Empirical Validation of the Bio-Cycle Fixation Model
Classical population genetics models systematically overpredict the rate of evolutionary change in species with overlapping generations. The math is straightforward: when grandparents, parents, and children coexist and compete for the same resources, not every “generation” represents a fresh opportunity for selection to act. The human population doesn’t reset with each breeding cycle, instead, people gradually age out of it as new children are born.
The Bio-Cycle Fixation Model isn’t a refutation of classical population genetics, but an extension. Kimura’s model assumes discrete generations (d = 1.0). The Bio-Cycle model adds a parameter for generation overlap (d < 1.0). When d = 1.0, the models are identical. The question is empirical: what value of d fits real organisms?
In this appendix, we present four tests. The first demonstrates why generation overlap matters by comparing predictions for organisms with different life histories. The remaining three validate the model against ancient DNA time series from humans, where we have direct observations of allele frequencies changing over thousands of years.
Test 1: Why Generation Overlap Matters
Consider two species facing identical selection pressure—a 5 percent fitness advantage for carriers of a beneficial allele (s = 0.05). How quickly does that allele spread?
For E. coli bacteria, the answer is straightforward. Bacteria reproduce by binary fission. When a generation reproduces, the parents are gone—consumed in the act of creating offspring. There is no overlap. Kimura’s discrete-generation model was built for exactly this situation.
Now consider red foxes. A fox might live 5 years in the wild and reproduce in multiple seasons. At any given time, the population contains juveniles, young adults, prime breeders, and older individuals—all competing, all contributing genes. When this year’s pups are born, last year’s pups are still around. So are their parents. The gene pool churns rather than resets.
Let’s model what happens over 100 years with the same selection coefficient (s = 0.05), starting from 1% frequency:
| Species | Nominal Generations | Effective Generations | Predicted Frequency |
| E. coli (Kimura d = 1.0) | 876,000 | 876,000 | 100% |
| Fox (d = 0.60) | 50 | 30 | 13.8% |
| Fox (Kimura d = 1.0) | 50 | 50 | 26.4% |
The difference is immediately observable. If we apply Kimura’s model to foxes (assuming d = 1.0), we predict the allele will reach 26.4 percent after 100 years. But if foxes have 60 percent generational turnover—a reasonable estimate for a mammal with 5-year lifespan and multi-year reproduction—the Bio-Cycle model predicts only 13.8 percent. The path to mutational fixation is significantly slowed.
This isn’t a refutation of Kimura’s model. It is merely recognizing when his generational assumptions apply and when they don’t. For bacteria, d = 1.0 is correct. For foxes, d < 1.0. For humans, with our even longer lifespans and extended reproduction, d should be lower still. The question is: what is the correct value?
Test 2: Lactase Persistence in Europeans
Ancient DNA gives us something unprecedented: direct observations of allele frequencies through time. We can watch evolution happen and measure how fast alleles actually spread, the consider which model best matches the way reality played out.
Lactase persistence—the ability to digest milk sugar into adulthood—is the textbook example of recent human evolution. The persistence allele was virtually absent in early Neolithic Europeans 6,000 years ago (less than 1 percent frequency). Today, about 75 percent of Northern Europeans carry it. Researchers estimate the selection coefficient at s = 0.04–0.10, driven by the ~500 extra calories per day available from milk.
Using the midpoint (s = 0.05), what does each model predict?
| Model | Final Frequency | Error |
| Actual (observed) | 75% | — |
| Kimura (d = 1.0) | 99.9% | +24.9 percentage points |
| Bio-Cycle (d = 0.45) | 67.4% | −7.6 percentage points |
Kimura predicts the allele should have reached near-fixation. It hasn’t. The Bio-Cycle model, with d = 0.45, predicts 67.4 percent—within 8 percentage points of the observed frequency. That’s a 69 percent reduction in prediction error.
Why d = 0.45? In Neolithic populations, average lifespan was 35–40 years. People reproduced between ages 15 and 30. At any given time, 2–3 generations were alive simultaneously. A 45 percent turnover rate per nominal generation is consistent with these demographics.
Test 3: SLC45A2 and Skin Pigmentation
Light skin pigmentation in Europeans evolved under strong selection for vitamin D synthesis at higher latitudes. SLC45A2 is one of the major genes involved. Ancient DNA from Ukraine shows the “light skin” allele was at 43 percent frequency roughly 4,000 years ago. Today it’s at 97 percent. Published selection coefficient: s = 0.04–0.05.
| Model | Final Frequency | Error |
| Actual (observed) | 97% | — |
| Kimura (d = 1.0) | 99.9% | +2.9 percentage points |
| Bio-Cycle (d = 0.45) | 95.2% | −1.8 percentage points |
Both models work reasonably here because the allele approached fixation. But Bio-Cycle is still more accurate—38% error reduction—using the same d = 0.45 that worked for lactase.
Test 4: TYR—A Secondary Pigmentation Gene
TYR is another pigmentation gene with smaller phenotypic effect—about half that of SLC45A2. Selection coefficient: s = 0.02–0.04. Ancient DNA shows TYR rising from 25 percent to 76 percent over 5,000 years.
| Model | Final Frequency | Error |
| Actual (observed) | 76% | — |
| Kimura (d = 1.0) | 99.3% | +23.3 percentage points |
| Bio-Cycle (d = 0.45) | 83.3% | +7.3 percentage points |
Once again, Kimura overshoots dramatically. Bio-Cycle reduces prediction error by 69 percent, using the same d = 0.45.
Summary: Three Scenarios, One d Value
| Locus | Observed | Kimura | Bio-Cycle | Error Reduction | d |
| Lactase | 75% | 99.9% | 67.4% | 69% | 0.45 |
| SLC45A2 | 97% | 99.9% | 95.2% | 38% | 0.45 |
| TYR | 76% | 99.3% | 83.3% | 69% | 0.45 |
Three different mutations. Three different selection pressures (dietary vs. UV/vitamin D). Three different time periods (4,000–6,000 years). Three different starting frequencies (1 percent to 43 percent). All fit well with a single value: d = 0.45. All errors in single digits.
The d values that would have correctly matched the observed frequencies are 0.48, 0.52, and 0.38 respectively. Our original estimate was 0.4, but that was based on modern life cycles, so it is unsurprising that ancient life cycles would require a higher value, as lifespans were shorter and first reproduction took place at younger ages.
What This Means
The Bio-Cycle Fixation Model extends Kimura’s framework to account for overlapping generations. For humans, the empirically validated correction is d = 0.45—meaning effective generations are 45 percent of nominal generations.
When we calculate the number of substitutions possible over evolutionary time, it is necessary to use effective generations rather than nominal ones. With d = 0.45 and 450,000 nominal generations since the human-chimp split 9 million years ago, we have approximately 202,500 effective generations for selection to act.
This isn’t theoretical speculation. Three independent ancient DNA time series converge on the same value. That’s not an accident. It’s a reflection of the real world.