A mechanistic model for thiol redox dynamics in the organogenesis stage rat conceptus

Abstract

Precise control of the glutathione/glutathione disulfide (GSH/GSSG) redox balance is vital for the developing embryo, but regulatory mechanisms are poorly understood. We developed a novel, mechanistic mass-balance model for GSH metabolism in the organogenesis stage (gestational day 10.0–11.13) rat conceptus predicting the dynamics of 8 unique metabolites in 3 conceptal compartments: the visceral yolk sac (VYS), the extra-embryonic fluid (EEF) and the embryo proper (EMB). Our results show that thiol concentrations in all compartments are well predicted by the model. Protein synthesis is predicted to be a major efflux pathway for all amino acid precursors of GSH synthesis and an essential model element. Our model provides quantitative insights in the transport fluxes and enzymatic fluxes needed to maintain thiol redox balances under normal physiological conditions. This is crucial to further elucidate the mechanisms through which chemical exposure can perturb redox homeostasis, causing oxidative stress, and potentially birth defects.

1. Introduction

Precise control of the cellular redox balance is vital for the developing embryo. The cellular redox balance determines cell fate decisions (e.g. proliferation, differentiation and apoptosis) that are critical to ensure normal embryonic development [1]. Disruption of redox homeostasis leads to conditions of oxidative stress (generally defined as a shift from a normal reducing to overly oxidizing intracellular environment), which alters embryo development, leading to organ malformations, behavioral deficits and/or embryonic lethality [1–3]. Several human teratogens are believed to elicit their toxic action through mechanisms of oxidative stress [3]. In addition, shifts towards an oxidizing environment during development have been associated with increased risk of postnatal chronic diseases including neurodegeneration, hypertension, cancer and type II diabetes [4]. Despite its importance, it is presently not well understood how the redox state is maintained, regulated, and modified over the course of mammalian embryonic development. Consequently, mechanisms through which the generation of endogenous or exogenous oxidants disturbs the redox balance are poorly known.

This partly results from the fact that the cellular redox state is maintained by a complex network of tightly coupled enzyme systems and small molecule- and protein-based redox couples, including glutathione (GSH)/glutathione disulfide (GSSG), cysteine (Cys)/cystine (CySS), and oxidized/reduced thioredoxin. Each of these redox couples are regulated independently, however, they are connected through substrates and products (e.g. NADPH, amino acids, etc.) [5]. Mathematical systems-biology based models are particularly well-suited to capture the complexity of the redox network, and the consequences of network perturbations, but quantitative approaches are still limited. Reed and co-workers [6] developed a mathematical model for GSH metabolism in the human liver. The Reed-Nijhout model was further developed and refined by several others to assess the reliability of biomarkers of glutathione metabolism, and to provide insight in the effect of vitamin B deficiency on GSH metabolism [7–9]. Similar models have been developed for plants [10] and other tissue/cell types including brain [11], erythrocytes [12] and sub-cellular compartments such as mitochondria [13], but a mechanistic model for conceptal redox dynamics is not yet available. Development of such a model would be highly useful to understand observed spatially-explicit redox patterns in the visceral yolk sac (VYS) and the embryo proper (EMB) (e.g. [14]), to develop a predictive framework that can be used to test the influence of system perturbations on GSH redox dynamics and to integrate 30 years of experimental research on conceptal GSH redox dynamics into a mathematical framework. Ultimately, the development of mathematical, systems-biology based models for GSH redox dynamics in unperturbed and perturbed (oxidative stress) conditions can help to elucidate connections between oxidative stress and observed adverse effects on embryo development, such as neural tube defects (e.g. [4,15]), which are presently largely unknown.

In a previous study, we characterized the ontogeny of thiol redox dynamics and protein mass dynamics in the organogenesis-stage rat embryo using well-established rat whole-embryo culture techniques (rWEC) [16]. We furthermore developed an overall, empirically-based mass-balance model for Cys, the rate-limiting precursor for GSH synthesis [16]. The mass-balance model quantifies protein uptake by the VYS via histiotrophic pathways and Cys use for protein synthesis in the VYS and EMB. Here, we build on this initial mass-balance to develop a predictive, mechanistic model that explicitly considers the biochemical reactions of GSH synthesis, degradation and transport kinetics of amino acid precursors. We investigate which components of the GSH redox network are most sensitive to perturbations and asses the importance of embryonic features, such as histiotrophic nutrition, protein synthesis and growth, in maintaining GSH redox homeostasis. Since, to the best of our knowledge, this is the first attempt to model GSH redox dynamics in the developing embryo, we also assess to what extent our current knowledge of conceptal redox biology can explain experimental data and where further research is needed.

Specifically our objectives are i) to develop and evaluate a mechanistic mass-balance model for overall GSH redox dynamics in the organogenesis stage rat conceptus (gestational day 10.0–11.13) under normal (unperturbed) conditions, and ii) to assess the importance of key embryonic features, i.e. histiotrophic nutrition, volumetric growth and protein synthesis, in maintaining GSH redox homeostasis.

2. Methods

2.1. Conceptual approach

An overview of the complete model is given in Fig. 1. The model consists of three compartments: the visceral yolk sac (VYS), an extraembryonic fluid compartment (EEF, combining yolk sac fluid and amniotic fluid) and the embryo proper (EMB). Glutathione dynamics is quantified in each of these compartments by a set of coupled ordinary differential equations describing the biochemical reactions of GSH metabolism and the transport processes that supply precursor amino acids for GSH formation and that remove GSH and oxidized GSH (GSSG) from the intracellular environment. The model accounts for two types of growth: volumetric growth and protein synthesis, which present pathways of amino acid dilution by growth and amino acid loss by incorporation into proteins, respectively. The model furthermore includes a quantification of histiotrophic nutrition, i.e. the endocytic uptake of serum proteins from the external medium by the VYS described as a function of the yolk sac surface area. Adult tissues, such as the liver, are able to create a substantial proportion of the Cys needed for GSH synthesis from methionine via the transsulfuration pathway [17]. Experimental research has however shown that this transsulfuration pathway is not yet active in the developing mammalian embryo [18–20]. This pathway is therefore not included in our model.

Fig. 1.

Schematic overview of GSH metabolism in the organogenesis stage rat embryo. The numbers denote the different steps of the gamma-glutamyl cycle described in Section 2.2, and the carrier-mediated transport of amino acids and the uptake of (maternal/external) proteins via receptor-mediated endocytosis, described in Section 2.3. We refer to the figure and these numbers when describing a specific process (e.g. as Fig. 1, #1). (Orange circles (P_x) represent protein mass in the external medium (ext), the visceral yolk sac (VYS) and the embryo proper (EMB), black circles represent amino acid transporters, and the grey ovals represent the two enzymes (dipeptidase (DP) and γ-glutamyltransferase (GGT)) involved in extracellular degradation of GSH. All flows are expressed in μmol substance per liter compartment per day (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

In total 21 coupled differential equations were developed describing the concentration dynamics of 8 unique substances (GSH, GSSG, Cys, CySS, glutamate (Glu), glycine (Gly), y-glutamylcysteine (yGluCys), cysteinylglycine (CysGly)) in the three conceptal compartments (VYS, EEF, EMB). All individual mass-balances are summarized in Table 1 and the following sections provide a more detailed description of all relevant (trans)formation, degradation (Section 2.2) and transport flows (Section 2.3). We subsequently describe the model parameterization, calibration and evaluation and the sensitivity analysis (Section 2.4).

Table 1.

Compartmental mass-balances for all metabolites.

Visceral Yolk Sac (VYS)

$\begin{array}{l}d{[Cys]}_{VYS}/dt={\nu }_{EXT\to VYS}^{Cys}+2\cdot {\nu }_{VYS,CySSred}-{\nu }_{VYS,GCL}-2\cdot {\nu }_{VYS,Cysox}-{\nu }_{VYS,G}^{\text{Cys}}-{\nu }_{VYS\to P}^{Cys}\\ -{\nu }_{VYS\to EEF}^{Cys}\end{array}$

$d{[Gly]}_{VYS}/dt={\nu }_{EXXT\to VYS}^{Gly}-{\nu }_{VYS,GSHase}-{\nu }_{VYS\to P}^{Gly}-{\nu }_{VYS,G}^{Gly}-{\nu }_{VYS\to EEF}^{Gly}$

$d{[Glu]}_{VYS}/dt={\nu }_{EXT\to VYS}^{Glu}-{\nu }_{VYS,GCL}-{\nu }_{VYS\to P}^{Glu}-{\nu }_{VYS,G}^{Glu}-{\nu }_{VYS\to EEF}^{Glu}$

$d{[\gamma GluCys]}_{VYS}/dt={\nu }_{VYS,GCL}-{\nu }_{VYS,GSHase}-{\nu }_{VYS,G}^{\gamma GluCys}$

$d{[GSH]}_{VYS}/dt={\nu }_{VYS,GSHase}+2\cdot {\nu }_{VYS,GSSGR}-2\cdot {\nu }_{VYS,GSHox}-{\nu }_{VSS\to EEF}^{GSH}-{\nu }_{VSS,G}^{GSH}$

$d{[GSSG]}_{VYS}/dt={\nu }_{VYS,GSHox}-{\nu }_{VYS,GSSGR}-{\nu }_{VYS\to EEF}^{GSSG}-{\nu }_{VYS,G}^{GSSG}$

$d{[CySS]}_{VYS}/dt={\nu }_{VYS,ox}^{CyS}-{\nu }_{VYS,red}^{CySS}-{\nu }_{VYS,G}^{CySS}-{\nu }_{VYS\to EEF}^{CySS}$

Extra-embryonic fluid compartment (EEF)

$d{[Cys]}_{EEF}/dt=({\nu }_{VYS\to EEF}^{Cys}\cdot V_{VYS}-{\nu }_{EEF\to EMB}^{Cys}\cdot V_{EMB})/V_{EEF}+({\nu }_{EEF,DP}-2\cdot {\nu }_{EEF,ox}^{Cys}-{\nu }_{EEF,G}^{Cys})$

$d{[Gly]}_{EEF}/dt=({\nu }_{VYS\to EEF}^{Gly}\cdot V_{VYS}-{\nu }_{EEF\to EMB}^{Gly}\cdot V_{EMB})/V_{EEF}+{\nu }_{EEF,DP}-{\nu }_{EEF,G}^{Gly}$

$\begin{array}{l}d{[Glu]}_{EEF}/dt=\left({\nu }_{VYS\to EEF}^{Glu}\cdot V_{VYS}-{\nu }_{EEF\to EMB}^{Glu}\cdot V_{EMB}\right)/V_{EEF}-{\nu }_{EEF,G}^{Glu}+{\nu }_{EEF,GGT}\\ +2\cdot {\nu }_{EEF,GGT,GSSG}\end{array}$

$d{[GSH]}_{EEF}/dt=({\nu }_{VYS\to EEF}^{GSH}\cdot V_{\nu ys}+{\nu }_{EMB\to EEF}^{GSH}\cdot V_{EMB})/V_{EEF}-{\nu }_{EEF,GGT}-2\cdot {\nu }_{EEF,ox}^{GSH}-{\nu }_{EEF,G}^{GSH}$

$\begin{array}{l}d{[GSSG]}_{EEF}/dt=({\nu }_{VYS\to EEF}^{GSSG}\cdot V_{VYS}+{\nu }_{EMB\to EEF}^{GSSG}\cdot V_{EMB})/V_{EEF}+{\nu }_{EEF,ox}^{GSH}-{\nu }_{EEF,G}^{GSSG}\\ -{\nu }_{EEF,GGT,GSSG}\end{array}$

$d{[CysGly]}_{EEF}/dt={\nu }_{EEF,GGT}+2\cdot {\nu }_{EEF,GGT,GSSG}-{\nu }_{EEF,DP}-{\nu }_{EEF,G}^{Cys-Gly}$

$d{[CySS]}_{EEF}/dt=(-{\nu }_{EEF\to VYS}^{CySS}\cdot V_{VYS}-{\nu }_{EEF\to E}^{CySS}\cdot V_{EMB})/V_{EEF}+{\nu }_{EEF,ox}^{Cys}-{\nu }_{EEF,G}^{CySS}$

Embryo proper (EMB)

$d{[Cys]}_{EMB}/dt={\nu }_{EEF\to EMB}^{Cys}+2\cdot {\nu }_{EMB,red}^{CySS}-{\nu }_{EMB,GCL}-2\cdot {\nu }_{EMB,ox}^{Cys}-{\nu }_{EMB,G}^{Cys}-{\nu }_{EMB\to P}^{Cys}$

$d{[Gly]}_{EMB}/dt={\nu }_{EEF\to EMB}^{Gly}-{\nu }_{EMB,GSHase}-{\nu }_{EMB,G}^{Gly}-{\nu }_{EMB\to P}^{Gly}$

$d{[Glu]}_{EMB}/dt={\nu }_{EEF\to EMB}^{Glu}-{\nu }_{EMB,GCL}-{\nu }_{EMB\to EEF}^{Glu}-{\nu }_{EMB,G}^{Glu}-{\nu }_{EMB\to P}^{Glu}$

$d{[\gamma GluCys]}_{EMB}/dt={\nu }_{EMB,GCL}-{\nu }_{EMB,GSHase}-{\nu }_{EMB,G}^{\gamma GluCys}$

$d{[GSH]}_{EMB}/dt={\nu }_{EMB,GSHase}+2\cdot {\nu }_{EMB,GSSGR}-2\cdot {\nu }_{EMB,GSHox}-{\nu }_{EMB\to EEF}^{GSH}-{\nu }_{EMB,G}^{GSH}$

$d{[GSSG]}_{EMB}/dt={\nu }_{EMB,GSHox}-{\nu }_{EMB,GSSGR}-{\nu }_{EMB\to EEF}^{GSS}-{\nu }_{EMB,G}^{GSSG}$

$d{[CySS]}_{EMB}/dt={\nu }_{EEF\to EMB}^{CySS}+{\nu }_{EMB,ox}^{Cys}-{\nu }_{EMB,red}^{CySS}-{\nu }_{EMB,G}^{CySS}$

The thiol concentrations in the VYS, EEF, and EMB are expressed in μmol/L_VYS, μmol/L_EEF, and μmol/L_EMB, respectively. Transport kinetics ν of amino acid precursors, Cys, Gly and Glu, from the VYS to the EEF are expressed in μmol/L_VYS/d, and transport kinetics of amino acid precursors from the EEF to the EMB are expressed in μmol/L_EMB/d. The transport fluxes are volume-corrected (i.e. multiplied by V_VYS/V_EEF or V_EMB/V_EEF) to correctly describe the concentration dynamics for the EEF.

2.2. Glutathione metabolism

In addition to transport fluxes of amino acid precursors, the GSH redox balance in the VYS and the EMB is controlled by a regulated series of enzymatic and plasma membrane transport steps that are collectively referred to as the gamma-glutamyl cycle [21,22] (Fig. 1). The gamma-glutamyl cycle ensures a balance between the mass-fluxes of GSH formation and GSH degradation and enables a continuous recycling of GSH. De novo GSH synthesis occurs exclusively intracellularly, i.e. in the VYS and the EMB, by the consecutive action of two enzymes, y-glutamylcysteine synthetase (GCL, glutamate-cysteine ligase, EC 6.3.2.2) and GSH synthetase (GSHase, EC 6.3.2.3), from three precursor amino acids, Glu, Gly and Cys [22] (Fig. 1, #1 & #2). Under normal physiological conditions, the rate-limiting factors of GSH synthesis are Cys availability and GCL activity [2]. In addition, GSH can reversibly inhibit GCL, creating a feedback mechanism to regulate cellular GSH levels [23]. While pathways and mechanisms of GSH synthesis are similar in the VYS and EMB, it is important to parameterize the VYS and EMB separately because they perform different functions, have unique needs concerning the status and requirements of GSH and related low molecular weight thiols, and have distinct enzyme activities and metabolic turnover rates [2,24–26].

The first step, the synthesis of y-glutamyl-cysteine (y-GluCys), can be characterized by a reversible, random, bi-reactant Michaelis-Menten mechanism with competitive inhibition by GSH against Glu [10] (Table 3):

$${\mathit{\nu }}_{x,GCL}=\left(\begin{array}{l}\frac{{\mathit{V}}_{max,x,GCL}\left({[\mathit{G}\mathit{l}\mathit{u}]}_x{[\mathit{C}\mathit{y}\mathit{s}]}_x-{[\mathit{\gamma }\mathit{G}\mathit{l}\mathit{u}\mathit{C}\mathit{y}\mathit{s}]}_x/{\mathit{K}}_{eq,GCL}\right)}{{\mathit{K}}_M^{Cys}{\mathit{K}}_M^{Glu}+{\mathit{K}}_M^{Cys}{[\mathit{G}\mathit{l}\mathit{u}]}_x+{\mathit{K}}_M^{Glu}{[\mathit{C}\mathit{y}\mathit{s}]}_x(1+{[\mathit{G}\mathit{S}\mathit{H}]}_x/{\mathit{K}}_i^{GSH}}\\ +[\mathit{G}\mathit{l}\mathit{u}]/{\mathit{K}}_M^{Glu})+{\mathit{K}}_M^{Cys}{\mathit{K}}_M^{Glu}({[\mathit{y}\mathit{G}\mathit{l}\mathit{u}\mathit{C}\mathit{y}\mathit{s}]}_x/{\mathit{K}}_p^{\gamma GluCys}\\ +{[\mathit{G}\mathit{S}\mathit{H}]}_x/{\mathit{K}}_i^{GSH})\end{array}\right)$$ (1)

Where ${\nu }_{\mathit{x}}$ is the overall rate in μmol/L_x/d, ${\text{V}}_{\max }$ is the maximum enzymatic rate in μmol/L_x/d, Km are the Michaelis-Menten constants in μmol/L_x, ${\text{K}}_{\text{eq}}$ is the equilibrium constant for the reversible reaction in μmol/L, K_p is the dissociation constant for the product in μmol/L, and K_i is the inhibition constant for GSH in μmol/L (see further description in Table 3).

Table 3.

Parameters, literature reported values and calibrated values (with initial values in parentheses) for γ-glutamyl-cysteine formation and GSH formation in the VYS membrane (VYS) and the embryo proper (EMB) and transport of GSH and GSSG from the VYS and the EMB.

Parameter	Explanation	Unit	Origin	References	Value VYS	Value EMB
${\mathbf{\nu }}_{\mathbf{x},\mathbf{G}\mathbf{C}\mathbf{L}}$	y-GluCys synthesis rate	μmol·Lx−1·d−1
${\text{V}}_{\text{max,x,GCL}}$	Maximal rate of forward reaction	μmol·Lx−1·d−1	rat embryo, VYS & EMB	[25]	42,669	39,926
${{\text{K}}_{\text{M,x}}}^{\text{Glu}}$	Michaelis constant for Glu with GCL	μmol·Lx−1	rat embryo, VYS & EMB	[25]	750	750
${{\text{K}}_{\text{M,x}}}^{\text{Cys}}$	Michaelis constant for Cys with GCL	μmol·Lx−1	rat embryo, VYS & EMB	[25]	30	30
${\text{K}}_{\text{eq,GCL}}$	Equilibrium constant for the overall reaction	μmol·Lx−1	theoretical	[6]	5,597	5,597
${{\text{K}}_{\text{i}}}^{\text{GSH}}$	Dissociation constant for GSH with GCL	μmol·Lx−1	planta	[6]	8,200	8,200
${{\text{K}}_{\text{p}}}^{\text{y}-\text{GluCys}}$	Dissociation constant of y-GluCys	μmol·Lx−1	planta	[6]	300	300
${\mathbf{\nu }}_{\mathbf{x},\mathbf{G}\mathbf{S}\mathbf{H}\mathbf{a}\mathbf{s}\mathbf{e}}$	GSH synthesis rate	μmol·Lx−1·d−1
${\text{V}}_{\max ,\text{x},\text{GSHase}}$	Maximal rate of forward reaction	μmol·Lx−1·d−1		Fitted	1.118.500	717.085
			human, erythrocyte	[57,58];	(212,565)	(212,565)
${{\text{K}}_{\text{M,x}}}^{\text{yGlyCys}}$	Michaelis constant for y-GluCys with GSHase	μmol·Lx−1	rat, kidney b	[59]	42	42
${{\text{K}}_{\text{M,x}}}^{\text{Gly}}$	Michaelis constant for Gly with GSHase	μmol·Lx−1	rat, kidney	[60,61,59];	833	833
${{\text{K}}_{\text{p,x}}}^{\text{GSH}}$	Dissociation constant for GSH with GSHase	μmol·Lx−1	planta	[6]	30	30
${\text{K}}_{\text{eq,x,GSHase}}$	Equilibrium constant for the overall reaction	μmol·Lx−1	planta	[6]	5600	5600
${\mathbf{\nu }}_{\mathbf{x},\mathbf{G}\mathbf{S}\mathbf{H}\mathbf{o}\mathbf{x}}$	GSH oxidation rate	μmol·Lx−1·d−1
${\text{v}}_{\text{x,GSHox}}$	Zero-order flux for GSH oxidation	μmol·Lx−1·d−1		Fitted	6.373.100	5.935.268 c
${\mathbf{\nu }}_{\mathbf{x},\mathbf{G}\mathbf{S}\mathbf{S}\mathbf{G}\mathbf{R}}$	GSSG reduction rate	μmol·Lx−1·d−1
${\text{V}}_{\text{max,VYS,GSSGR}}$	Maximal rate of forward reaction	μmol·Lx−1·d−1	rat embryo, VYS & EMB	[24]	7,429,100	7,136,900
${{\text{K}}_{\text{M}}}^{\text{GSSG}}$	Michaelis constant for GSSG	μmol·Lx−1	rat, kidney, liver	[62,63]	37.37	37.37
${{\text{K}}_{\text{M}}}^{\text{NADPH}}$	Michaelis constant of NADPH	μmol·Lx−1	rat, kidney	[64,62,63]	11.06	11.06
$\text{NADPH}$	NADPH concentrationd	μmol·Lx−1	typical cellular concentration	[65]	100	100
${\mathbf{\nu }}_{\mathbf{x},\mathbf{C}\mathbf{y}\mathbf{S}\mathbf{S}\mathbf{r}\mathbf{e}\mathbf{d}}$	(Net) CySS reduction rate	μmol·Lx−1·d−1
${\text{k}}_{\text{CySSred}}$	First-order rate constant for net CySS reduction	d−1		Fitted	67.3	9.6 e
		d−1		[34]	(9.6–3360)
${{\mathbf{\nu }}_{\mathbf{y},\mathbf{M}\mathbf{R}\mathbf{P}}}^{\mathbf{G}\mathbf{S}\mathbf{H}}$	GSH efflux rate	μmol·Lx−1·d−1
${\text{k}}_{\text{y,}}{{}_{\text{MRP}}}^{\text{GSH}}$	First-order rate constant for GSH efflux by MRP from VYS to EEF or EMB to EEF	d−1		Fitted	2.67	0.88
${{\mathbf{\nu }}_{\mathbf{y},\mathbf{M}\mathbf{R}\mathbf{P}}}^{\mathbf{G}\mathbf{S}\mathbf{S}\mathbf{G}}$	GSSG efflux rate	μmol·Lx−1·d−1
${{\text{v}}_{\text{y},\text{MRP}}}^{\text{GSSG}}$	First-order rate constant for GSSG efflux by MRP from VYS to EEF or EMB to EEF	d−1		Fitted	5.30	8.37

^a)From Reed et al. [6]. These are plant data, but were used in GSH metabolism model for human liver by Reed et al. [6].

^b)Km for gamma-L-Glu-L-alpha-aminobutyrate, which is an analogue of y-glutamyl-cysteine Luo et al. [59].

^c)The calibrated value for alpha is 0.93.

^d)The NADPH concentration is assumed to remain constant, and is therefore added as a parameter rather than a state variable.

^e)The calibrated value for beta is 0.14.

The second step, the synthesis of GSH, catalyzed by (GSHase) (EC 6.3.2.3), is assumed to follow a reversible, random, bi-reactant Michaelis-Menten mechanism as well, although there is not yet a kinetic mechanism described for GSHase [10] (Table 3):

$${\mathit{\nu }}_{x,GSHase}=\left(\begin{array}{l}\frac{{\mathit{V}}_{max,x,GSHase}\left({[\mathit{\gamma }\mathit{G}\mathit{l}\mathit{u}\mathit{C}\mathit{y}\mathit{s}]}_x{[\mathit{G}\mathit{l}\mathit{y}]}_x-{[\mathit{G}SH]}_x/{\mathit{K}}_{eq,x,GSHase}\right)}{{\mathit{K}}_M^{\gamma GluCys}{\mathit{K}}_M^{Gly}+{\mathit{K}}_M^{Gly}{[\mathit{\gamma }\mathit{G}\mathit{l}\mathit{u}\mathit{C}\mathit{y}\mathit{s}]}_x}\hfill \\ +{\mathit{K}}_M^{yGluCys}{[\mathit{G}\mathit{l}\mathit{y}]}_x(1+{[\mathit{\gamma }\mathit{G}\mathit{l}\mathit{u}\mathit{C}\mathit{y}\mathit{s}]}_x/{\mathit{K}}_M^{yGluCys})\hfill \\ +{\mathit{K}}_M^{\gamma GluCys}{\mathit{K}}_M^{Gly}\left({[\mathit{G}\mathit{S}\mathit{H}]}_x/{\mathit{K}}_p^{GSH}\right)\hfill \end{array}\right)$$ (2)

The antioxidant function of GSH is accomplished largely by GSH peroxidase (GPx)-catalyzed reactions, which reduce hydrogen peroxide and lipid peroxide as GSH is oxidized to GSSG (Fig. 1, #3). Under physiological conditions, the enzyme-catalyzed oxidation of GSH is expected to show near-zero-order kinetics [27], i.e. GSH oxidation is independent of the GSH concentration and can be quantified with a flux ${\text{v}}_{\text{X,GSHox}}$.

Glutathione regainment from GSSG occurs in a reaction catalyzed by GSSG reductase at the expense of NADPH (Fig. 1, #4). Under normal physiological conditions, GSSG reductase activity and NADPH availability are sufficient to maintain GSH redox homeostasis (Griffith, 1999). GSSG reductase has been characterized as having a branched mechanism, such that at physiological (low) GSSG concentrations a ping-pong mechanism dominates and at high GSSG concentrations an ordered mechanism takes over [28]. We therefore use a ping-pong mechanism to characterize GSSG reduction:

$${\mathit{\nu }}_{x,GSSGR}=\frac{{\mathit{V}}_{max,x,GSSGR}{[\mathit{G}\mathit{S}\mathit{S}\mathit{G}]}_x{[\mathit{N}\mathit{A}\mathit{D}\mathit{P}\mathit{H}]}_x}{\left({\mathit{K}}_M^{GSSG}{[\mathit{N}\mathit{A}\mathit{D}\mathit{P}\mathit{H}]}_x+{\mathit{K}}_M^{NADPH}{[\mathit{G}\mathit{S}\mathit{S}\mathit{G}]}_x+{[\mathit{N}\mathit{A}\mathit{D}\mathit{P}\mathit{H}]}_x{[\mathit{G}\mathit{S}\mathit{S}\mathit{G}]}_x\right)}$$ (3)

The predominant pathway of GSH degradation is an active efflux from the intracellular environment to the extracellular fluid compartment [29]. Both GSH and GSSG are actively transported out of the embryo into the amniotic fluid by exporter pumps of the multidrug resistance protein (MRP) family [29] (Fig. 1, #5). The efflux of GSH and GSSG from the VYS and EMB to the EEF is presently characterized by first-order transport kinetics:

$${\nu }_{y,MRP}^S=k_{y,MRP}^S\cdot {[S]}_x$$ (4)

where ${{\nu }_{\text{y},\text{MRP}}}^{\text{S}}$ is the efflux rate of GSH or GSSG in μmol/L/d, ${{\text{k}}_{\text{y},\text{MRP}}}^{\text{S}}$ is the first-order kinetic rate constant for transport of GSH or GSSG (S) from the VYS to the EEF or the EMB to the EEF (y) in d⁻¹.

Although GSH is stable inside the cell, it is readily hydrolyzed in the extracellular space by the plasma-membrane enzymes γ-glutamyltransferase (GGT) and dipeptidase (DP) [30,31], (Fig. 1, #6). The reaction catalyzed by GGT is the hydrolysis of the γ-glutamyl bond of GSH, which results in the formation of Glu and cysteiny-glycine (Cys-Gly) [29,30]. The reaction mechanism is a modified ping-pong reaction as, next to hydrolysis, GGT is involved in a transpeptidation reaction as well. The transpeptidation reaction uses a standard amino acid as substrate and forms Cys-Gly and a (γ-L-glutamyl)-L-amino acid. This second reaction is not yet included in the model as pH and amino acid concentrations in extracellular fluids where GGT is localized favor the hydrolysis reaction [31]. Next to GSH, GSSG is a substrate of GGT. According to Wickham et al. [31], GSH and GSSG have a similar access to the binding site with similar catalytic activity. It is therefore assumed that GSH breakdown and GSSG breakdown can be quantified following a competitive inhibition mechanism. Breakdown of one mole of GSH results in formation of one mole of Cys-Gly, whereas breakdown of one mole of GSSG results in the formation of two moles of Cys-Gly. The degradation of GSH and GSSG can thus be quantified as:

$${\nu }_{EEF,GGT,S}=\frac{V_{\max ,EEF,GGT}{[S]}_{EEF}}{\left(\alpha K_{M,S}^{GGT}+{[S]}_{EEF}\right)}$$ (5)

where $\alpha =\left(1+\left[\text{I}\right]/{\text{K}}_{\text{i,s}}\right)$; ${\text{K}}_{\text{i,s}}$ is GSH or GSSG, and S = GSH or GSSG, depending on the modelled substance.

Cysteinyl-glycine is subsequently cleaved by a dipeptidase (EC 3.4.11.2, membrane alanyl aminopeptidase) into Cys and Gly (Fig. 1, #6):

$${\mathit{\nu }}_{EEF,DP}=\frac{{\mathit{V}}_{max,EEF,DP}{[\mathit{C}\mathit{y}\mathit{s}-\mathit{G}\mathit{l}\mathit{y}]}_{EEF}}{\left({\mathit{K}}_M^{\mathit{D}P}+{[\mathit{C}\mathit{y}\mathit{s}-\mathit{G}\mathit{l}\mathit{y}]}_{EEF}\right)}$$ (6)

The enzyme catalyzed hydrolysis of GSH and GSSG regenerates the amino-acid precursors for GSH synthesis. Particularly important is the regeneration of the rate-limiting amino acid precursor Cys via this pathway. In the extra-cellular environment, Cys rapidly auto-oxidizes to CySS [32]. This auto-oxidation is characterized by a second-order rate constant $\left(\text{k}{`}_{\text{EEF,Cysox}}\right)$ (Fig. 1, #7):

$${\nu }_{EEF,Cysox}=\text{k}{`}_{EEF,Cysox}\cdot {[Cys]}_{EE{F}^2}$$ (7)

Cystine is transported into cells were it is reduced to Cys [33]. At least two enzyme systems are known to catalyze the reduction of CySS into Cys: thioredoxin-1/thioredoxin reductase 1 (Trx1/TR1) and glutaredoxin-1/GSH disulfide reductase (Grx1/GSH GR) [34]. The mechanism(s) of enzyme-catalyzed reduction of CySS and the mechanism behind Cys oxidation in the intracellular environment are not yet fully understood. We assume that there is a net reduction of CySS to Cys in the intracellular environment, which can be characterized by a first-order reaction rate (Fig. 1, #8):

$$v_{x,CySSred}=k_{x,CySSred}\cdot {[CySS]}_x$$ (8)

2.2.1. Time-dependent enzymatic rates for VYS

Choe et al. [24] demonstrated a significant, 6-fold increase in the activity of glutathione peroxidase (GSH-Px) and glutathione disulfide reductase (GSSG-Rd) in the VYS during early development (GD9-13), whereas no significant change in both enzyme activities was observed in the embryo proper. To account for the ontogeny of these enzymatic rates, we assumed that the increase in activity can be attributed to an increase in enzyme concentration and thus an increase in V_max, rather than a change in enzyme configuration (and a lowering of K_m). For GSSG-Rd, this results in the following change in V_max:

$$V_{\max ,VYS,GSSGR}(t)=V_{\max ,VYS,GSSGR}(0)\cdot \left(1+k_{inc,GSSGR}\cdot (t-t(0))\right)$$ (9)

As GSH oxidation by GSH-Px is modelled by a second-order reaction rate (see Eq. 3), the incremental increase in enzyme activity is characterized by an increase in the second-order reaction rate ${\text{k}}_{\text{VYS,GSHox}}$:

$$k_{VYS,GSHox}(t)=k_{VYS,GSHox}(0)\cdot \left(1+k_{inc,GSHox}\cdot (t-t(0))\right)$$ (10)

The value of k represents the average, daily increment in activity; first calculated per enzyme (GPx and GSSGR) and then averaged across the two enzymes. This gives a k of 0.67 d⁻¹ (see Appendix A1.1). Note that these equations are valid for a specific developmental time-frame, i.e. GD9 to GD13, and that they cannot be used to estimate maximum enzymatic rates at later developmental stages.

2.3. Transport kinetics, protein synthesis and volumetric growth

2.3.1. Transport kinetics

In addition to the enzymatic steps of the gamma-glutamyl cycle, GSH dynamics in the developing conceptus are influenced by transport rates of amino acid precursors, protein synthesis rates and embryonic growth. The VYS captures proteins from the maternal fluid (or external medium) via endocytosis and breaks these down into individual amino acids using lysosomal proteolysis (Fig. 1, #9). In previous work, we showed that the protein influx into the rat conceptus, during early organogenesis, is a function of the yolk sac surface area (A(t) in mm²) and a time-dependent protein uptake rate constant (${\text{k}}_{\text{EXT}\to }^{\text{P}}{}_{\text{VYS}}(\text{t})$ in μg_P mm⁻²·d⁻¹) [16]. As endocytosis is generally considered to be the rate-limiting step in the degradation of exogenous proteins [35], the influx of individual amino acid precursors for GSH synthesis equals the protein uptake rate, and can be characterized as:

$${\nu }_{EXT\to VYS}^{AA}=\left(k_{EXT\to VYS}^P(t)\cdot A_{VYS}(t)\cdot {\alpha }_P^{AA}\right)/V_{VYS}$$ (11)

where ${\text{V}}_{\text{EXT}}^{\text{AA}}\to \text{VYS}$ is the influx of amino acid (AA) precursors into the visceral yolk sac in μmol/L_VYS/d; ${\alpha }_{\text{P}}^{\text{AA}}$ is the amino acid (Glu,Gly,Cys) content of proteins in μmol_AA/μg_P and ${\text{V}}_{\text{VYS}}$ is the yolk sac volume (time-dependent) (in μL).

The individual amino acids are used in the VYS for protein synthesis as well as for GSH synthesis. The remaining amino acids are actively transported from the VYS to the EEF, and from the EEF to the EMB, using transporters from the amino acid transport system (Fig. 1, #10). At present, the exact identity and activity of these transporters has not yet been identified and characterized in the developing conceptus. We therefore did not include specific amino acid transporters in our model, but we did quantify amino acid transport via carrier-mediated kinetics instead of first-order kinetics, i.e. as:

$${\nu }_y^{AA}=\frac{{V_{\max y}}^{AA}\cdot {[AA]}_x}{{K_{My}}^{AA}+{[AA]}_x}$$ (12)

where ${{\text{v}}_{\text{y}}}^{\text{AA}}$ is the amino acid (Cys, Gly, Glu) transport rate from the (y) VYS to the EEF (in μmol/L_VYS/d) and the EEF to the EMB (in μmol/L_EMB/d), ${{\text{V}}_{\text{maxy}}}^{\text{AA}}$ is the maximum transport rate for amino acid AA and transport direction y (in μmol/L_x/d), ${{\text{K}}_{\text{my}}}^{\text{AA}}$ is the Michaelis-Menten constant (in μmol/L_x), and [AA]_x is the amino acid concentration in compartment x (VYS, EEF, EMB) (in μmol/L_x).

We chose to use carrier-mediated kinetics instead of first-order kinetics or fixed transport fluxes, as this provides more flexibility for model parameterization: depending on the values of V_max and K_m, carrier-mediated kinetics can describe first-order kinetics (S < < K_m) or fixed transport fluxes (S > > K_m, v = V_max). In this case, the K_m and V_max values represent composite values, as individual amino acids are commonly transported by multiple transporters [36].

Once liberated by the VYS, amino acids are known to accumulate in the EEF that bathes the embryo. The EEF serves as a storage pool for amino acids that can be captured and later recovered, when required by the EMB, for its GSH synthesis and protein synthesis needs [37]. The transport kinetics of Glu, Gly and Cys are therefore assumed to be unidirectional, i.e. from the VYS to the EEF and from the EEF to the EMB prior to the establishment of active vitelline circulation (GD10.6). After establishment of vitelline circulation, amino acids are additionally transported through the vitelline vessels. Cysteine can be imported into cells either directly or in its oxidized form, CySS, via the cystine/glutamate antiporter system x_c⁻ [38]. At present, the direction and relative magnitude of Cys and CySS transport is not well understood. Jones [39] suggested a Cys:CySS shuttle in which CySS is transported preferentially into cells and reduced as Cys as a potential mechanism to maintain the redox state of the plasma Cys:CySS pool. We follow this suggestion, and assume that CySS is taken up from the EEF by both the VYS and EMB, which is characterized by Michaelis-Menten kinetics (Fig. 1, #11). This is in addition to the transport of Cys from the VYS to the EEF and from the EEF to the EMB.

2.3.2. Protein synthesis and volumetric growth

Organogenesis stage embryos are characterized by an extremely rapid growth (e.g. [40]), both in terms of compartmental volume and protein synthesis, which present pathways of amino acid loss by growth dilution and incorporation into proteins, respectively. We use our previously developed exponential relationships for volumetric growth and protein synthesis in the organogenesis stage rat embryo to characterize amino acid loss via growth dilution (Eq. 13) and amino acid incorporation into proteins (Eq. 14).

$${\nu }_{x,G}^s=a_{x,V}\cdot {[S]}_x$$ (13)

where ${\text{v}}_{\text{x},\text{G}}^{\text{S}}$ is the growth dilution rate for substance S (GSH, GSSG, Cys, CySS, Glu, Gly, CysGly, γGluCys) in compartment x (VYS, EEF, EMB) in μmol/L/d), ${\text{a}}_{\text{x},\text{V}}$ is the growth dilution rate constant for compartment x (VYS, EEF, EMB) in 1/d, and ${[\text{S}]}_{\text{x}}$ is concentration of substance S in compartment x in μmol/L.

$$v_{x\to P}^{AA}={\alpha }_P^{AA}\cdot a_{x,V}\cdot V_{p,x}\cdot {10}^6$$ (14)

where ${\text{v}}_{\text{x}\to \text{P}}^{\text{AA}}$ is the amino acid incorporation rate in newly synthesized proteins in μmol/L_x/d and ${\text{V}}_{\text{p},\text{x}}$ is the compartment specific protein-to-volume ratio in μg/μL.

2.4. Model parameterization, calibration, evaluation and sensitivity analysis

2.4.1. Parameterization

Values for Michaelis-Menten parameters (K_m, V_max) and rate constants (k; first- or second order) for the gamma-glutamyl cycle were obtained from an extensive literature search (Tables 3 and 4). If available, we collected and used data for the organogenesis stage rat embryo. In most cases, no data was available for the rat embryo and data from other tissue types and species were used. In a similar approach as used by Reed et al. [6] to parameterize a GHS redox dynamics for the human liver, we assume that Michaelis-Menten constants (K_m) are species- and tissue-independent. We test this assumption by conducting a sensitivity analysis (see below Sensitivity Analysis). Maximum velocities (V_max) can vary widely across species and tissues as they depend on the enzyme concentration. We therefore choose to calibrate the V_max parameters for the VYS and the EMB, if no values could be found in the literature. Overall, our literature search yielded values for 20 out of 32 parameters of the gamma-glutamyl cycle. The remaining 12 parameters were calibrated (see below Calibration).

Table 4.

Parameters, literature reported values and calibrated values (with initial values in parentheses) for GSH and GSSG degradation, Cys-Gly degradation and Cys oxidation in the extra-embryonic fluid compartment (EEF).

Parameter	Explanation	Unit	Origin	Reference	Model EEF
${\text{v}}_{\text{x,GGT}}$	Rate of production of Cys-Gly	μmol·LEEF−1·d−1
${\text{V}}_{\max ,\text{GEF},\text{GGT}}^{\text{GSH}}$	Maximal rate of forward reaction (GSH and GSSG)	μmol·LEEF−1·d−1		Fitted	930.8
			Rat epididymis (duct behind testis)	[66]	(705,600)
${{\text{K}}_{\text{M}}}^{\text{GSH}}$	Michaelis constant of GSH	μmol·LEEF−1	human (isoform GGT1 and GGT5);	[31]	10.55
${{\text{K}}_{\text{M}}}^{\text{GSSG}}$	Michaelis constant of GSSG	μmol·LEEF−1	human (isoform GGT1 and GGT5);	[31]	25.43
${\nu }_{\text{EEF,DP}}$	Rate of production of Cys and Gly	μmol·LEEF−1·d−1
${\text{V}}_{\text{max,EEF,DP}}$	Maximal rate of forward reaction	μmol·LEEF−1·d−1		Fitted	45.941
				[7]	(3,600,000)
${{\text{K}}_{\text{M}}}^{\text{Cys-Gly}}$	Michaelis-Menten constant for Cysteinyl-Glycine	μmol·LEEF−1	rat, microvillus membrane from kidney, jejunum	[67]	2500
${\nu }_{\text{x,Cysox}}$	Cys oxidation rate	μmol·LEEF−1·d−1
${\text{k}}_{\text{Cysox}}$	Second order rate constant for Cys oxidation	d−1		Fitted	1.03

Parameters and values pertaining to protein influx, protein synthesis and volumetric growth were obtained from a previous study (Table 2) [16]. Typical Glu, Gly and Cys contents in protein (expressed in μmol amino acid per μg protein) were derived from data obtained from peerreviewed literature [37,41–45]. Koszalka et al. [45] showed that the amino acid composition of proteins in the EMB and VYS are remarkably constant during organogenesis. It is therefore assumed that the amino acid content in protein is invariable between GD10 and GD11.13. Also, as amino acid composition of serum proteins, VYS proteins and EMB proteins is similar [37], the same value for Cys, Glu and Gly content in protein was used for all proteins (Table 2).

Table 2.

Parameters, values and references for characterizing protein (and amino acid) influx, protein synthesis and growth dilution throughout early organogenesis.

Parameter	Description	Value	Unit	Ref.
Protein uptake
${\text{k}}_{\text{EXT}\to \text{VYS}}^{\text{P}}(\text{t})$	Time-dependent protein uptake rate constant	14.1·exp(−0.07 · t)	μgp·mm−2·d−1	[16]
${\text{A}}_{\text{VYS}}(\text{t})$	Yolk sac surface area growth	8.7·10−5·exp(1.2 · t)	mm2	[16]
${\text{α}}_{\text{P}}^{\text{Cys}}$	Cys content of protein	3.98·10−4	μmolCYS· μgp−1	[41–45]
${\text{α}}_{\text{P}}^{\text{Glu}}$	Glu content of protein	6.63·10−4	μmolGLU· μgp−1	[41–45,37]
${\text{α}}_p^{\text{Gly}}$	Gly content of protein	1.10·10−3	μmolCLY· μgp−1	[41–45,37]
Protein synthesis
${\text{P}}_{\text{VYS}}(\text{t})$	Protein mass dynamics in the VYS	1.8·10−04 · exp(aVYS,V · t)	μg	[16]
${\text{P}}_{\text{EMB}}(\text{t})$	Protein mass dynamics in the EMB	7.0·10−05 · exp(aEMB,V · t)	μg	[16]
Volumetric growth
${\text{a}}_{\text{VYS,V}}$	Growth dilution rate constant for the visceral yolk sac	1.18	d−1	[16]
${\text{a}}_{\text{EEF,V}}$	Growth dilution rate constant for the extra-embryonic fluid compartment	1.83	d−1	[16]
${\text{a}}_{\text{EMB,V}}$	Growth dilution rate constant for the embryo	1.31	d−1	[16]
${\text{V}}_{\text{VYS}}(\text{t})$	Visceral yolk sac volume at time t	(l/Vp,VYS) · PVYS(t)	mm3	[16]
${\text{V}}_{\text{EEF}}(\text{t})$	Extra-embryonic fluid volume at time t	4.8–10−08 · exp(aKEF,V · t)	mm3	[16]
${\text{V}}_{\text{EMB}}(\text{t})$	Embryonic volume at time t	(1/Vp,EMB) · PEMB(t)	mm3	[16]
${\text{V}}_{\text{p,VYS}}$	Protein-to-volume ratio (time-invariant)	123	μg·μL−1	[16]
${\text{V}}_{\text{p,EMB}}$	Protein-to-volume ratio (time-invariant)	173	μg·μL−1	[16]

Note that the regressions are developed and applicable for GD10.0 to GD11.13.

2.4.2. Calibration data

The remaining model parameters for the gamma-glutamyl cycle and the parameters describing the inter-compartmental transport kinetics of precursor amino acids (Cys, Gly, Glu) were calibrated on measured time-course data of Cys, CySS, GSH and GSSG in the organogenesis stage rat conceptus between GD10 and GD11.13 [16]. This rich dataset provides temporally- and spatially-explicit measurements of the main low molecular weight thiols in three conceptal compartments, i.e. VYS, EEF, and EMB, under unperturbed conditions. Four metabolites were not measured, cysteinyl glycine (CysGly) in the extra-embryonic fluids, y-glutamyl cysteine (yGluCys) in the EMB and VYS, and Gly and Glu concentrations in all 3 compartments. Hansen et al. [25] estimated basal yGluCys levels of 0.55 nmol/mg protein and 0.67 nmol/mg protein in the EMB and VYS of organogenesis stage rat embryos. We used these measurements here assuming that yGluCys levels per mg protein remain constant over time. An estimate of CysGly levels in the YSF and AF was obtained based on a calculated GSH:CysGly ratio of 0.77 in human plasma (based on measurements from Zhang et al. [46]. For the VYS and EMB, typical intracellular Gly and Glu concentrations were obtained from Brosnan et al. [47] and Brosnan [48]. It is assumed that the concentration of Gly and Glu in the VYS and EMB are constant over time as the concentration of all previously measured thiols except Cys is relatively constant [16]. We obtained concentrations of Gly and Glu in the EEF from Rowe and Kalaizis [37], who measured Gly and Glu concentrations in extraembryonic fluids of GD11 Sprague-Dawley rat conceptuses. These data were converted to time-course data, assuming that the mass of Gly and Glu in the EEF is constant over time, as we previously showed that the total thiol mass in the EEF during early organogenesis is relatively constant. This implies that the concentration of Gly and Glu in the EEF decreases with time. The time-course data are provided in the Appendix (Section A2.1, Table A1).

2.4.3. Calibration approach

We implemented the model in Matlab (R2016b) and numerically integrated the ODEs using the ODE15 s solver. Unknown model parameters were optimized by minimizing the least-square difference between spatially- and temporally-explicit predictions of GSH, GSSG, Cys, CySS, CysGly, γGluCys, Gly and Glu and the experimental time-course data, using a global solver, Pattern Search, from Matlab’s Global Optimization Toolbox. As metabolite concentrations differ by two orders of magnitude, we normalized all errors by dividing the sum of squared errors by the square of the maximum measured value for a specific metabolite in a specific compartment. We employed a weighted fitting procedure to account for differences in data source (i.e. measured vs. literature-collected) and number of data points (Appendix, section A2.2, Table A2). We also gave lower weights to Cys concentrations in the EMB proper after GD10.6 as our previous study shows that the observed sudden depletion of Cys in the EMB after GD10.6 is not found in other studies by Harris and co-workers [16]. To reduce the dimensionality of the parameter space, we incorporated the following constraints: 1). we allow V_max to vary between the VYS and the EMB, but we set the boundaries at a factor 1000 from the initial values collected from literature, 2.) As it is unlikely that the VYS and the EMB possess vastly different oxidation and reduction rates for GSH and Cys, we tied the GSH oxidation rate in the EMB to the GSH oxidation rate in the VYS as ${\text{k}}_{\text{EMB},\text{GSHox}},=\text{alpha}*{\text{k}}_{\text{VYS},\text{GSHox}}$. Analogously, we tied the net CySS reduction rate in the EMB to the net CySS reduction rate in the VYS as ${\text{k}}_{\text{EMB},\text{CySSred}},={\text{beta}}^{*}{\text{k}}_{\text{VYS},\text{CySSred}}$. The factors alpha and beta are constrained between 1 and 0.1 as experimental studies have shown that the VYS generally has a superior enzymatic capacity in comparison to the EMB [24,49,50,26].

To estimate parameter values for amino acid transport kinetics, we derived initial values for the maximum transport rate (V_max) from a first mass-balance analysis of required total Cys, Gly and Glu by the VYS and EMB during early organogenesis (see also Veltman et al. [16]). For transport from the EEF to the EMB, initial V_max’s were set equal to the required amino acid flux by the EMB. This transport rate is expressed in μmol per liter embryo per day (rather than in μmol per liter EEF per day) as the EMB probably controls its amino acid requirements. The EEF and the EMB grow with distinct growth rates (see Table 2), and this adjustment ensures that the EMB maintains a constant amino acid influx. For the transport rate from the VYS to the EEF, initial V_max’s were derived from the known amino acid uptake rate from the external medium via histiotrophic nutrition and the required amino acids by the VYS.

2.4.4. Evaluation

In a first evaluation step, we compared model predictions of thiol metabolite concentrations (Cys, CySS, GSH, GSSG) with measured thiol levels not used in the model calibration. We therefore collected measured thiol levels in the organogenesis stage rat embryo from literature (i.e. [51–54,26,55,56] (see Appendix, Section A2.3, Table A3). This evaluation exercise can provide an indication of model applicability, i.e. we evaluate if the model is generally applicable to predict GSH redox dynamics in organogenesis-stage rat embryos or if the model is only applicable to the current dataset. We also assessed the goodnessof-fit by calculating the root-mean-square error (RMSE) on the log and the squared geometric standard error (GSD², a.k.a. coefficient of variation) for both the calibration and evaluation dataset.

2.4.5. Sensitivity analysis

We performed a local sensitivity analysis to assess the robustness of the calibrated parameters, to evaluate the assumption of species-independent K_m’s, and to identify the parameters with the largest influence on the model output. We therefore calculated local sensitivities, s_ij(t), as:

$$s_{ijx}(t)=\left(\frac{\left|y_{p,i,x}-y_{o,i,x}\right|/y_{o,i,x}}{\left|p_{p,j}-p_{o,j}\right|/p_{o,j}}\right)$$ (17)

Where s_ij(t) is the local sensitivity of substance i (GSH, GSSG, Cys, CySS, CysGly, γGluCys, Glu, Gly) in compartment x (VYS, EEF, EMB) at time point t due to a perturbation in parameter j. y_p,i = concentration of substance i in compartment x at time point t due to a perturbation in parameter j, y_p,i = original concentration (unperturbed conditions) of substance i in compartment x at time point t, ${\text{p}}_{\text{p,j}}$= value of parameter j after a perturbation p, ${\text{p}}_{\text{0,j}}$= original value of parameter j.

We first varied the values of the optimized parameters by ± 2%, 10%, and 50% and in a second run, we varied the values of all parameters by ± 2%. Sensitivity coefficients (s_ij) were calculated for 55 specific time points between GD10 and GD11.13. These time-specific coefficients were averaged to obtain one value of s_ij for the entire timespan. For the calibrated parameters, we also evaluated the influence of parameter perturbation on the overall least-square error (as lsq_new – lsqoriginal).

3. Results

3.1. Thiol concentrations

Fig. 2 shows the measured and predicted concentration dynamics of the 8 metabolites (GSH, GSSG, Cys, CySS, Gly, Glu, CysGly, yGluCys) in the 3 conceptal compartments (VYS, EEF, EMB) during early organogenesis. Thiol dynamics in the VYS and EMB is well predicted by the model, except for the observed sudden depletion of Cys in EMB and Cys and CySS in the VYS upon activation of the heart-beat (GD10.6). For the EEF, the model correctly predicts a decrease in amino acid and thiol levels from GD10 to GD11.2. Similar to the VYS and the EMB, the model does not yet capture the observed sudden depletion in Cys levels in the EEF at GD10.6. The depletion in GSH and GSSG levels in the EEF at GD10.6 is reasonably well captured, although the model predicted depletion in GSH and GSSG levels is not as sudden as observed in the experimental data.

Fig. 2.

Concentration dynamics of 8 metabolites in 3 conceptal compartments (VYS, EEF, EMB) during early organogenesis (GD9.8–11.2). Lines represent model predictions and markers represent measurements.

Overall there is, however, a good agreement between model predictions and empirical data used for calibration (Fig. 3a, RMSE = 0.27, GSD² = 3.5), with 95% of the predictions within a factor 3.5 of the empirical data. There is also a good agreement between model predictions and the additional empirical data collected from the literature (Fig. 3b, RMSE = 0.24, GSD² = 3.0), with 95% of the predictions within a factor 3.0 of the empirical ‘evaluation’ data. In contrast to the calibration data, the concentration of Cys in the EMB is underestimated for the evaluation data.

Fig. 3.

Model predictions vs. empirical values (1:1 plot) for a) calibration data and b) literature-collected data.

3.2. Transport kinetics, protein synthesis and volumetric growth

Fig. 4 shows the predicted mass-fluxes for the transport of precursor amino acids from the external medium to the VYS (Fig. 4A) and from the VYS to the EEF and the EEF to the EMB (Fig. 4B). The calibrated parameter values for transport kinetics of amino acids and oxidized and reduced GSH are presented in Table 5. The VYS plays an important role in transport of amino acid precursors to the EMB. While the initial mass-fluxes of Glu and Gly from the yolk sac to the EEF are larger than embryonic demands, after GD10.8 and GD10.5 embryonic demand is predicted to exceed the yolk sac supply (Fig. 4B). For Cys (Cys+2CySS), embryonic demands are predicted to exceed the yolk sac supply from the start (GD10.0). Next to the transport of amino acid precursors, the model also predicted an export of both reduced GSH and oxidized GSH (GSGG) from the VYS to the EEF and from the EMB to the EEF. In terms of mass-fluxes, export fluxes of oxidized and reduced GSH are substantially smaller (> 20-fold) than the transport fluxes of amino acids.

Fig. 4.

A. Predicted mass-fluxes (ν in μmol/d) of Glu, Gly and Cys from the external medium into the VYS via histiotrophic nutrition, B. Predicted mass-fluxes (ν in μmol/d) of Glu, Gly and netCys (VYS: νCys-2*νCyss; EMB: νCys + 2*νCySS), from the VYS to the EEF and from the EEF to the EMB.

Table 5.

Transport kinetics: calibrated values for v, Vmax, Km and k for transport of amino acids and oxidized and reduced GSH.

Transport	ν [μmol·LEMB−1·d−1]	Vmax [μmol·LVYS −1·d−1]	Km [μmol·L−1]	k [d−1]
$\mathrm{G}\mathrm{l}{\mathrm{u}}_{\mathrm{V}\mathrm{Y}\mathrm{S}\to \mathrm{E}\mathrm{E}\mathrm{F}}$		572,936	2,253
${\text{Gly}}_{\text{VYS}\to \text{EEF}}$		2%, 890	682
${\text{Cys}}_{\text{VYS}\to \text{EEF}}$		147,250	19
${\text{CySS}}_{\text{EEF}\to \text{VYS}}$		7,045	271
${\text{GSH}}_{\text{VYS}\to \text{EEF}}$				2.7
${\text{GSSG}}_{\text{VYS}\to \text{EEF}}$				5.3
${\text{Glu}}_{\text{EEF}\to \text{EMB}}$	274,290
${\text{Gly}}_{\text{EEF}\to \text{EMB}}$	169,490
${\text{Cys}}_{\text{EEF}\to \text{EMB}}$	103,130
${\text{CySS}}_{\text{EEF}\to \text{EMB}}$
${\text{GSH}}_{\text{EMB}\to \text{EEF}}$				0.9
${\text{GSSG}}_{\text{EMB}\to \text{EEF}}$				8.4

Fig. 5 presents the predicted, average mass-fluxes of amino acid precursors (Cys, Glu, Gly) in the VYS and the EMB between GD10 and GD11.13. Fig. 5 clearly shows that the mass-fluxes of amino acid precursor transport to the VYS and to the EMB are much larger than the mass-fluxes needed for GSH synthesis in these tissues. In both the VYS and the EMB, protein synthesis represents the dominant use pathway for all amino acid precursors.

Fig. 5.

Predicted, average (GD10-11.13) mass-fluxes (ν in μmol/d) of Cys, Glu and Gly in the VYS (A) and the EMB (B). Influx represents the fluxes creating Cys, Glu and Gly in the VYS or EMB, whereas ‘efflux’ represents the use fluxes of Cys, Glu and Gly in the VYS or EMB. The ‘use fluxes’ include amino acid fluxes for protein synthesis and de novo GSH synthesis as well as growth dilution.

As early organogenesis-stage embryos are characterized by an extremely rapid growth, we also explicitly considered volumetric growth in our model. We, however, find that volumetric growth is a negligible “dilution” pathway for Cys, Gly and Glu in both the VYS and the EMB (Fig. 5). In addition, volumetric growth is found to be a minor (negligible) “dilution” pathway for all other metabolites, i.e. GSH, GSSG, CySS, γGluCys, in the VYS and the EMB as well (Appendix Fig. A2–A4). For the EEF, growth dilution is, however, of importance (Appendix Fig. A2–A4).

3.3. Sensitivity analysis

We performed a local sensitivity analysis to assess the uncertainty in the optimized parameters and to identify the most sensitive components in the system. The sensitivity analysis shows that five optimized parameters, i.e. the GSH oxidation fluxes $\left({\text{v}}_{\text{VYS},\text{GSHox}},{\text{beta}}_{\text{GSHox}}\right)$ and the transport fluxes of Cys, Gly and Glu from the EEF to the EMB, are sensitive to a 2% increase and a 2% decrease in their values, suggesting that these parameters are accurately estimated (Appendix Fig. A5, Table A4). Two additional parameters, ${\text{V}}_{\text{maxCyS,VYS,EEF}}$ and ${\text{V}}_{\text{max,EEF,GGT}}$, are sensitive to a 5% and 10% perturbation in their original values, suggesting that these parameters are also relatively well estimated (Appendix Table A4). The remaining parameters, except ${\text{V}}_{\max ,\text{EEF},\text{DP}}$, are all identifiably within a factor 2 of their optimized value (Appendix Table A4).

The overall sensitivity analysis shows that embryonic γGluCys levels are the most sensitive to parameter perturbations. Gamma-glutamylcysteine (γGluCys) concentrations in the EMB are very sensitive to a 2% perturbation in the transport fluxes of Gly and Cys from the EEF to the EMB. In addition, all amino-acid precursors for GSH synthesis (Cys, Gly, Glu) are relatively sensitive to a perturbation in their influx rates (from the EEF to the EMB). This 2% perturbation in the amino acid influx rates does not yet affect GSH and GSSG concentrations in the EMB (Appendix: Fig. A8). For the VYS, GSSG is the most sensitive metabolite in terms of parameter perturbations. GSSG is sensitive to an increase or decrease in the GSH oxidation flux $\left({\text{v}}_{\text{VYS},\text{GSHox}}\right)$ (Appendix: Fig. A6). For the EEF, GSH and GSSG levels are most sensitive to parameter perturbations. These levels are affected by a perturbation of the GSH oxidation rates in both the VYS and the EMB (Appendix: Fig. A7).

4. Discussion

4.1. Thiol concentrations

We developed a kinetic mass-balance model that describes thiol redox dynamics in three distinct compartments of the developing rat conceptus. Our model provides quantitative insights in the mass-fluxes needed to maintain the GSH/GSSG redox balance in the developing embryo in normal unperturbed conditions, which were previously poorly understood. Overall, the model accurately predicts thiol concentration dynamics in the VYS, the EEF and the EMB, but it does not capture the observed sudden depletion in several thiols in the VYS, EEF and EMB at GD10.6. While the observed depletion is consistent, the physiological reason for this depletion is presently unclear. A potential explanation could be that at GD10.6 the embryonic heart-beat is initiated and the embryo starts transitioning from anaerobic metabolism to aerobic metabolism, which would naturally result in a higher production of reactive oxygen species. However, as this transition occurs gradually, rather than abrupt [68], this may not have such an immediate, dramatic effect on thiol concentrations. Also, a comparison of model predictions to other data showed that the model under predicts embryonic Cys concentrations in these cases, which may suggest that the observed low Cys concentrations are experiment-related.

We evaluated our model predictions with experimental data from other ratWEC studies that were not used for model calibration. We find a good correlation between model predictions and the evaluation dataset as well, indicating that the model is not only applicable to the particular experimental dataset used in this study, but it is generally applicable to predict GSH redox dynamics in the organogenesis stage rat embryo.

4.2. Transport kinetics, protein synthesis and volumetric growth

Whereas previous models for GSH redox dynamics have focused on adult tissues or organs and are generally one-compartmental [6–13], we here present a multi-compartment model for GSH redox dynamics in the developing embryo. We thus needed to account for inter-compartmental transport of amino-acid precursors and relevant embryonic features, such as protein uptake via histiotrophic nutrition, volumetric growth and protein synthesis. In a previous study, we derived an equation to quantify Cys uptake in the VYS via histiotrophic pathways [16]. We here show that this approach works well for the other two precursor amino acids, Glu and Gly, as well. As no empirical information is available on the transport kinetics of Cys, Glu and Gly from the VYS to the EEF and the EEF to the EMB, we had to fit these parameters.

Although amino acid transport kinetics are known to be carrier-mediated, transport rates of Cys, Gly and Glu from the EEF to the EMB were best characterized with a constant flux (zero-order kinetics), which may indicate that the growing EMB tightly controls its influx of amino-acid precursors for GSH synthesis. The sensitivity analysis showed that all amino acid fluxes to the EMB are sensitive to a 2% perturbation in their optimized parameter values, suggesting that these fluxes are accurately estimated. For the VYS, amino acid transport kinetics could not be predicted with constant fluxes because the VYS volume was predicted to grow faster than its surface area. This would have resulted in a predicted decrease in the influx of amino acids into the VYS via histiotrophic nutrition between GD10 to GD11.13, which, in turn, would have resulted in unstable dynamics of Glu, Gly and Cys in the VYS. The use of Michaelis-Menten kinetics was found to stabilize the dynamics of the precursor amino acids in the VYS and was used here. Further research, including an identification of present amino acid transporters is needed to better characterize and quantify amino acid transport kinetics from the VYS.

A unique aspect of our modelling approach is that we accounted for key embryonic features, such as volumetric growth and protein synthesis. While volumetric growth was found to have a negligible impact on thiol concentration dynamics in the VYS and the EMB, we find that the inclusion of protein synthesis in a model for conceptal redox dynamics is essential, as protein synthesis is the dominant efflux rate for all GSH-precursor amino acids in the EMB and the second dominant efflux rate for GSH-precursor amino acids in the VYS. Considering protein synthesis would be particularly important in stressed conditions, as Stipanuk et al. [69] showed that protein synthesis has a higher priority for Cys than does GSH synthesis (in the adult rat liver) with tissue GSH levels becoming depleted at Cys intakes that are marginal but adequate for protein synthesis.

Finally, in contrast to adult tissues, enzyme kinetics in the developing embryo are not necessarily stable in normal, unperturbed conditions. We included the experimentally-observed increase in GSH oxidation and reduction rates in the VYS with gestational age [24], by assuming a linear increase in enzyme concentrations between GD10.0 and GD11.2 (see Section 2.2.1). This increase in enzyme concentrations resulted in a predicted time-dependent increase in GSH oxidation and reduction rates in the VYS (Appendix Fig. A2). However, as both the oxidation and the reduction rate were predicted to increase in an equal manner, the increase in enzyme activity did not result in a change in GSH and GSSG concentrations (Fig. 2). The ontogenetic increase in enzyme activity may therefore have little effect on metabolite concentrations under normal physiological conditions. The increase in enzyme activity can, however, be important for stressed conditions, as the systems response is largely determined by the magnitude of the fluxes.

4.3. Uncertainty, limitations and recommendations

We find that some pathways in the GSH redox network are not yet well characterized. Specifically, we find that the oxidation and reduction reactions and transport kinetics of the Cys/CySS redox couple are not well known. We assumed that CySS is taken up from the EEF into the VYS and the EMB, and once internalized, CySS is reduced to Cys, which we characterized with a net reduction flux. These assumptions are consistent with empirical evidence [70], however, as both the transport rates and the reduction rate are unknown, these parameters cannot be uniquely identified and their values should be regarded as uncertain.

We partly parameterized our model by minimizing the (normalized) difference between model predictions and experimental measurements of thiol concentrations in the three conceptal compartments. We find that most model parameters can be estimated from the data. An exception is the value for GSH and GSSG degradation in the EEF by DP. This is not unexpected, as DP catalyzes the second step in the extracellular degradation of GSH and GSSG and the literature-collected data (Table 4, [7,66]) suggest that the first enzymatic step is rate-limiting. Also, the calibrated values for GSH and GSSG degradation in the EEF by GGT and DP are substantially lower than reported in the literature for adult tissues. We estimated a V_max of 931 μmol/L/d for GGT (for GSH and GSSG) and a V_max of 45,941 μmol/L/d for DP (for GSH and GSSG), which is ~750 fold and ~75 fold lower than reported in the literature for rat epididymis and rat kidney, respectively [7,66]. While Miki and Kugler [71] demonstrated the occurrence of both enzymes in the yolk sac endoderm, the enzymatic rates in the developing embryo are presently unknown. Using the literature-reported values in our model resulted in a rapid depletion of GSH and GSSG in the EEF, which would need to be compensated by a greater influx of GSH and GSSG from the VYS and the EMB into the EEF. As GSH synthesis is restricted by known enzymatic rates for GCL [25], we find that these high values are not supported. This may suggest that the developing embryo has a much lower capacity to breakdown GSH and GSSG than adult tissues.

The current model is parameterized to quantify GSH redox dynamics under normal physiological conditions. To use the model for perturbed conditions, a few modifications are needed: First, we quantified GSH oxidation rates using zero-order kinetics. This is appropriate for normal physiological conditions [27], but inadequate to assess perturbed conditions when the system is under oxidative stress. In oxidative stress conditions, the GSH oxidation rate could be quantified by a second-order rate constant for GSH oxidation as in Raftos et al. [12] or using random order Michaelis-Menten kinetics as in Reed et al. [6]. Dedicated perturbation experiments that induce oxidative stress in the VYS and/or EMB are needed to identify and to parameterize the appropriate expression.

Second, we assumed that ATP and NADPH levels are constant and for ATP, present at saturating levels. Again, while these assumptions are most likely valid for normal, unperturbed conditions, it is not necessarily correct for perturbed conditions that may affect the ATP- and/ or NADPH-generating pathways. For example, Sant et al. [72] showed that exposure to MEHP induced selective alterations in EMB and VYS for oxidative phosphorylation and energy metabolism pathways in the organogenesis-stage mouse embryo and Komalapriya et al. [73] showed that NADPH production via the pentose phosphate pathway is induced under oxidative stress conditions in the fungal pathogen Candida albicans. Thus to use the model for perturbed conditions, it may be necessary to expand the model scope, e.g. by including the pentose phosphate pathway as in [73] or by implementing an empirically-based function for NADPH (or ATP) concentration dynamics.

Finally, it should be emphasized that our model is parameterized for GD10 to GD11.13. It is valid for this specific gestational period, but it cannot be used directly, that is without adaptations, to predict thiol concentrations in earlier or later gestational stages as, by nature, the developing embryo is a highly dynamic system.

5. Conclusion

Mathematical systems-biology models are considered essential tools to further advance our understanding of GSH redox dynamics (e.g. [5]), but quantitative approaches are scarce. Here, we present a novel, mechanistic model describing GSH redox dynamics in the developing rat conceptus. Our model accounts for the mass-fluxes of GSH precursor uptake, utilization, turnover, and enzyme kinetics, and includes key embryonic features, such as protein uptake via histiotrophic nutrition, volumetric growth and protein synthesis. Our study indicates that it is feasible to develop systems-biology based models for the developing embryo and provides a good foundation for further model development to assess redox dynamics and oxidative stress responses in the developing embryo caused by chemical exposure.