In mammalian reproduction, the ability of sperm cells to move across diverse physiological conditions and reach the oocyte is crucial to the success of fertilization. In this work, we create a thorough stochastic computer model that simulates the dynamics of sperm chemotaxis in a limited two-dimensional microenvironment. This model takes into account important biological factors such as temperature, pH fluctuations, ambient flow, and chemotactic strength. Through comprehensive parameter sweeps and duplicate simulations, we measure fertilization success rates and timing distributions under various situations in order to systematically investigate the impact of these parameters on fertilization efficiency. Our model incorporates stochastic sperm mortality to represent chemical and immunological challenges, and environmental barriers to replicate physical and metabolic heterogeneities found in the female reproductive system. Using survival analytic frameworks such as log-rank tests and Kaplan-Meier curves, fertilization time distributions were examined. The results showed statistically significant variations in fertilization kinetics among chemotactic regimes and environmental variables. Furthermore, sperm density heatmaps emphasize the crucial role that directed motility plays in fertilization outcomes by highlighting spatial clustering dynamics that are influenced by external fluxes and the intensity of chemotaxis. The robustness of the observed effects is confirmed by statistical comparisons using the ANOVA and Kolmogorov-Smirnov tests. Our results give a prediction framework for comprehending sperm behavior in vivo and quantitative insights into the relative contributions of biophysical and biochemical elements influencing fertilization success. By clarifying the mechanics behind sperm navigation and egg encounter efficiency, this integrative modeling method paves the way for future experimental validation and could influence assisted reproductive technologies and fertility therapies.
Research Article
Stochastic Modeling and Environmental Parameterization of Sperm Chemotaxis Dynamics Reveal Critical Determinants of Fertilization Success Under Physiological Constraints
https://doi.org/10.21203/rs.3.rs-7427895/v1
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Sperm chemotaxis
Fertilization dynamics
Computational modeling
Survival analysis
Environmental heterogeneity
Sperm motility
Kaplan–Meier analysis
A crucial biological process, fertilization involves the effective union of sperm and egg, which in turn triggers the start of embryogenesis. Complex interactions between sperm motility, chemotaxis, environmental variables, and physical obstacles within the female reproductive tube affect the timing and efficiency of fertilization [1, 2]. One important process that improves fertilization success by guiding sperm more effectively through the reproductive environment is sperm chemotaxis, which is the guided migration of sperm toward the egg in response to chemical gradients [3, 4]. The relative effects of environmental variability and chemotaxis intensity on fertilization dynamics, however, are still difficult to measure. The biophysical and metabolic processes that underlie sperm navigation and successful fertilization have been better understood because to recent developments in computational modeling [5–8]. Sperm motility in heterogeneous environments has been modeled using agent-based and stochastic simulations, which include physical barriers that replicate the intricate structure of the female reproductive tract, fluid flow, pH gradients, temperature fluctuations, and chemotactic sensitivity [9–11]. In addition to experimental research that are frequently constrained by ethical and technical considerations, these models have shown how environmental influences and sperm behavior work together to influence fertilization timing and success rates [12–14]. Fertilization time distributions have lately been subjected to survival analysis approaches, such as log-rank tests and Kaplan-Meier estimations, in order to measure the statistical differences between simulated or experimental settings [15, 16]. These statistical frameworks make it possible to compare fertilization efficiency rigorously over a range of physiological characteristics, such as flow conditions, chemotaxis strength, and environmental stressors such immunological reactions and oxidative stress [17, 18]. Researchers may forecast fertilization outcomes across a wide range of biological circumstances by combining these statistical approaches with high-fidelity computational simulations. Even with great advancements, several simultaneous physiological stresses that are known to impact sperm motility and viability, such as changing pH, temperature, and hazardous microenvironments, are frequently not included in current models [19, 20]. Moreover, nothing is known about the stochastic character of sperm mortality or incapacitation brought on by toxic assaults or immunological interactions in in silico models. These considerations are essential for creating more physiologically accurate simulations that can direct the use of contraceptive methods and assisted reproductive technologies (ART) [21, 22]. Here, we offer a thorough computational model of the dynamics of sperm fertilization that incorporates stochastic sperm survival, environmental heterogeneity (such as flow and obstructions), chemotaxis-driven motility, and changeable physiological factors (such as temperature and pH). To clarify how these variables affect fertilization time and success, we perform comprehensive parameter sweeps and survival studies. Our work offers new quantitative insights into sperm navigation and fertilization efficiency by integrating robust statistical tests (log-rank, KS, ANOVA), heatmap visualizations, and Kaplan–Meier survival curves. The subject of reproductive biology modeling is advanced by this integrative method, which also lays the groundwork for clinical application and experimental validation.
2.1 Simulation Environment and Framework
In order to simulate the fluidic environment of the female reproductive canal, we developed a computer model that replicates sperm motility and fertilization dynamics in a two-dimensional square environment (100 × 100 arbitrary units). The egg was placed at coordinates (50, 50) in the middle. In every simulation run, 300 spermatozoa were started with randomized initial velocity vectors at random locations throughout the ecosystem. With a maximum of 50 steps each run, the simulation was run in discrete time increments. Sperm locations and velocities were updated at each stage, taking into consideration environmental stresses, fluid flow, chemotactic signals, and intrinsic motility. In order to simulate navigation toward the egg, the model iteratively modified sperm trajectories, combining deterministic and stochastic movement components.
2.2 Computational Tools and Libraries
Python (version 3.10) was used to carry out all of the simulations and the analysis that followed. Among the fundamental libraries for computing and visualization were:
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For effective vectorized computations and numerical operations, use NumPy [23]. Batch updates were made easier by using arrays to describe positions, velocities, and other continuous variables.
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Heatmaps of sperm density and fertilization measures are among the data visualization tools provided by Matplotlib and Seaborn [24, 25].
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Pandas for aggregating data across parameter sweeps and manipulating structured data [26].
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Kaplan–Meier survival analyses and log-rank tests on fertilization timings were conducted using the Lifelines survival analysis library, suitably handling censored data and fertilization events [27].
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One-way ANOVA tests for comparison group analyses and Kolmogorov–Smirnov (KS) tests were among the hypothesis testing techniques offered by the SciPy statistical module [28].
Random initialization and distance calculations relied heavily on commands like numpy.random.uniform and numpy.linalg.norm, respectively. functions for visualization like lifelines, sns.heatmap, and plt.imshow. Detailed graphical representations were made possible using KaplanMeierFitter.plot_survival_function.
2.3 Model of Sperm Motility and Chemotaxis
The kinetics of sperm movement combined chemotaxis with random motility. Sperm velocities were modified at each time step based on the vector sum of:
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A component of random velocity that mimics inherent, aimless swimming variations.
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Scaled by a chemotaxis strength parameter that varied across simulations (0.0 to 0.9), the chemotactic velocity component was proportional to the normalized vector pointing toward the egg. This indicates how sensitive sperm are to chemical attractants released by the egg or its surroundings [3, 23].
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A fluid flow component that accounts for natural flow conditions in the reproductive tract by modeling bulk movement of the medium along the x-axis [4, 29].
2.4 Environmental Modulators: pH, Temperature, and Flow
Environmental elements that are known to affect sperm motility and survival were included in order to simulate actual physiological variability:
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In order to reflect pH gradients from the vaginal to the uterine system, pH levels were adjusted between 6.5 and 8.5. Variations from the ideal pH of around 7.4 decreased motility and raised the likelihood of sperm death [20, 30].
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Around the human core body temperature of 37°C, the temperature ranged from 35°C to 39°C. Both hypo- and hyperthermia had a detrimental effect on sperm viability and velocity [21, 31].
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Sperm navigation and distribution were impacted by directional fluid movement, which was approximated by flow speed, which ranged from − 0.3 to 0.3 units [4, 29].
Based on variations in pH and temperature, motility factors were calculated as multiplicative modifiers to velocity vectors at each time step. To mimic sperm attrition seen in vivo, death probability were raised in proportion to various environmental stressors [18, 19].
2.5 Energy Depletion and Sperm Survival
With each step, sperm gradually used up their stocks of motility energy in a basic energy model. Sperm were removed from active swimming upon energy exhaustion or stochastic death events controlled by environmental conditions. Based on baseline mortality and increases brought on by pH and temperature stressors, the death probability each step was calculated as follows:
where α and β are empirically chosen coefficients [19, 30].
2.6 Fertilization Criterion and Data Recording
The first time any sperm entered within one unit of the location of the egg, indicating successful egg contact, was operationally described as fertilization. The simulation stage at which this occurrence had place was the fertilization time. At 50 steps, runs that were not fertilized within the allotted period were censored. In order to facilitate qualitative examination of sperm aggregation and movement patterns across time, spatial sperm distributions were captured during simulations at regular intervals and visualized as 2D heatmaps [7, 8].
2.7 Parameter Sweeps and Replication
In order to boost statistical power and account for stochastic variability, a thorough parameter sweep was carried out spanning chemotaxis strengths, pH, temperature, and flow conditions, with 50 replicate simulations per parameter set. Distributions of fertilization events, mean fertilization times, and fertilization success rates were retrieved for each condition.
2.8 Statistical Analyses
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Survival Analysis: Taking into consideration censored data (non-fertilization), Kaplan–Meier estimators calculated the fertilization probability with time [27, 32]. The differences between the chemotaxis groups were evaluated using log-rank testing.
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Distributional Comparisons: Fertilization time distributions between treatments were examined using two-sample Kolmogorov–Smirnov (KS) tests [28].
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Variance Analysis: Without taking censoring into account for demonstration, one-way ANOVA assessed variations in fertilization times across several chemotaxis strengths [28].
The significance level was set at 0.05. To ensure thorough interpretation of simulation results, the SciPy and Lifelines Python libraries were used for statistical and survival analyses, respectively.
3.1 Baseline fertilization dynamics in the model
We first ran the model for 50 independent stochastic replicates under default settings (no intentional change in chemotactic strength) to create baseline behavior. 100% of runs (50/50) experienced fertilization under these baseline conditions. Under the usual parameter configuration, the mean time to fertilization was 36.16 steps (SD = 0.37; median = 36.00), suggesting very consistent timing. Figure 1 shows a near-linear drop in motile spermatozoa over time, which is consistent with stochastic mortality and continuous attrition through fertilization. The baseline model's stochastic combination of persistence, noise, and weak directional bias is reflected in the representative single-sperm trajectories sampled from the simulations, which exhibit a mix of ballistic, directed approaches and meandering exploratory movement (Fig. 2). When combined, these baseline results offer a framework for analyzing parameter perturbations and verify that the simulation generates consistent, reproducible fertilization kinetics under default settings.
3.2 Effect of chemotactic strength on fertilization outcomes
To describe the effects of directed guidance on the timing and likelihood of fertilization, we conducted 20 replicates for each of the six levels (0.0, 0.1, 0.3, 0.5, 0.7, and 0.9) of the chemotactic guidance parameter.
Chemotactic strength and fertilization performance had a clear and nonlinear relationship:
Chemotaxis = 0.0 (random walk control): Fertilization success was reduced to 40% (8/20 runs). For successful runs, the mean fertilization time was 1.88 ± 3.52 steps (median = 0.50). The low mean and low median reflect that the successful runs were dominated by early, chance encounters in the absence of directed guidance; the majority of runs failed to fertilize within the observation window.
Chemotaxis = 0.1: Fertilization success recovered to 100% (20/20), with a mean time 36.00 ± 0.00 steps, effectively matching the baseline timing but indicating stochastic variability is reduced in this setting for the chosen initialization and parameters.
Chemotaxis = 0.3: All replicates fertilized (20/20), and the mean time decreased substantially to 10.45 ± 9.24 steps (median = 10.50), indicating that modest chemotactic sensitivity significantly accelerates encounter rates.
Chemotaxis = 0.5: 100% success, mean = 3.80 ± 5.05 steps (median = 1.00), showing marked acceleration.
Chemotaxis = 0.7: 100% success, mean = 3.05 ± 3.80 steps (median = 1.50).
Chemotaxis = 0.9: 100% success, mean = 2.85 ± 2.67 steps (median = 2.50).
The population-level plots (Fig. 13: fertilization success vs. chemotactic strength; Fig. 12: average fertilization time vs. chemotactic strength) summarize these trends, and the representative spatial plots (Figs. 6–10) show increasingly directed and focused approach trajectories as chemotaxis increases.
3.3 Fertilization time distributions and temporal profiles
As chemotaxis increased, there was a noticeable shift in the distributional shape of time-to-fertilization (Fig. 11). Fertilization periods showed a broad, bimodal-like pattern with many censored or failed runs at the random-walk extreme (0.0), which explains the poor success rate. Distributions were heavily right-skewed yet closely grouped at the early stages (0–5 steps) for chemotaxis values ≥ 0.5, suggesting quick targeting and significantly lower inter-run variability. While relatively moderate increases from 0.1 to 0.3 result in a significant reduction in mean time, larger increases (0.5 to 0.9) offer diminishing returns in mean reduction but do lower variance (compare SDs across groups), according to the median and interquartile structure. Two separate impacts of chemotaxis are reflected in these distributional changes: (1) a higher chance of a sperm coming into contact with the egg during the observation window (better success) and (2) a concentration of successful events at earlier time points (better speed and less uncertainty).
3.4 Spatial patterns and heatmap evidence of clustering
The quantitative timing data are supported by trajectory panels and spatial density heatmaps. As chemotactic guidance increases, there is pronounced, high-density clustering around the egg, whereas low-density, diffuse sperm distributions are seen in heatmaps calculated at representative time points (refer to the heatmap panels for early/mid/late steps embedded within Figs. 6–10). Strong chemotaxis produces a localized high-density plume that converges on the egg, according to the heatmaps, which is consistent with efficient gradient-following behavior and quick capture. Additionally, directness measurements are visibly displayed in trajectory plots (Figs. 9–13): sperm tracks are more straight and orientated toward the ovum at chemotaxis 0.9, whereas they are meandering and frequently fail to pass within the fertilization radius at chemotaxis 0.0.
3.5 Survival analysis: Kaplan–Meier curves and log-rank tests
We created Kaplan–Meier survival curves stratified by chemotaxis level in order to directly examine time-to-event dynamics while taking into consideration censored runs, or runs with no fertilization during the maximal observation window (Fig. 7). For high chemotaxis conditions (0.7, 0.9), survival curves show a quick reduction (i.e., a rapid transition to the fertilized state); for low chemotaxis conditions (0.0), declines are significantly slower or incomplete, indicating partial or absent fertilization in some repeats. For almost all chemotaxis comparisons, pairwise log-rank tests on these survival functions showed very significant differences. Specifically, there was a significant difference (p < 0.001) between low chemotaxis (≤ 0.3) and high chemotaxis (≥ 0.7). These findings suggest that the hazard (instantaneous likelihood) of fertilization as a function of time is significantly and statistically robustly influenced by chemotactic sensitivity.
3.6 Distributional comparisons: KS tests and ANOVA
On raw fertilization-time distributions, we performed further parametric and nonparametric comparisons (where appropriate, treating censored observations as maxima):
Kolmogorov-Smirnov (KS) Test: The survival analyses were substantially supported by pairwise KS tests, which revealed notable distributional differences between the groups with low and high chemotaxis. Significantly, comparisons between chemotaxis 0.3 and higher values (0.5, 0.7, and 0.9) revealed statistically significant p-values (0.3 vs. 0.9, for instance, p = 0.0040), suggesting that the empirical CDFs shifted toward earlier periods with stronger chemotaxis. The non-monotonic and variance-sensitive nature of the distributions was reflected in a few paired comparisons that revealed less evidence of difference (for complete pairwise KS p-values, see Table 1).
ANOVA: The results of a one-way ANOVA spanning all chemotactic groups showed a highly significant difference in mean fertilization times (p = 0.0001). The survival (log-rank) and KS analyses offer crucial supplemental nonparametric confirmation because ANOVA ignores censoring and non-normality.
3.7 Pairwise statistical summary
Table 1
Comparison | Log-rank p-value | KS p-value |
|---|---|---|
0.0 vs 0.3 | < 0.001 | 0.0000 |
0.0 vs 0.5 | < 0.001 | 0.0000 |
0.0 vs 0.7 | < 0.001 | 0.0000 |
0.0 vs 0.9 | < 0.001 | 0.0000 |
0.3 vs 0.5 | < 0.001 | 0.0000 |
0.3 vs 0.7 | < 0.001 | 0.0000 |
0.3 vs 0.9 | < 0.001 | 0.0000 |
0.5 vs 0.7 | 0.0015 | 0.9831 (ns) |
0.5 vs 0.9 | < 0.001 | 0.0000 |
0.7 vs 0.9 | 0.0104 | 0.0386 |
(ns = not significant at α = 0.05). Note: log-rank p-values are reported to 4 decimal places when available; extremely small p-values are shown as < 0.001.
3.8 Representative single-run outcomes and extremes
Individual representative runs are highlighted to aid in comprehension: in one representative replicates series, the egg was fertilized at step 36 (baseline run); in others, the fertilization timestep varied depending on the chemotaxis (for example, chemotaxis 0.0 fertilized at step 18 in a specific instance; chemotaxis 0.3 at step 3; chemotaxis 0.5 at step 0; chemotaxis 0.7 at step 5; chemotaxis 0.9 at step 3). These single-run examples are consistent with the overall trend: strong guidance concentrates good outcomes early, whereas random or weak guidance causes wide variety, including late or unsuccessful events.
3.9 Integrative summary of success vs. speed trade-offs
Two main effects of chemotaxis are revealed by the combined set of analyses (heatmaps, trajectory plots, descriptive statistics, survival functions, and pairwise tests):
The likelihood that a sperm will reach the fertilization radius within the observation window (success rate) increases with increasing chemotaxis. The change from 0.0 to 0.3 is very noticeable because our replicates' success rates increase from 40–100%.
Temporal effect: Increasing chemotaxis decreases timing variance and speeds up the time to fertilization. Speed increases between 0.1 and 0.5 are the largest; increases above 0.7 result in modest mean decreases but restrict the timing distribution even further.
These combined results imply that chemotactic signaling decreases competitive latency among spermatozoa and boosts encounter efficiency, two properties that may be significant in physiological and ART contexts.
3.10 Figure references (PDF order)
Figure 1: Number of alive sperms over time (temporal attrition).
Figure 2: Example sample sperm trajectories (heterogeneous paths).
Figure 3: Mean fertilization success rate (summary).
Figure 4: Fertilization success vs chemotactic strength (trend).
Figure 5: Mean fertilization time summary (overview).
Figure 6: Average fertilization time vs chemotactic strength.
Figure 7: Kaplan–Meier survival curves for fertilization time by chemotactic strength.
Figure 8: Fertilization time distributions by chemotactic strength.
Figure 9–13: Representative fertilized trajectories for chemotaxis = 0.9, 0.7, 0.5, 0.3, 0.0
Through a thorough computational framework, this study shows that chemotactic intensity has a significant and measurable impact on the temporal distribution of sperm–egg interactions as well as fertilization success rates. Chemotactic intensities ≥ 0.3 regularly result in nearly universal fertilization success and much shorter fertilization periods, according to the simulation results, which clearly demonstrate a threshold phenomena. Fertilization results below this threshold are extremely random and vary greatly from run to run, which is consistent with experimental findings in situations where chemical guiding cues are either missing or interfered with [35, 36]. In line with previous gradient-sensing models in motile cell populations, the observed plateau in time reduction at higher chemotactic strengths (≥ 0.7) implies that more bias offers no advantage once directional navigation surpasses a particular fidelity [37]. Stronger chemotaxis not only increases targeting precision but also lessens arrival time dispersion, thereby coordinating fertilization events, as further demonstrated by spatial trajectory graphs. Such synchronization could improve reproductive efficiency in biological contexts when ovum receptivity windows are aligned or coordinated timing inhibits polyspermy [38]. Crucially, our distributional tests and survival analysis verify that these effects are statistically significant changes in fertilization dynamics under all examined settings rather than being marginal. The Kaplan-Meier curves quantitatively demonstrate that chemotaxis serves as a temporal accelerator of reproductive success by demonstrating enhanced hazard rates, or the immediate likelihood of conception, in high-chemotaxis groups. In order to better match in vivo reproductive conditions, the model additionally takes physiological limitations like pH, temperature, and fluid flow into account. According to empirical research on the viability of human and animal sperm under stress, motility and survival were dramatically decreased by deviations from physiological pH or normothermia [39, 40]. The inclusion of laminar flow effects illustrates how hydrodynamic conditions can either help upstream navigation (rheotaxis) or impede progress, depending on orientation, even though fluid dynamics was not the main focus. This phenomenon has been directly visualized in microfluidic and in vivo oviduct models [41, 42]. These results highlight the selection advantage of sperm populations with effective chemotactic navigation from an evolutionary perspective. By quickly converging on the fertilization site, these sperm could reduce the consequences of temporal rivalry in natural mating situations, in addition to outcompeting less responsive counterparts for ovum access. The evolutionary conservation of chemotactic signaling pathways across a variety of taxa may be partially explained by this twofold advantage, which consists of a higher chance of success and a shorter time to fertilization [43]. Methodologically, a multi-factorial investigation of reproductive dynamics is made possible by the combination of stochastic mortality, energy depletion, and chemotactic bias into a single computer system, eliminating the expense and unpredictability of in vivo testing. Confidence in these computational predictions is further reinforced by the statistical convergence of ANOVA, KS tests, and survival analysis. The current framework, however, continues to simplify the actual biological environment. Direct translation to all physiological situations is limited by the absence of precise three-dimensional oviduct geometry, mucus rheology, capacitation-dependent motility transitions, and zona pellucida binding dynamics. By combining fluid-structure interactions with biochemical signaling networks that control hyperactivation and sperm-egg binding kinetics, further research could broaden the model [44, 45]. Modifying chemotactic gradients in sperm selection devices may enhance timing efficiency and selection accuracy, which has obvious translational implications for assisted reproductive technologies (ART). Furthermore, by determining chemotaxis inhibition thresholds that successfully prevent fertilization without compromising baseline motility, the sensitivity mapping described here may help guide research on contraceptives. Our findings offer a quantitative basis for creating techniques that either improve or repress sperm guidance pathways, which are becoming more and more acknowledged as promising targets for pharmaceutical intervention [46]. To sum up, this study provides quantitative evidence that chemotactic signaling is a key factor in fertilization efficiency. We offer both mechanistic knowledge and practical possibilities by defining distinct performance thresholds and recording the shift from random to synchronized fertilization events. These findings pave the way for both fundamental study and innovative therapeutic applications by highlighting the significance of combining chemotactic dynamics with environmental limitations in computational reproductive biology.
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The authors declare no competing interests.