, where 
is the number of trials (1,000 in this case) and q is the fitness.MacCarthy and Bergman. 10.1073/pnas.0705455104. |
Fig. 2. Graphs showing variable comparisons for combinations of asexual/sexual initial phases and dominant/recessive dominance modes. (a) Graphs are for asexual (reduction modifier) wins. (b) Corresponds to sexual (modifier for increased recombination) wins.
Fig. 3. Modeling overview. (a) Representation of a gene-regulatory network. Each gene (horizontal arrow) produces a transcription factor (open circles, squares, or diamonds). These factors influence the expression of each gene via upstream cis-regulatory elements (solid circles, squares, or diamonds, shown with different gray levels to reflect inherent differences in affinity and capacity for activation or repression). Each genotype is represented by a matrix W of regulatory interactions, which produce the gene expression dynamics s(t). Selection involves checking for dynamic stability in s(t). (b) Example of reproduction in dominant mode. The left parent has an R modifier, and the parent on the right has an r modifier. After sexual reproduction (as dictated by Rr in dominant mode), the offspring has randomly inherited the r modifier from the parent on the right and random matrix rows from either parent (dotted and continuous lines, respectively). (c) Summary of life cycle. Initial population is generated by cloning M copies of a random stable individual. Subsequent generations occur via reproduction/mutation/development (producing the gene expression dynamics)/selection (only stable offspring enter the next generation).
Table 2. Experimental results for a wide variety of conditions
Initial phase† | Type initial phase | Dominance mode‡ | Reduction modifier wins | |
No | No | N/A | Recessive | 884 |
No | No | N/A | Dominant | 932 |
No | Yes | Sexual | Recessive | 761 |
No | Yes | Sexual | Dominant | 878 |
No | Yes | Asexual | Recessive | 892 |
No | Yes | Asexual | Dominant | 990 |
Yes | No | N/A | Recessive | 883 |
Yes | No | N/A | Dominant | 892 |
Yes | Yes | Sexual | Recessive | 754 |
Yes | Yes | Sexual | Dominant | 828 |
Yes | Yes | Asexual | Recessive | 862 |
Yes | Yes | Asexual | Dominant | 966 |
The results are the number of reduction modifier wins (out of 1,000 trials). All results are significant, P < 10-16 (binomial test).
*Single-point segregation indicates an alternative segregation model in which a single crossover point is chosen.
†
Experiments were performed both with and without an initial phase. Where the initial phase was implemented, it was either sexual or asexual.
‡
Dominance mode was either recessive or dominant.
Table 3. Experimental results for low mutation rate (μ = 0.002)
Initial phase | Recombination model | r allele fixations in 1,000 trials | Inclusive robustness | Difference | P value (before vs. after) | |
Before | After | |||||
Asexual | Dominant | 544 | 0.9995 | 0.9910 | -0.0085 | 4.11 × 10-33 |
Asexual | Recessive | 504 | 0.9995 | 0.9967 | -0.0028 | 5.17 × 10-32 |
Sexual | Dominant | 511 | 0.9994 | 0.9994 | 3.73 × 10-5 | 0.056 |
Sexual | Recessive | 497 | 0.9994 | 0.9995 | 0.0001 | 9.9 × 10-11 |
Shown is the number of r allele fixations in 1,000 trials. None of these outcomes is significantly higher than chance (binomial test). We also present inclusive robustness measures both before and after introduction of the modifier allele. The difference between these (after/before) follows the expected sign from the original results (μ = 0.1). A difference of means (t test) shows that the differences between before and after in inclusive robustness are significant in three cases, marginally significant in one.
Table 4. Experimental results using alternative recombination levels
Initial phase | Recombination model | r allele fixations in 1,000 trials |
Asexual | Dominant | 969 |
Asexual | Recessive | 958 |
Sexual | Dominant | 835 |
Sexual | Recessive | 885 |
Results using low (PL = 0.01) and high (PH = 0.5) recombination levels, where the probability that sexual reproduction occurs is PL and PH for homozygotes rr and RR, respectively. In dominant mode, the heterozygotes use PH, whereas in recessive mode, heterozygotes use PL. All results are significant (P < 10-16).
Table 5. Details of results for each experiment
Data Set 1
Initial phase | Dominance mode | Modifier win |
Asexual | Dominant | Asexual |
Number of wins = 990 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 110.519 | 0.989 | 0.822 | 0.068 | 0.016 |
Median | 65.000 | 0.990 | 0.826 | 0.062 | 0.011 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | 0.678 | 0.459 |
log(fixation time) | initial epistasis | -0.490 | 0.240 |
Initial epistasis | IR (after) | -0.466 | 0.217 |
P value for t test comparing initial and final epistasis | 3.367 E-70 |
Data Set 2
Initial phase | Dominance mode | Modifier win |
Asexual | Recessive | Asexual |
Number of wins = 892 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 390.481 | 0.989 | 0.934 | 0.069 | 0.061 |
Median | 288.500 | 0.990 | 0.936 | 0.063 | 0.059 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | 0.235 | 0.055 |
log(fixation time) | Initial epistasis | -0.159 | 0.025 |
Initial epistasis | IR (after) | -0.474 | 0.225 |
P value for t test comparing initial and final epistasis | 0.005 |
Data Set 3
Initial phase | Dominance mode | Modifier win |
Sexual | Dominant | Asexual |
Number of wins = 878 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 297.521 | 0.988 | 0.990 | -0.131 | -0.047 |
Median | 245.000 | 0.989 | 0.991 | -0.122 | -0.043 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | 0.163 | 0.027 |
log(fixation time) | Initial epistasis | -0.040 | 0.002 |
Initial epistasis | IR (after) | -0.320 | 0.102 |
P value for t test comparing initial and final epistasis | 3.030 E-136 |
Data Set 4
Initial phase | Dominance mode | Modifier win |
Sexual | Recessive | Asexual |
Number of wins = 761 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 578.353 | 0.988 | 0.995 | -0.129 | 0.039 |
Median | 452.000 | 0.989 | 0.995 | -0.120 | 0.039 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | -0.017 | 0.000 |
log(fixation time) | Initial epistasis | 0.081 | 0.007 |
Initial epistasis | IR (after) | -0.515 | 0.265 |
P value for t test comparing initial and final epistasis | 1.822 E-305 |
Data Set 5
Initial phase | Dominance mode | Modifier win |
Asexual | Dominant | Sexual |
Number of wins = 10 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 672.300 | 0.991 | 0.893 | 0.013 | -0.064 |
Median | 485.000 | 0.992 | 0.888 | 0.006 | -0.047 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | 0.504 | 0.254 |
log(fixation time) | Initial epistasis | -0.174 | 0.030 |
Initial epistasis | IR (after) | -0.509 | 0.259 |
P value for t test comparing initial and final epistasis | 1.088 E-04 |
Data Set 6
Initial phase | Dominance mode | Modifier win |
Asexual | Recessive | Sexual |
Number of wins = 108 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 833.259 | 0.990 | 0.935 | 0.057 | -0.106 |
Median | 782.000 | 0.990 | 0.935 | 0.051 | -0.102 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | -0.173 | 0.030 |
log(fixation time) | Initial epistasis | 0.244 | 0.059 |
Initial epistasis | IR (after) | -0.538 | 0.289 |
P value for t test comparing initial and final epistasis | 2.630 E-44 |
Data Set 7
Initial phase | Dominance mode | Modifier win |
Sexual | Dominant | Sexual |
Number of wins = 122 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 394.393 | 0.989 | 0.991 | -0.118 | -0.117 |
Median | 326.000 | 0.989 | 0.991 | -0.096 | -0.102 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | -0.103 | 0.011 |
log(fixation time) | Initial epistasis | -0.087 | 0.008 |
Initial epistasis | IR (after) | -0.338 | 0.114 |
P value for t test comparing initial and final epistasis | 0.915 |
Data Set 8
Initial phase | Dominance mode | Modifier win |
Sexual | Recessive | Sexual |
Number of wins = 239 |
|
|
| Fixation time | IR (before) | IR (after) | Initial epistasis | Final epistasis |
Mean | 826.498 | 0.988 | 0.994 | -0.129 | -0.117 |
Median | 676.000 | 0.989 | 0.995 | -0.120 | -0.108 |
Measurements compared | r | R 2 | |
log(fixation time) | IR (after) | -0.079 | 0.006 |
log(fixation time) | Initial epistasis | 0.109 | 0.012 |
Initial epistasis | IR (after) | -0.479 | 0.230 |
P value for t test comparing initial and final epistasis | 0.050 |
Results are separated by modifier win (r allele fixation implies asexual win, R allele fixation implies sexual win). Each data set is described by four tables. (1) Experiment parameters are combinations of asexual/sexual initial phase and dominant/recessive dominance mode, also the number of wins in each category is shown. (2) Mean and median values for five measurements, as described in Methods (IR = inclusive robustness), "before" indicates before introduction of the modifier, and "after" indicates after introduction of the modifier. (3) Correlation coefficient (r) and corresponding linear regression R2 value for the measurements shown. (4) P value for t test comparing initial (i.e., immediately after introduction of the modifier) and final epistasis.
Table 6. Comparing the variation in fitness of the recombined (Vr) vs. nonrecombined offspring (Vn)
Initial phase | Recombination model | Difference in fitness variance (Vr -Vn) | ||
Minimum | Mean | Maximum | ||
Asexual | Dominant | 30.5 | 69.3 | 86.5 |
Asexual | Recessive | 30.9 | 69.4 | 87.0 |
Sexual | Dominant | - 0.2 | 11.4 | 24.1 |
Sexual | Recessive | - 0.2 | 11.4 | 24.2 |
Classical population genetics models predict that under synergistic epistasis the recombined offspring will have higher variance in fitness (compared to the nonrecombined), whereas under antagonistic epistasis the opposite should occur. We would therefore expect to observe positive (Vr - Vn) after the sexual initial phase and negative (Vr - Vn) after the asexual initial phase. However, our results show that, not only are both positive, but that (Vr - Vn) actually is greater after the asexual initial phase. The table shows the difference (Vr - Vn) under four conditions. Note that fitness (inclusive robustness) in the model is binomially distributed, and therefore variance is, where is the number of trials (1,000 in this case) and q is the fitness.