Mathematical modeling of human oocyte aneuploidy

Abstract

Aneuploidy is the leading contributor to pregnancy loss, congenital anomalies, and in vitro fertilization (IVF) failure in humans. Although most aneuploid conceptions are thought to originate from meiotic division errors in the female germline, quantitative studies that link the observed phenotypes to underlying error mechanisms are lacking. In this study, we developed a mathematical modeling framework to quantify the contribution of different mechanisms of erroneous chromosome segregation to the production of aneuploid eggs. Our model considers the probabilities of all possible chromosome gain/loss outcomes that arise from meiotic errors, such as nondisjunction (NDJ) in meiosis I and meiosis II, and premature separation of sister chromatids (PSSC) and reverse segregation (RS) in meiosis I. To understand the contributions of different meiotic errors, we fit our model to aneuploidy data from 11,157 blastocyst-stage embryos. Our best-fitting model captures several known features of female meiosis, for instance, the maternal age effect on PSSC. More importantly, our model reveals previously undescribed patterns, including an increased frequency of meiosis II errors among eggs affected by errors in meiosis I. This observation suggests that the occurrence of NDJ in meiosis II is associated with the ploidy status of an egg. We further demonstrate that the model can be used to identify IVF patients who produce an extreme number of aneuploid embryos. The dynamic nature of our mathematical model makes it a powerful tool both for understanding the relative contributions of mechanisms of chromosome missegregation in human female meiosis and for predicting the outcomes of assisted reproduction.

Keywords: aneuploidy; chromosome missegregation; maternal age effect; mathematical modeling; meiosis.

Fig. 1.

Graphical representation of the model. (Legend) C describes the number of chromatids with respect to a haploid genome (N). Shortly before ovulation, a dictyate stage oocyte [2N(4C)] will complete MI, giving rise to a euploid egg [N(2C)] and the first polar body, PB1 [also N(2C)]. When fertilized, the egg will complete second stage of meiosis, MII, giving rise to a euploid zygote [2N(2C)] and a secondary polar body, PB2 [N(1C)]. Parental nuclei are separate entities at this stage [N(1C) each]. Possible outcomes of meiosis are enumerated for either error-free MI (A) or MI errors NDJ (B), PSSC (C), or RS (D). For zygotic maternal pronuclei (blue circles), only a deviation from normal haploid status N(1C) is noted. (A) Following error-free MI, a euploid egg [N(2C)] is fertilized and gives rise to either a euploid zygote (with probability 1 − 23q₁, where q₁ stands for the probability of an MII-NDJ error on a single chromosome; Table 1) or to an aneuploid zygote (with a probability 23q₁; Table 1). (B) There is a probability d of an MI-NDJ error. An affected dictyate oocyte will then give rise to an egg that either carries an extra chromosome copy (i.e., two more chromatids, 2C + 2) or lacks it entirely (2C − 2). (C) There is a probability p of an MI-PSSC error in MI. An affected dictyate oocyte will then give rise to an egg that either carries an extra chromatid (2C + 1) or lacks one (2C − 1). (B and C) In MII, the sister chromatids can fail to separate due to NDJ. MII-NDJ can either affect that same chromosome (depicted in red, with a probability q₂ in B and q₃ in C) or a different one (depicted in blue, with a probability 22q₂ in B and 22q₃ in C). (D) There is a probability r of an RS error in MI, but the affected dictyate oocyte will give rise to an egg that carries normal chromosome content (2C). Following MII, the zygote will either carry normal chromosome number (77% of the times) or not (23% of the times) (18). The pairs of homologous chromosomes are colored in black/red and gray/blue. The dots represent centromeres. Complex events affecting multiple chromosomes during meiosis occur with an overall probability c, which is included in model equations in Table 1.

Fig. 2.

Model estimates. (A) Model selection. Rows represent different karyotypes in the dataset (from euploid embryos to embryos with one trisomic and one monosomic chromosome, with frequencies provided in Table 2), and columns indicate the different models. For each karyotype, relative differences between model simulation and data were color-coded on a gray scale, where white represents a perfect fit of the model to the data, and black means that the model returned a value different from the observation (see color legend for precise values). AICc and χ² below the array plot represent the AICc and the significance obtained from χ² goodness-of-fit test performed on each model estimates compared with observations. ns, not significant; ***P < 2.2 × 10⁻¹⁶. (B) Four major karyotype categories simulated with Model 2. “No error” category (blue bars) represents the count for euploid embryos. “One error” category (red bars) represents single-monosomy or single-trisomy embryos. “Two errors” category (yellow bars) represents embryos with two aneuploid chromosomes. “Complex events” category (green bars) accounts for the complex error karyotypes. For reference, all possible meiosis outcomes are shown to the right and indicate which meiosis outcome was used to explain the category plotted on the left. (C and D) Estimated error rates for Model 2 in each age category. The MI-PSSC error rate (p) and MI-NDJ error rate (d) are shown in yellow and blue, respectively (C). The adjusted MII-NDJ error rate in a euploid egg (q) or in an aneuploid egg (q*) are shown in black and red, respectively (D). Adjusted MII rates are used to account for the MI outcome (Methods).

Fig. 3.

Model simulations detect individuals with extreme aneuploidy rates and can be used to calculate overall probability of euploid conception. (A) The dots represent 1,292 patients who had a minimum of 4 embryos evaluated for chromosomal abnormalities on day 5. Red and black dots highlight the patients with aneuploidy rates with probabilities less than 0.05, based on 10,000 model simulations of each patient, considering her age and the number of tested embryos (186 unique combinations of age and number of embryos in total). (B and C) Major error categories among normal individuals (gray bars collapsed from gray dots in A; n = 1,220) and individuals with extreme-high aneuploidy rates (red bars, collapsed from red dots in A; n = 48) (B) or the individuals with extreme-low aneuploidy rates (black bars collapsed from black dots in A; n = 24) (C). (D) Mean probability of obtaining a euploid embryo in a single IVF cycle, stratified by patient’s age and the number of available eggs. Reported are raw P values from the proportion test. ns, not significant; *P < 0.05; ***P < 1 × 10⁻⁸.

Fig. 4.

Karyotypes and underlying error mechanisms stratified by maternal age categories. (A) Euploid embryo source. Light gray bars represent euploid embryos obtained from normal meiosis (MI-Normal/MII-Normal). Dark gray bars indicate euploid embryos as a result of MI-PSSC/MII-Normal. Black bars represent a sum of euploid embryos from MI-NDJ/MII-NDJ and MI-PSCC/MII-NDJ. (B) Relative contribution of errors in A to euploid embryos. (C) Single-trisomy embryo source. Green and gray bars represent trisomic embryos as an outcome of MI-NDJ/MII-Normal and MI-PSSC/MII-Normal, respectively. Yellow bars represent outcomes of MI-Normal/MII-NDJ. Blue and red bars represent outcomes of MI-PSSC/MII-NDJ on the same chromosome (depicted in red in Fig. 1) and a different one (depicted in blue in Fig. 1), respectively. (D) Relative contribution of errors in C to trisomic embryos. MI-NDJ/MII-NDJ outcomes can give rise to pronuclei with “+2” DNA content. Upon simulation of the model, these outcomes occurred at a frequency of <1% across the entire age span and are thus omitted from the figure for clarity.