Ethics statement

All experiments were performed according to the guidelines established by the Institutional Animal Care and Use Committee (IACUC) of PNNL.

Background

The central focus of the model is on the progression of events that lead to oocyte growth through several stages, followed by final oocyte maturation (FOM) and then ovulation, whose timing is taken here to be the time point at which DHP first exceeds a threshold and when almost all of the oocytes are in FOM. We do not include actual spawning in the model because the release of eggs is different in the wild (where it is triggered by interactions with male fish) and in aquaculture (where the eggs are usually “stripped” by hand). In the model, concentrations of FSH, LH and VTG are given for separate tissue compartments. To differentiate between compartments subscripts are used: blood plasma is denoted with a P, the pituitary is denoted with a Pit, the ovary with a O and the liver is denoted with an L. Other tissues not specified are denoted with an N. The description of the equations for a hormone’s plasma levels occurs in the compartment associated with its synthesis and secretion. A clearance–volume approach is used to describe the pharmacokinetic behavior of VTG and circulating hormones, with a plasma referenced volume of distribution and a clearance parameter that represents loss due to excretion or sequestration (e.g. VTG accumulation in oocytes). The volume of distribution of a hormone is denoted by V_i and plasma clearance of a hormone is denoted by Cl_i, where i is replaced with the hormone name, e.g. FSH, LH, E2. In addition to the previously mentioned parameters, a consistent notation is used for parameters performing the same biological function, e.g. synthesis and degradation, to help maintain model uniformity.

Functions

The model uses Hill and Heaviside (a.k.a. the Unit Step function) functions to describe nonlinear hormone interactions. The Hill function is a sigmoidal function with values ranging from 0 to 1 and is used to model stimulatory (H₊) and inhibitory (H_-) effects that have a threshold, (1)

In each equation x is the concentration of the hormone, protein, or mRNA, T is the threshold value at which x exerts half of its maximal feedback and n determines the range of concentrations x must fall in to switch between no effect and maximal effect. Values of x not close to T will fall outside of the interval T±ε_n, where ε_n is a positive constant depending on n and is a mathematical representation of the maximal distance a value can be from T and still be considered close. For values of x outside this interval the Hill function remains relatively close to zero, no effect, or relatively close to one, maximal effect. As n becomes larger ε_n becomes smaller and the change between no effect and the maximal effect becomes more sudden; letting n go to infinity results in an instantaneous switch between no effect, zero, and the maximal effect, one, when concentration levels of x reach T. This on/off switch can be modeled using the Heaviside function, (2) by evaluating it at x-T when letting n go to infinity in H₊ or T-x when letting n go to infinity in H_-.

To incorporate time delays in interactions between model variables, transit compartments are used (reviewed in [24]). Transit compartments have been shown to be well suited to account for delays associated with signal transduction processes [25], which for example, occurs in the present model with FSH stimulation of E2 synthesis. This approach allows for a less complex model but still retains the capacity to incorporate more details of the signal transduction process, which can directly replace transit compartments. An individual transit compartment is restricted to short time delays, so for longer time delays that indicate multiple distinct processes are occurring, more than one transit compartment is used. Furthermore, the number of distinct processes can increase or decrease based on level of detail describing the biological process. This allows us to not only find the optimal number of transit compartments to use to represent the delay of hormone A’s influence on hormone B by D_A,B hours, denoted by m, but it is also reasonable to assume the delay produced by an individual transit compartment is equal to the average delay of the compartments, D_A,B/m. A superscript of TC_m is used for a system of m transit compartments; for example, if the growth of B starting at time 0 is modeled by (3) then incorporating the time delay using a system of m transit compartments will change the formula to (4) where the transit compartments are given by (5)

The use of transit compartments, Hill functions and the Heaviside function help describe delays and nonlinear interactions between hormones and key reproductive processes.

Hypothalamus—Pituitary

The hypothalamic region of the mid-brain is well established as the signal initiation site that stimulates the pituitary gland at the base of the brain to release GTHs that act on the gonads to cause their sexual development. The endocrine control of the HPOL pathway begins in neurosecretory fibers from the hypothalamus that project directly into the adenohypophysis region of the pituitary gland [26]. These fibers release GnRH in the vicinity of the gonadotrophs, the pituitary cells that produce the GTHs, increasing the basal synthesis rate of the beta subunit mRNA for FSH (mFSH) and LH (mLH). The GTHs are heterodimeric proteins consisting of a common α and GTH specific β subunit.

The regulation of GnRH is partly due to kisspeptins (KISS), which are the links from environmental cues to the HPOL axis [2]. The complexity of the GnRH-KISS system is still being characterized in fish and therefore we have chosen to not include it in this version of the model. We have assumed that FSH beta subunit mRNA levels primarily reflect changes in GnRH released into the pituitary during the reproductive cycle. Thus, an empirical equation for GnRH production was derived based on previously measured mFSH levels in the trout pituitary. The equations for mFSH and mLH are based on the equations found in [11] and begin with a basal synthesis rate, k_s,mFSH and k_s,mLH, and a degradation rate, k_d,mFSH and k_d,mLH. The positive feedback of GnRH is represented through a stimulatory factor, α_mFSH,GnRH and α_mLH,GnRH. E2 is known to increase the expression of mLH and pituitary content of LH [27,28,29]. This positive feedback effect of E2 on mLH is represented though a stimulatory factor, α_mLH,E2, and delayed through the use of transit compartments by D_E2,mLH hours to account for signal transduction and other biological processes.

(6)

(7)

After FSH is synthesized from mFSH it is immediately released into the blood. The tight coupling of mFSH levels and blood FSH levels ([30], and data presented in this study) allows the model to have a constant rate of synthesis and release of FSH from the pituitary, k_s,FSH, adjusted by the weight of the pituitary, w_Pit. As in [11], the clearance-volume approach then leads to the following equation: (8)

Plasma levels of FSH are detectible throughout the reproductive cycle, implying a continuous release from the pituitary; in contrast, plasma levels of LH remain largely undetectable until the end of the reproductive cycle. Though plasma levels of LH remain undetectable, pituitary levels continue to increase [30] implying a block on the release of LH from the pituitary into the plasma. The release of LH from the pituitary is known to be controlled by a dopamine (D2) inhibition, which is strengthened by E2 through positive feedback on D2 receptors (D2r) in the pituitary [31,32]. Declining levels of E2 remove the block on LH release [32] only after levels surpass a threshold, T_E2,LH, [33]. Besides E2, rising DHP levels also appear to block LH release. In some fishes, DHP causes an increase of D2 receptors in the pituitary [31], and in rainbow trout, DHP significantly decreased the release of LH in-vitro [34]. To model the release of LH from the pituitary we have chosen a release function that assumes the following: for each ng/ml of LH in the pituitary there exists a minimum amount of D2 receptors needed to maintain the block, the amount of LH released depends on the pituitary levels of LH that do not have the necessary amount of D2r to be blocked, the quantity of D2 receptors is proportional to levels of E2 and DHP and for the block to be removed, E2 levels must be higher than a threshold T_E2,LH. To better understand how these assumptions lead to the final release function found in Eq (12) an initial release function will be created from the first assumption, Eq (9), and built upon from the remaining assumptions, Eq (10) through Eq (12). The first assumption can be interpreted as the release of LH occurs when the amount of LH in the pituitary exceeds the minimum amount of D2-receptors per ng/ml of LH in the pituitary, ξ where ξ>0, e.g. (9) Applying the second and third assumption to the release function gives (10) where (11) and [D2r]/ ξ = N_E2[E2]+N_DHP[DHP]. Finally, the fourth assumption gives the final release function of (12)

With the exception of the release function the equations modeling the pituitary and plasma levels of LH are similar to those found in [11]. In the pituitary LH is synthesized from mLH at a constant rate of k_s,LH. The stored LH is then subject to degradation at a rate of k_d,LH before it is released at a rate of k_r,LH. The LH released into the plasma is adjusted by the weight of the pituitary, w_Pit, and follows the clearance-volume approach.

(13)

(14)

After release into the plasma, the GTHs travel to the ovaries to stimulate production of the sex steroids.

Ovary

The process of oocyte growth can be initially separated according to the capacity to absorb vitellogenin: pre-vitellogenic, vitellogenic and post-vitellogenic. These stages can be further subdivided according to specific morphological and biochemical changes occurring during growth [6]. Our model divides the reproductive cycle into seven stages; stages 1 and 2 are pre-vitellogenic, stages 3 through 6 are vitellogenic and the last stage is FOM. The proportion of oocytes in stage j, j = 1,2,…,6,FOM, at time t will be denoted by S_j(t). The model in [11] characterizes the stage of an oocyte based on length of time in the reproductive process and did not contain feedback from other variables. In the present model the stage of an oocyte is based on its size, which is defined by its diameter, and interacts with other variables. This is a convenient metric that is commonly reported in experimental studies and has been used to determine oocyte staging in trout (see Appendix S1 for ways to translate diameter to other measurements of size). The distribution of diameters at time t is given by a Weibull distribution with time dependent parameters l(t) and k(t). The distribution's parameters are chosen such that the mean and variance of the Weibull distribution at time t is equal to the mean and variance of oocyte sizes given by Eqs (16) and (18), respectively. Hence the stages are given by (15) where s_j is the boundary size between stages j-1 and j and the random variable X_t is the size of an oocyte at time t with X_t ~ Weibull(l(t),k(t)).

When describing oocyte growth it is important to recognize that approximately 84% of the increase in oocyte diameter occurs during vitellogenesis [5]. Therefore, the model assumes that oocyte growth rate is dependent on the amount of VTG being sequestered, Seq(t), and the conversion of diameter increase for every unit of VTG sequestered,. The growth rate in the non-vitellogenic stages is described by a constant growth rate,, as only a small portion of growth occurs during these stages.

(16)

The rate at which the oocytes absorb VTG is determined in part by circulating levels of FSH, oocyte stage, and VTG levels [35,36,37]. The increase in basal sequester rate, Cl_VTG,seq, due to circulating FSH is represented with a Hill function with a half threshold of T_Seq,FSH. Since oocytes can only sequester VTG when they exceed a certain size, [37], and have not entered FOM, the sequester period will only be from stage 3 until stage 6.

(17)

The growth rate of individual oocytes can vary substantially, presumably due to differential uptake of VTG but other, unknown factors may also be involved. At the start of the reproductive cycle, all primary oocytes (stage 1) appear to be uniform in size. As oocytes mature towards the mid-vitellogenic stage, size variance increases before becoming more uniform as final maturation nears [38]. To model this, each stage will have a maximal variance assigned to it,, where the variances increase as the oocytes near the mid-vitellogenic stage, Stage 4, and decrease as the oocytes approach FOM.

(18)

During stages 2 through FOM oocytes produce E2 and DHP when FSH and LH, respectively, are introduced to the ovaries. Once in the ovaries FSH and LH stimulate the thecal and granulosa cells, the layers of somatic cells surrounding the oocytes, to produce E2 and DHP, respectively [1]. When FSH or LH binds to their respective receptors, the ensuing stimulation of steroid synthesis is not instantaneous. To account for this delay, transit compartments are introduced in place of FSH and LH in Eqs (19) and (20), respectively, with time delays D_FSH,E2 and D_LH,DHP, respectively. The sensitivity of the oocytes' responses to LH and FSH is stage specific and the amount of steroid secreted into the plasma can be thought of as a clearance from the follicle. Hence a follicle in stage j will produce () ml of E2 (DHP) for each ng/ml of FSH (LH) that stimulates. Therefore, the total amount of E2 produced during Stage j at time t is , where n_oocyte is the average number of oocytes per kg of fish. Similarly, the total amount of DHP produced during Stage j at time t is . In addition to FSH stimulated E2 production, we have a basal production rate for each follicle, k_E2. We do not include a basal production rate for DHP since the follicles ability to produce DHP is only a short time period.

(19)

(20)

Once produced, E2 and DHP are then released into the blood and stimulate the production of VTG and promote ovulation, respectively.

Liver

The liver plays an essential role in the reproductive process because it synthesizes the egg yolk precursor protein VTG, which is necessary for oocyte growth and development [39]. Our model for VTG synthesis is largely based on the model described by [12]. VTG synthesis is initiated when E2 binds to a nuclear estrogen receptor, R, at a binding rate of k_on,ER, which is produced by transcription of the estrogen receptor mRNA, mR. The resulting E2-receptor complex, ER, stimulates the production of mR in a positive auto-regulatory feedback loop and stimulates the production of the mRNA for VTG, mVTG. Based on the experimental data of [40] we assume that activation of transcription happens on a one-to-one ratio, meaning a given dose of E2 yields similar increases in the E2-receptor complex, ER, and mVTG levels. ER then dissociates at a rate of k_off,ER increasing the number of unbound E2-receptors, R.

(21)

(22)

(23)

(24)

Once synthesized, each molecule of mVTG can be translated into VTG multiple times; this ability is represented by the amplification factor γ. Furthermore, N_mVTG represents a scaling factor for mVTG concentration. The VTG in the liver, denoted by VTG_L, is secreted into the plasma at a rate of k_r,VTG, the amount of which is denoted by VTG_P. The amount of VTG released into the plasma is measured in g/kg of liver weight, w_L. VTG is then exchanged between the plasma and compartments other than the liver and ovaries, VTG_N, taken up by the ovaries through the oocytes, Eq (17), or cleared from the body, Cl_VTG. The VTG in compartments other than the ovary are assumed to be in reversible equilibrium with blood with a transfer clearance of Cl_VTG,trans.

(25)

(26)

(27)

Parameter identification, model calibration and sensitivity analysis

Most parameter values were estimated directly from previously published experimental studies, prior modeling efforts [11,12] or modified slightly from literature values. These parameters and their sources are identified in Table 2. Other parameters were estimated using observed data collected from female rainbow trout during a second reproductive cycle. The latter data was previously published in [41] (oocyte diameter; plasma E2 and VTG; liver ER and VTG mRNA) or measured for this study (plasma FSH, LH, DHP and pituitary mRNA for beta subunits of FSH and LH). The data for FSH and LH was analyzed by radioimmunoassay as described by [42]. DHP was determined by enzyme immunoassay using reagents purchased from Cayman Chemical (Ann Arbor, MI). The β subunit mRNA for FSH was measured by q-RT-PCR as described in [43]. The β subunit mRNA for LH was also measured by q-RT-PCR using a similar method except the forward and reverse primers were from the LH-β gene (GenBank accession number AB050836). These primer sequences were: forward- AAACGATCCGCCTACCTGACT and reverse- AGCCACAGGGTAGGTGACATG. From these data, input curves were created using MATLAB’s cubic spline program and used to decouple the system of ODEs into smaller subsystems with fewer unknown parameters. Using the remaining input curves and the solution of the subsystem of ODEs, the error was then optimized around a least squares cost function with respect to the unknown parameters. After various decouplings, the solution for the original system of ODEs was compared to the input equations using a least squares cost function. The final cost function was then optimized by making small perturbations in all parameters not obtained explicitly through the literature. The parameters gained from this optimization process are also in Table 2. After the parameters were obtained, a global sensitivity analysis was performed using the PAWN method, outlined in [44], where the cumulative distribution functions created by a one-dimensional output distribution are used to compute density-based sensitivity indices. The model predictions of mFSH, mLH, FSH_P, LH_Pit, LH_P, O_Avg, E2, DHP, mR, R, ER, mVTG, VTG_L, VTG_P and VTG_N using the original parameters found in Table 2 and a set of altered parameters were compared using the Euclidean norm to measure the relative changes at every hour over a 350 day time period. Summing the relative changes of the individual functions provides a one-dimensional representation of the HPOL model that normalizes the effects the altered parameters have on the individual functions. The set of altered parameters was created using a Sobol’ quasi-random sequence to obtain a uniform set of parameters that vary between ±20% of the original values listed in Table 2. For the HPOL model varying the parameters between ±20% of the original values provided a large diverse set of model predictions while lowering the proportion of similar results. These results are displayed in Table 2 with larger numbers indicating a more sensitive parameter that exerts greater influence. Many of the more sensitive parameters are those associated with the synthesis and pharmacokinetics of E2 and VTG and growth of oocytes (Table 2; highest sensitivity values). Within the pituitary, the most sensitive parameters are associated with FSH synthesis. The greater sensitivity of these parameters reflects the model’s emphasis on oocyte growth and staging. This corresponds with the biology in that oocyte growth over the entire cycle is driven by FSH through its effect on E2, which stimulates VTG production.

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Table 2. HPOL model parameters, values and sensitivity ranking (sens.).

https://doi.org/10.1371/journal.pcbi.1004874.t002

Application and future work

An important value of quantitative biological models is the ability to perform simulations that vary model parameters in ways that are associated with various physiological conditions, real or hypothetical. In this manner, the rainbow trout HPOL model can be a tool to explore the effects of diverse physical and chemical stressors on reproduction. As an example, we consider trenbolone exposure to female rainbow trout. Trenbolone is a synthetic testosterone analogue that is used to promote growth in cattle. Concerns over the possible release of trenbolone to aquatic environments have prompted several studies on its effects to fish. In fathead minnows, trenbolone exposures (27 ng/L and higher) for up to three weeks while fish are actively spawning, caused a rapid and significant decline in circulating E2 and VTG levels and eventual cessation of spawning [51,63,64]. Subsequent in vitro experiments with fathead minnow ovary tissue suggest the decrease was due in part to decreased expression of enzymes involved in E2 synthesis [65]. In short-term exposures (one–four weeks) to female rainbow trout, trenbolone exposure stimulated FSH release in pre-vitellogenic fish, but also caused decreases in E2 levels in both pre- and mid-vitellogenic trout [51]. Other experiments with trout indicated plasma clearance of E2 was increased by 50% [51]. We were interested in estimating whether these effects in rainbow trout alone or in various combinations would be sufficient to prevent reproduction if trout were assumed to be continuously exposed throughout the reproductive cycle. We adjusted the following parameters based on the prior experimental evidence: FSH release (k_s,FSH; increased 4x), E2 synthesis (SE2O4-5; decreased 46%), E2 plasma clearance (Cl_E2; increased 41%) and VTG expression (k_s,mVTG; decreased 40%). A summary of model simulations for E2 and oocyte diameter is shown in Fig 7. Increasing the release of FSH alone or in combination with up to 46% decreases in E2 synthesis did not affect overall oocyte growth (Fig 7). If a decrease in VTG expression or increased E2 clearance occurs simultaneously with decreased E2 synthesis, then oocyte growth is slowed to an extent that ovulation would be unlikely to occur (Fig 7). These simulations suggest trenbolone effects on E2 synthesis by itself are not sufficient to cause reproductive failure in trout and additional effects occurring simultaneously elsewhere in the system are needed. Additional experimental research is needed to verify these observations but the exercise illustrates the potential for the HPOL model to incorporate in vivo and in vitro results for predictions of reproductive performance. This is a valuable attribute because long-term or complete reproductive cycle exposures are costly to perform and require many animals.

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Fig 7. Model predicted profiles of plasma levels of (A) E2 and (B) average oocyte growth for rainbow trout exposed to trenbolone.

Previously published studies suggest trenbolone exposure may decrease E2 synthesis to varying degrees depending on exposure levels, and also increase the clearance of E2 from plasma and increase the secretion of FSH in trout. The solid black line is the model predictions for fish not exposed to trenbolone and uses the parameter values found in Table 2. The dashed lines depict model predictions of known effects of trout that are exposed to trenbolone and use parameter values found in Table 2 except for Cl_E2,Sj = 0.54*Cl_E2,Sj for j = 4 and 5, k_s,FSH = 4*k_s,FSH during Stages 1 and 2. The blue dashed line also uses Cl_E2 = 1.41*Cl_E2 and the green dashed line uses k_s,mVTG = 0.6*k_s,mVTG. The model predicts that ovulation is unlikely to occur when an increase in E2 clearance and/or a decrease in mVTG synthesis is added to the effects because the delay in ovulation is more than 50 days.

https://doi.org/10.1371/journal.pcbi.1004874.g007

An emerging concept in environmental health research is the adverse outcome pathway (AOP) [66]. An AOP seeks to extrapolate effects of stressors measured at the cellular / sub-cellular level of biological organization to whole organism and population-level effects. It is anticipated that future research efforts will increasingly rely on in vitro methods to generate health effects data. Thus, mathematical models such as the trout HPOL model presented here will be useful for translating in vitro based observations into whole fish effects. As we consider future efforts to develop the next-generation HPOL model, emphasis will be placed on linking model outputs more closely to the types of input data used for population models. This would include expanding the description of oocyte development to provide more detail on egg quality and lifetime fecundity of rainbow trout over multiple spawning cycles.