Modeling Diffusive Search by Non-Adaptive Sperm: Empirical and Computational Insights

Abstract

During fertilization, mammalian sperm undergo a winnowing selection process that reduces the candidate pool of potential fertilizers from ~10⁶-10¹¹ cells to 10¹-10² cells (depending on the species). Classical sperm competition theory addresses the positive or 'stabilizing' selection acting on sperm phenotypes within populations of organisms but does not strictly address the developmental consequences of sperm traits among individual organisms that are under purifying selection during fertilization. It is the latter that is of utmost concern for improving assisted reproductive technologies (ART) because 'low fitness' sperm may be inadvertently used for fertilization during interventions that rely heavily on artificial sperm selection, such as intracytoplasmic sperm injection (ICSI). Importantly, some form of sperm selection is used in nearly all forms of ART (e.g., differential centrifugation, swim-up, or hyaluronan binding assays, etc.). To date, there is no unifying quantitative framework (i.e., theory of sperm selection) that synthesizes causal mechanisms of selection with observed natural variation in individual sperm traits. In this report, we reframe the physiological function of sperm as a collective diffusive search process and develop multi-scale computational models to explore the causal dynamics that constrain sperm 'fitness' during fertilization. Several experimentally useful concepts are developed, including a probabilistic measure of sperm 'fitness' as well as an information theoretic measure of the magnitude of sperm selection, each of which are assessed under systematic increases in microenvironmental selective pressure acting on sperm motility patterns.

Figure 1:

Sperm-Agent Search is a Function of Ensemble Motility Pattern. (A) Representative model simulation of 250 sperm with equal proportions of each motility type searching a closed space. Color scale is blue (early) to white (late) frames in the video. (B) Root mean squared displacement (μm) for simulations involving the indicated composition of motility types. Mixed populations consisted of 50 sperm of each motility type. (C) Search progress (%) for the simulations described in subpanel (B).

Figure 2:

Sperm number and search properties. (A) Time to first contact with an egg in microenvironments with TCw=1 (maze A), TCw= 22(maze B), and TCw=54 (maze C). (B) Area (μm2) searched at first contact with the egg for microenvironments with increasing weighted complexity (top to bottom as in subpanel A). Lines indicate the median. N = 100 simulations for each condition.

Figure 3:

A Time-Homogeneous Markov Model of Sperm Phenotype Heterogeneity: (A) Linear regression curves for different calcium ion selective electrode filling solutions used to calculate the free Ca2+ concentrations in HEPES buffered assay media in the presence of 1mM EGTA. (B) Representative heat map showing Indo-1 fluorescence ratios for sperm under the indicated Ca2+ and HCO3− pseudo-titration conditions. Iono = ionomycin. Free calcium concentrations (bottom) are in micromolar units. T = time since the beginning of the assay in minutes. (C) Probability density estimate from spectral flow cytometry for approximately 105 live cells per indicated condition. Dead cells were excluded from analysis based on ToPro3 fluorescence intensity. (D) Representative intracellular calcium oscillations derived from a squared sine function assigned to each sperm in the model simulations. Teal bar at the top of the graph indicates the upper 5% of the concentration range during which the cells were allowed to transition motility states according to a Markov probability transition table. (E) Relative proportion of sperm in each indicated motility state over time (in model-timestep units). In the long run, sperm in the models absorbed into a weak motility state.

Figure 4:

Impact of Sperm Phenotype Heterogeneity on Diffusive Search. (A) Histogram of the sperm intracellular calcium oscillation frequencies randomly drawn from a Poisson distribution with the indicated means (λ). (B) Search time for sperm populations with different phenotype distributions in microenvironments with increasing total weighed complexity (TCw). (C) Logarithmically transformed search times from subplot B used for statistical analysis to satisfy 2-way ANOVA assumptions. Ns = not significant, *p<0.05, ****p<0.0001. Lines indicate medians. Simulations consisted of N = 100 agents.

Figure 5:

Measures to Infer Sperm ‘Fitness’ as well as Quantify the Magnitude of Sperm Selection. (A) Cumulative distributions total probability P(qi) for each oscillation frequency in the initial sperm population for each simulation condition. (B) The Bayesian likelihood (frequency of sperm for each oscillation frequency that contacted the egg). (C) Cumulative posterior probability of egg contact for each oscillation frequency. Note, a prior distribution of 1/N, where N is the total number of sperm in the simulation, was used in the calculation. This can be interpreted to mean that each sperm had an assumed equal chance of contacting the egg. (D) Relative information gain (a.k.a. Kullback-Leibler divergence) calculated for each simulation condition. ****p<0.0001. TCw = total weighted complexity. N = 100 sperm in each simulation. Points in A-C represent the median of 100 simulations. Points in D represent relative information gain for each of 100 simulations.