exception – see next section). However, not all parameters were estimable from the main data set due to a lack of information about, e.g., elemental biomass composition and some conversion efficiencies, and therefore had to be fixed; similar values were used for fixed parameters that occur in both model variants. The values of eight fixed parameters, as marked in Table 2c, were taken or calculated from the rainbow trout entry in the DEB parameter database (AmP 2017). Among those was the somatic maintenance rate parameter, which could not be estimated as it strongly covaried with other parameters, notably the general reserve turnover rate. Since the value of the somatic maintenance rate parameter is relatively invariant across species (Kooijman 2010), it was fixed at the value in the DEB parameter database, while the latter was treated as a free parameter.

Table 2.

Parameters and variables used in the analysis of the main set of rainbow trout data. (a) Dynamic model quantities; (b) Experimental variables; (c) fixed parameters; (d) estimated parameters.

(a) Dynamic model quantities
	Interpretation	Units
jV	Specific growth rate	day−1
mE	Density of general reserves	-
mF	Density of reproductive buffer (dDEB)	-
mG	Density of reproductive matter (stDEB+)	-
mRE	Density of reproductive reserves (stDEB+)	-
MV	Amount of structural biomass	C-mole
S	Shape correction factor, (MVm / MV)1/3	-
λ	Fraction of reserves allocated to reproduction (dDEB)	-
(b) Experimental variables
	Interpretation	Units
E2	Plasma estradiol content	ng ml−1
LF	Follicle diameter	mm
WB	Wet weight total body	kg
WL	Wet weight liver	g
WO	Wet weight ovaries	kg
VT	Plasma vitellogenin content	mg ml−1
(c) Fixed parameters (T=11°C)
	Interpretation	Value	Source
dM	C-mole to dry weight conversion	24.6 g C-mole−1	AmP*
dW	Wet weight to dry weight conversion	0.2	AmP
f	Scaled food density	0.9	See text**
jM	Specific maintenance rate	0.025 day−1	AmP
kJ	Maturity maintenance coefficient	0 day−1	See text
mF0	Initial density of reproductive matter (stDEB+)	0***	See text
mGm	Maximum density of reproductive matter (stDEB+)	6.60	See text
MVm	Maximum structural biomass	1.12 C-mole	AmP
yF	Conversion efficiency general reserves to reproductive buffer	0.95	AmP
yG	Conversion efficiency reproductive reserves to gonads (stDEB+)	1	See text
yRE	Conversion efficiency general to reproductive reserves	0.95	AmP
yV	Conversion efficiency reserves to structure	0.88	AmP
κ	Fraction reserves allocated to soma (stDEB+)	0.56	AmP
(d) Estimated parameters
	Interpretation	dDEB		stDEB+		Units
		Value	95% CI	Value	95% CI
dT	Vitellogenin conversion factor	131.6	71.5–339.3	102.2	56.4–404.7	mg day−1
kE	General reserve turn-over rate	3.37	2.99–3.71	3.63	3.25–3.99	× 10−3 day−1
kRE	Reproductive reserve turn-over rate	NA	NA	1.11	1.00–1.24	× 10−3 day−1
kT	Vitellogenin clearance rate	0.044	0.016–0.142	0.032	0.012–0.166	day−1
mF0	Initial density of reproductive buffer	1.67	0.66–3.64	NA*	NA*	× 10−3
mFm	Maximum density of reproductive buffer	3.67	3.20–4.21	NA	NA	-
mG0	Initial density of reproductive matter	NA	NA	9.28	4.60–17.5	× 10−3
MV0	Initial amount of structural biomass	0.846	0.787–0.915	0.827	0.770–0.890	C-mole
n	Number of eggs	4.43	3.57–5.48	5.15	4.10–6.52	× 103 #
p	Compound parameter, (κVL + κELmE)/(1 − κOV)	5.50	3.52–10.89	5.76	3.63–11.6	-
q1	Estradiol conversion factor	56.0	44.6–66.9	NA	NA	ng ml−1
q2	Estradiol conversion factor	NA	NA	3.40	2.58–4.24	ng ml−1
VT0	Initial plasma vitellogenin content	102.3	66.1–142.6	96.7	57.2–144.9	mg ml−1
κOV	Fraction of reproductive matter in ovaries	0.971	0.957–0.984	0.967	0.951–0.982	-
λm	Maximum fraction of reserves to reproduction	0.761	0.684–0.839	NA	NA	-

^*‘Add my Pet’ DEB parameter data base (2017)

^**Parameterization section in Materials and Methods

^***Free parameter in dDEB – see Table 2d

^*Fixed parameter in stDEB+ - see Table 2c

Figure 4.

Application of a simplified version of dDEB to production in the common bean, Phaseolis vulgaris. (A) An empirical third degree polynomial describes the dynamics of the leaf area index, defined as the total green leaf surface are per unit area ground cover, an important determinant of the photosynthetic capacity (p₁ = 30.5 min⁻¹, p₂ = 5.2 min⁻², p₃ = −0.08 min⁻³). (B) The simplified dDEB model fits above ground vegetative biomass (open circles, solid curve) and pod mass (closed circles, dotted curve) with mean bean mass as the initial amount of structural biomass, observed mean time of first flowering (34 d) as starting point of photosynthate allocation to reproduction, m_F = 0.01 and negligible losses in converting photosynthate into vegetative and reproductive biomass. Parameter estimates (with 95% confidence intervals) are λ_m = 0.52 (0.30–0.87), m_Fm = 1.09 (0.95–1.24), j_M = 0.08 (0.03–0.16) d⁻¹ and c = 0.12 (0.08–0.17); $J_{Pm}^{°}$ = 65.2 g dry weight m⁻² d^−1- based on the net photosynthesis rate estimated by Sale (1975). Data are from Lima et al. (2005) and represent the means of four replicates of six cultivars grown from large seeds. See Supplemental Information for model description.

The reasoning for the remaining five fixed values is as follows. First, the value for the scaled food density was set at 0.9, which is close to its maximum of 1.0, as the fish were well fed. Second, according to the parameter database, maturity maintenance costs would have been an insubstantial fraction of the total energy budget of the fishes and were therefore ignored. Third, the initial density of reproductive reserve in stDEB+ was assumed negligible, since there was no information available that could be used to identify the reproductive reserve pool as a pool separate from general reserve and reproductive matter in this model variant (in contrast, this parameter could be estimated for dDEB – see Table 2d). This assumption is supported by the fact the fish had recently matured and were stripped before the experiment. Fourth, the maximum density of reproductive matter in stDEB+ strongly covaried with other parameters and was therefore fixed; it was identical to the density of reproductive matter in a female of ultimate size at optimal conditions after one year according to the parameter database. Fifth, the conversion efficiency of reproductive reserves to reproductive matter in stDEB+ was set at unity, implying that all the conversion overheads were subsumed in the conversion of general into reproductive reserve.

Free parameters were estimated by maximizing likelihood considering all data types in a set simultaneously, while assuming that discrepancies between data and model predictions were due to normally distributed homoscedastic error in the data. These estimations were done with a modified version of the BYOM platform coded in Matlab (www.debtox.info/byom). Confidence intervals were estimated from the likelihood profile of each parameter. Universally suitable goodness-of-fit measures are lacking for nonlinear models (see e.g. Shcherbakov et al. 2013), which problem was compounded by the composite nature of the trout data sets analyzed in this study. Therefore, in the analysis of trout data sets, in addition to likelihood values, two goodness-of-fit measures were used to evaluate model performance: the symmetric mean scaled error, SMScE_i, and the model efficiency, ME - see Supplemental Information for equations.

Results

The dDEB and stDEB+ models are relatively parameter sparse. The dDEB model needed 21 parameters, of which 12 were estimated, to describe the patterns in the main data set by Gillies et al. (Gillies et al. 2016), including total body, ovaries, total body less ovaries and liver wet weight, mean follicle diameter and vitellogenin and estradiol plasma content. The stDEB+ model required two more parameters, 23 in total, of which 11 could be estimated from the main data set. Thus, on average, less than two parameters were estimated from each data type.

Despite this relative parameter sparseness, both models fit the trends in the main data set well (see Figure 2 and Table 2). The fits to the weight and follicle diameter data are virtually indistinguishable between the two models (see Figure 2A–E). The goodness-of-fit measures are also similar for the two models (see Table 2). In addition, the estimated values for the general reserve turnover rate k_E, the only free core DEB parameter, are statistically indistinguishable at the 95% level (see Table 2d), though the value implied by the parameters published in the DEB parameter database for rainbow trout (AmP 2017) is about 10–20% lower (2.92e-3 day⁻¹ at 11°C). More divergence in model performance is seen in the predictions of plasma vitellogenin and estradiol contents, notably during the last third of the reproductive cycle (see Figure 2F–G). The peaks of those plasma contents in this period are substantially better described by dDEB than by stDEB+, as the latter cannot capture the drop in plasma vitellogenin and estradiol levels near the end of the reproductive cycle. The goodness-of-fit measures for those plasma contents also favor dDEB over stDEB+ (see Table 3). In addition, the overall goodness-of-fit measures point to dDEB as the superior model. The log likelihood of dDEB is 21.9 higher than that of stDEB+, which difference is significant given that dDEB has only one more free parameter than stDEB+; any difference between log likelihoods beyond two would indicate dDEB is the better model according to the AIC criterion.

Table 3.

Statistics of model fits to Main data set¹.

Data type	Figure	dDEB $\ln \mathcal{L}=-1551.2$			stDEB+ $\ln \mathcal{L}=-1573.1$
		σ	SMScE	ME	σ	SMScE	ME
E2	2G	11.8 ng. ml−1	0.592	0.556	14.2 ng ml−1	0.724	0.363
LF	2E	0.445 mm	0.162	0.906	0.491 mm	0.298	0.885
WB	2A	170 g	0.077	0.811	157 g	0.071	0.838
WL	2D	5.66 g	0.176	0.449	5.70 g	0.175	0.439
WO	2B	44.7 g	0.304	0.871	47.7 g	0.298	0.853
WB – WO	2C	305 g	0.109	0.071	299 g	0.107	0.110
VT	2F	35.2 mg ml−1	0.542	0.644	40.3 mg ml−1	0.635	0.534
Overall	2		0.280	0.615		0.309	0.575

¹A perfect fit implies SMScE = 0 and ME = 1.

Although cultivation conditions were roughly similar among the three experiments, the fish in the supplementary data sets SD1 and SD2 grew more vigorously than those in the main data set. This can be clearly seen in Fig. 3A, which shows that the model predictions by dDEB and stDEB+ with the parameters estimated from the main data set (bottom two curves) underestimate growth of fish in set SD1. The predictions are greatly improved, however, by adjusting the general reserve turnover rate parameter. Increasing this value by 25% (dDEB) or 20% (stDEB+) yields curves that are virtually indistinguishable and represent the growth data well. Similarly, with the value of the general reserve turnover rate parameter from the main data set, both models estimate the predictions of end weights in data set SD2 about 25–30% lower than actually observed. Also with this data set, satisfactory estimates of final body weights are obtained by increasing the value of the general reserve turnover rate parameter with 35% (dDEB) or 20% (stDEB+) (results not shown).

Figure 3.

The ability of dDEB and stDEB+ parameterized with values estimated from the main data set (see Fig. 2 and Table 1d) to predict production in rainbow trout was evaluated with supplementary data set SD1 (A) and set SD2 (B). (A) With the estimated parameter values, both dDEB (dotted curve) and stDEB+ (dot-dashed curve) underestimated the gain in weight in set SD1 (circles). Predictions are greatly improved by increasing the reserve turnover rate by 25% (dDEB, solid curve) or 20% (stDEB+, broken curve) relative to the value estimated from the main data set. (B) dDEB (solid curve, reserve turnover rate 35% higher than the one in the main data set) and stDEB+ (broken curve, reserve turnover rate 20% higher than the one in the main set) predict measured total egg mass versus body weight (symbols) from data set SD2 about equally well.

The analysis of reproductive data from SD1 and SD2 comes with two caveats. First, the exact moment of spawning in these experiments is unknown. This hinders the comparison of model predictions of reproductive endpoints with observed values, as the former depend strongly on timing, given the relatively steep increase in ovary weight during the final weeks of the reproductive cycle (cf. Fig 2C). Second, the models predict the weight of ovaries, whereas the data report egg mass. With these caveats in mind, we take the census time to be 355 days into the reproductive cycle and assume the final weight of the ovaries equals that of eggs. Then, with the reserve turnover rate from the main data set, the models overestimate the reproductive effort in data set SD1 by about a third (see Table 4). With the general reserve turnover rate adjusted (see above), this overestimation increases to 45–70%, though the gonadosomatic index (GSI) remains relatively unaffected as body masses are also predicted higher. Relative to data set SD2, the models underestimate reproduction 25–30%, assuming general reserve turnover rates estimated from the main data set. With those estimates adjusted as before, underestimates shrink to 2% and 20% for dDEB and stDEB+, respectively, while predicted GSI values change relatively little. The models predict reproductive effort at day 355 as a function of total body mass about equally well, given the scatter in the data (see Fig. 3B). With general reserve turnover rates adjusted, the measured mean mass and diameter of single eggs in data set SD2, 105.7 (±14.5) mg and 5.54 (±0.36) mm, respectively, are close to the values predicted by dDEB (93.3 mg and 5.62 mm, respectively), whereas the predictions by stDEB+ differ more (65.3 mg and 4.96 mm, respectively).

Table 4.

Measured and predicted body and egg masses supplementary data sets on day 355.

Set		Body mass	Egg mass	GSI
SD1	Data	2608 (±393)	274 (±84)	0.105
	dDEB, kE from main set	2096 (±188)	373 (±34)	0.178
	dDEB, kE 25% higher	2660 (±220)	470 (±40)	0.177
	stDEB, kE from main set	2177 (±197)	370 (±42)	0.170
	stDEB, kE 20% higher	2629 (±223)	400 (±46)	0.152
SD2	Data	2483 (±663)	419 (±161)	0.169
	dDEB, kE from main set	1732 (±201)	296 (±35)	0.171
	dDEB, kE 35% higher	2428 (±251)	412 (±43)	0.170
	stDEB, kE from main set	1849 (±213)	308 (±67)	0.167
	stDEB, kE 20% higher	2263 (±246)	336 (±80)	0.149

Discussion

We have formulated and evaluated two models of feedback control on the production of reproductive matter. The models provide a key to quantitatively connecting molecular level processes to organismal performance, a major challenge in biology. In particular, they describe growth and reproduction as processes subject to hormonal regulation, and thus provide a link between detailed physiologically-based models about the endocrine system (see e.g. Gillies et al. 2016) to the DEB modeling framework.

Important strengths of DEB include its generality and relative simplicity. The core dynamics of the standard DEB model for a healthy animal consist of only three state equations and involve universal processes, such as feeding, maintenance, development, reproduction and growth, with similarly general formulae relating these processes to measurable rates, such as respiration, waste and heat production. The additional equations required for modeling particular species and context specific measurable quantities (e.g. Equations 8–19 in Table 1) are somewhat narrower in applicability, but still have considerable generality. For example, we would expect these equations to be applicable to most fishes, albeit with species-specific values for their parameters.

Our representation of demand-driven energy allocation to the production of reproductive matter focuses on a general dynamic mechanism, namely feedback control of gonads. We used this mechanism to develop two extensions of the standard DEB model, stDEB+ and dDEB (see Figure 1). These extensions share the feature that, depending on the nutritional state of an adult, growth may occur concurrently with the accumulation of reproductive matter; this contrasts with other simple models, often used in optimality arguments, in which an adult commits either resources to growth or to reproduction at any given time (see e.g. Cohen 1971; Quince et al. 2008). However, a dDEB organism may cease to grow, and may even shrink, while it continues to allocate resources to reproduction (see below). We evaluated these extensions in depth with data on a single fish species, i.e. rainbow trout, due to the availability of extensive, time-resolved information on whole organism performance as well as on suborganismal processes related to the endocrine system.

Our models describe the production of biomass and reproductive matter in female rainbow trout in the three data sets analyzed here about equally well (see Fig. 2A–D, 3 and Table 3). Values of the core DEB parameter quantifying the rate of general reserve turnover estimated from these data sets differ 20–35% from each other, and they are 10–55% higher than the value published in the DEB parameter database (AmP 2017), though are rather similar in dDEB and stDEB+ (see Table 2d). Rainbow trout are a remarkably adaptable species with a long history of domestication and wide geographic distribution, existing as both anadromous and land locked varieties and have a relatively high level of genetic variation among different populations (Maccrimmon 1971; Hershberger 1992). Thus, it is not surprising that the general reserve turnover rate parameter varies among strains. The dDEB variant performs better in describing the dynamics of plasma estradiol and vitellogenin contents as well as the development of individual eggs (see Fig. 2E–G), and overall dDEB fits the main data set significantly better than stDEB+, as judged from likelihood values (see Table 3). It could be argued, though, that the types of data being superiorly described by dDEB are of relative minor importance for many modeling purposes, in particular those centering on whole organism performance. However, there are potentially decisive conceptual differences between the model variants.

The major conceptual difference between dDEB and stDEB+ lies in the timing of (somatic) reserve allocation to reproduction. In stDEB+, a well-fed adult allocates a constant fraction of mobilized reserves to reproduction plus maturity maintenance throughout the reproductive cycle and grows at a rate that is independent of the size of the reproductive buffer. This contrasts with the dynamic allocation of reserves in dDEB, in which the allocation is under the control of the size of the reproductive buffer relative to that of the animal. Consequently, this allocation can vary a great deal over a reproductive cycle (see Fig. S1 in the Supporting Information). Concurrently, growth follows an opposite trend. In a constant environment, dDEB predicts that most of the growth of a species with a seasonal reproduction pattern occurs before the gonads start developing substantially, whereas growth in stDEB+ is of the von Bertalanffy type. Consequently, size data could discriminate between the two models. Unfortunately, the total body weight measurements analyzed in this study contain too much scatter to be of much help. Length measures typically are relatively precise and could therefore be used to evaluate the merits of dDEB and stDEB+. It should be noted, though, that dDEB reduces to stDEB in a hypothetical adult animal that releases gametes nearly continuously, as the density of the reproductive buffer would be almost constant.

Both dDEB and stDEB+ predict the growth of the gonads occurs primarily during the later parts of the reproductive cycle, which is a common observation for synchronous annually spawning fishes like rainbow trout (Tyler & Sumpter 1996) as well as many marine invertebrates, notwithstanding the time-invariant fraction of reserves being allocated to reproduction in the latter model variant. In stDEB+, this is made possible by separating the reproductive buffer into two sequential pools, of which the first, reproductive reserves, receives somatic reserves according to the kappa rule of standard DEB, whereas the second containing actual reproductive matter (e.g. eggs) exerts positive and negative feedback control on the rate at which it is being filled with reserves from the first pool (see Equations 10–11 and Fig. 1). A potentially unrealistic consequence of separating the reproductive buffer into two pools is that although the gonad pool may be completely emptied during spawning, an animal following stDEB+ may be left with a substantial amount of reproductive reserves at the time of spawning. Indeed, in stDEB+ parameterized with the main data set, a three year old female rainbow trout releases only a little over 50% of the total amount of somatic reserves allocated to reproduction at spawning, despite its negligible reproductive buffer at the beginning of the reproductive cycle (see Figure S2 in the Supporting Information). In addition, stDEB+ recognizes two reserve pools, reproductive and somatic, with different dynamics; this begs the question how an animal following stDEB+ would be able to tell apart those reserve pools, given their likely large overlap in chemical nature and storage location.

A particular characteristic of dDEB is that reproduction can induce starvation symptoms, even when environmental resources are abundant. Due to the demand driven positive feedback of the reproductive buffer on reserve allocation in dDEB, the energy flow to the somatic branch may become insufficient to meet somatic maintenance demands. At that point, an organism has several options (Kooijman 2010). For instance, it could increase the reserve mobilization rate, give maintenance requirements priority over reproduction, reabsorb reproductive matter, skimp on maintenance, or use structural biomass as an energy source to meet maintenance, i.e. shrink. All these options may be realistic, depending on the life history strategy of the organism. For instance, reabsorption of gonads under stress conditions occurs in parasitoid wasps (Richard & Casas 2009; Richard & Casas 2012), bivalves (Gosling 2003) and fishes (Schreck, Contreras-Sanchez & Fitzpatrick 2001), among other groups. Here we allowed structural biomass to be recycled for maintenance purposes, but did so in a provisional manner (the thermodynamic implications of shrinking are rather intricate and fall beyond the scope of this paper). This mechanism of structure recycling may be of use to describe the degeneration of structures and the loss in vitality before and after spawning in semelparous fishes, such as species of eel and salmon.

In addition, this recycling mechanism is relevant for species with marked biomass turnover processes, such as holometabolous insects and annual plants. In the pupa stage, holometabolous insects degrade most tissues and build new structures. Without demand-driven feedback mechanisms and implied recycling mechanisms for structural biomass, such as in dDEB, the modeling of holometabolous insects within a DEB context is cumbersome (Llandres et al. 2015). Many annual plants feature strategies in which vegetative structures wither while seed mass is still increasing. The common bean, P. vulgaris, for instance, clearly displays this pattern (see e.g. Lima et al. 2005). In order to illustrate the ability of dDEB to capture this pattern, we used a stripped-down dDEB model without reserves, added an empirical relationship describing the dynamics of relative leaf cover (see Figure 4A) and a simple standard model describing photosynthesis as a function of leaf cover (see Supplemental Information for a full description of the model). This modified dDEB model describes the dynamic allocation of resources to above ground vegetative biomass and reproductive matter in this particular data set quite well (see Fig. 4B). It should be noted that the apparent relocation of structural biomass to seeds is due to an indirect mechanism: structural biomass is metabolized to meet the maintenance demands of the remaining structure, while an increasing fraction of photosynthate is invested in seed production.

Our models are designed to serve as pivots connecting Adverse Outcome Pathways (AOP) for endocrine disruptors to processes at ecological levels of organization. AOPs conceptualize the transfer of information from molecular to organismal levels of organization as the first step in scaling up to inform ecological risk assessment (Ankley et al. 2010). Starting with one or more molecular initiating events, i.e. perturbations caused by a chemical stressor, AOP models quantify the impacts of that stressor on molecular, cellular and/or organ-level processes. However, these models currently lack the ability to further these impacts to projections of those adverse effects on individual growth, reproduction, and survival, which are in the realm of the DEB modeling framework. Thus, the AOP framework could provide the mechanistic basis for modeling toxic effects within the DEB modeling framework, and thereby opening the door to process-based risk assessments in ecotoxicology (Murphy et al. 2017).

In conclusion, by including gonadal feedback control on energy allocation to reproduction and somatic processes we obtain three major benefits. Firstly, through this mechanism, the formation of reproductive matter can take on a marked seasonal, semelparous or batch-mode pattern with a minimum of mathematical complexity. Secondly, it facilitates the modeling of growth and reproduction as processes subjected to endocrine regulation, that is, it enables a connection between organismal and suborganismal level processes. Thirdly, since the control variable, i.e. the density of reproductive matter, has a generic form, species and sex specific attributes of endocrine regulation can be added without changing the core of the model. We anticipate that this mechanism, and our two model extensions that follow from it, will provide a gateway for incorporating molecular-level mechanisms of endocrine disruption into organismal-level models of individual performance, such as those in the DEB framework.