============================================================================== MULLER'S RATCHET: the click rate, the classical prediction, and the rate of sex that stalls it. Science Journaling Club, Volume 2 Issue 1, Fall 2025. ============================================================================== This is a simulation. Nothing here was observed in an organism. Python : 3.12.3 numpy : 2.4.2 MASTER SEED : 20250921 (PCG64, one independent stream per cell via SeedSequence.spawn) Replicates : 200 independent populations per parameter cell Burn-in : 400 generations discarded before any statistic is recorded ============================================================================== PART 0. NULL CHECK. With U = 0 the ratchet must never click and mean fitness must stay at exactly 1.0. ============================================================================== N s reps gens clicks mean load mean fitness ------------------------------------------------------------------------------ 100 0.02 200 1000 0 0.0 1.000000000000000 PASS 100 0.10 200 1000 0 0.0 1.000000000000000 PASS 500 0.02 200 1000 0 0.0 1.000000000000000 PASS 500 0.10 200 1000 0 0.0 1.000000000000000 PASS 2000 0.02 200 1000 0 0.0 1.000000000000000 PASS 2000 0.10 200 1000 0 0.0 1.000000000000000 PASS ------------------------------------------------------------------------------ Null check on the load-class engine: PASS With no mutation there is no least-loaded class to lose. kmin stays 0 in all 200 replicates of all six cells and mean fitness is exactly 1. Null check on the explicit-genome engine (N=200, s=0.05, U=0, R=1, 40 reps, 300 generations): upward clicks = 0, downward steps = 0, mean load = 0.0 -> PASS ============================================================================== PART 1. THE GRID. Measured click rate beside the classical prediction built on n0 = N exp(-U/s). ============================================================================== Columns: n0 pred = N exp(-U/s), Haigh's equilibrium size of the least-loaded class n0 seen = mean size of the least-loaded class in the moving frame, all gens rate = ratchet clicks per generation, mean over the 200 replicates SE = standard error of that mean, computed across the 200 replicates exp(-n0) = the classical one-generation extinction estimate of the rate meas/pred = the ratio this study is actually testing N U s U/s n0 pred n0 seen rate SE exp(-n0) meas/pred ------------------------------------------------------------------------------ 100 0.10 0.02 5.0 0.6738 11.72 0.03691 5.68e-04 0.50977 0.0724 200 0.10 0.02 5.0 1.348 15.2 0.02804 4.94e-04 0.25987 0.108 500 0.10 0.02 5.0 3.369 21.61 0.01907 3.01e-04 0.03442 0.554 1000 0.10 0.02 5.0 6.738 29.43 0.01385 2.29e-04 0.00119 11.7 2000 0.10 0.02 5.0 13.48 40.06 0.00951 1.55e-04 1.40e-06 6.77e+03 100 0.20 0.02 10.0 0.00454 6.278 0.09030 8.55e-04 0.99547 0.0907 200 0.20 0.02 10.0 0.00908 7.564 0.07374 7.37e-04 0.99096 0.0744 500 0.20 0.02 10.0 0.0227 9.468 0.05900 6.12e-04 0.97756 0.0604 1000 0.20 0.02 10.0 0.0454 11.28 0.04651 4.85e-04 0.95562 0.0487 2000 0.20 0.02 10.0 0.0908 13.63 0.04005 4.76e-04 0.91320 0.0439 100 0.40 0.02 20.0 2.06e-07 3.608 0.20983 0.00120 1.00000 0.21 200 0.40 0.02 20.0 4.12e-07 4.054 0.18526 9.88e-04 1.00000 0.185 500 0.40 0.02 20.0 1.03e-06 4.673 0.15222 9.06e-04 1.00000 0.152 1000 0.40 0.02 20.0 2.06e-06 5.202 0.13548 8.64e-04 1.00000 0.135 2000 0.40 0.02 20.0 4.12e-06 5.723 0.12166 6.78e-04 1.00000 0.122 100 0.80 0.02 40.0 4.25e-16 2.322 0.47074 0.00166 1.00000 0.471 200 0.80 0.02 40.0 8.50e-16 2.5 0.43163 0.00157 1.00000 0.432 500 0.80 0.02 40.0 2.12e-15 2.72 0.38270 0.00150 1.00000 0.383 1000 0.80 0.02 40.0 4.25e-15 2.903 0.35385 0.00129 1.00000 0.354 2000 0.80 0.02 40.0 8.50e-15 3.101 0.32493 0.00117 1.00000 0.325 ------------------------------------------------------------------------------ 100 0.10 0.05 2.0 13.53 18.09 0.01438 2.74e-04 1.33e-06 1.08e+04 200 0.10 0.05 2.0 27.07 29.22 0.00722 1.53e-04 1.76e-12 4.1e+09 500 0.10 0.05 2.0 67.67 62.62 0.00176 5.32e-05 4.10e-30 4.3e+26 1000 0.10 0.05 2.0 135.3 124.8 2.34e-04 2.18e-05 1.68e-59 1.4e+55 2000 0.10 0.05 2.0 270.7 259.2 6.00e-06 3.45e-06 2.81e-118 2.1e+112 100 0.20 0.05 4.0 1.832 8.98 0.04765 6.28e-04 0.16016 0.298 200 0.20 0.05 4.0 3.663 12.24 0.03139 5.01e-04 0.02565 1.22 500 0.20 0.05 4.0 9.158 19.16 0.01909 3.00e-04 0.00011 181 1000 0.20 0.05 4.0 18.32 27.44 0.01135 1.89e-04 1.11e-08 1.0e+06 2000 0.20 0.05 4.0 36.63 42.54 0.00588 1.10e-04 1.23e-16 4.8e+13 100 0.40 0.05 8.0 0.03355 4.647 0.12832 9.30e-04 0.96701 0.133 200 0.40 0.05 8.0 0.06709 5.617 0.10562 7.96e-04 0.93511 0.113 500 0.40 0.05 8.0 0.1677 7.251 0.07654 6.84e-04 0.84558 0.0905 1000 0.40 0.05 8.0 0.3355 8.812 0.06066 6.49e-04 0.71501 0.0848 2000 0.40 0.05 8.0 0.6709 10.96 0.04870 5.26e-04 0.51124 0.0953 100 0.80 0.05 16.0 1.13e-05 2.742 0.33434 0.00147 0.99999 0.334 200 0.80 0.05 16.0 2.25e-05 3.054 0.29079 0.00120 0.99998 0.291 500 0.80 0.05 16.0 5.63e-05 3.551 0.24125 0.00110 0.99994 0.241 1000 0.80 0.05 16.0 0.0001125 3.977 0.21163 0.00100 0.99989 0.212 2000 0.80 0.05 16.0 0.0002251 4.447 0.18477 9.28e-04 0.99977 0.185 ------------------------------------------------------------------------------ 100 0.10 0.10 1.0 36.79 33.61 0.00150 4.85e-05 1.05e-16 1.4e+13 200 0.10 0.10 1.0 73.58 69.64 8.80e-05 1.36e-05 1.11e-32 7.9e+27 500 0.10 0.10 1.0 183.9 180.1 0 0 1.31e-80 0 1000 0.10 0.10 1.0 367.9 364.6 0 0 1.71e-160 0 2000 0.10 0.10 1.0 735.8 732.2 0 0 2.91e-320 0 100 0.20 0.10 2.0 13.53 15.01 0.01501 2.80e-04 1.33e-06 1.13e+04 200 0.20 0.10 2.0 27.07 25.75 0.00549 1.07e-04 1.76e-12 3.1e+09 500 0.20 0.10 2.0 67.67 62.37 3.92e-04 2.78e-05 4.10e-30 9.6e+25 1000 0.20 0.10 2.0 135.3 130.6 4.00e-06 2.82e-06 1.68e-59 2.4e+53 2000 0.20 0.10 2.0 270.7 266 0 0 2.81e-118 0 100 0.40 0.10 4.0 1.832 6.534 0.06613 7.32e-04 0.16016 0.413 200 0.40 0.10 4.0 3.663 8.985 0.04365 5.95e-04 0.02565 1.7 500 0.40 0.10 4.0 9.158 14.18 0.02309 4.19e-04 0.00011 219 1000 0.40 0.10 4.0 18.32 21.61 0.01199 2.17e-04 1.11e-08 1.1e+06 2000 0.40 0.10 4.0 36.63 35.65 0.00451 8.77e-05 1.23e-16 3.7e+13 100 0.80 0.10 8.0 0.03355 3.402 0.21480 0.00114 0.96701 0.222 200 0.80 0.10 8.0 0.06709 4.019 0.17229 9.83e-04 0.93511 0.184 500 0.80 0.10 8.0 0.1677 5.148 0.12562 8.67e-04 0.84558 0.149 1000 0.80 0.10 8.0 0.3355 6.255 0.09904 7.96e-04 0.71501 0.139 2000 0.80 0.10 8.0 0.6709 7.527 0.07670 6.14e-04 0.51124 0.15 ------------------------------------------------------------------------------ Cells: 60. Total replicate-generations in the grid: 11,220,000 Worst probability lost off the top of the load window, any cell: 4.44e-16 Generations in which the minimum jumped by more than one, all cells: 33605 Cells that clicked at least once: 56. Cells that never clicked: 4. ============================================================================== PART 1a. HAIGH'S EQUILIBRIUM, tested only on replicate-generations in which the ratchet has not clicked at all, since a deterministic equilibrium is meaningless once the minimum has moved. ============================================================================== Predictions: mean load = lambda = U/s, variance of load = lambda, size of class zero = N exp(-lambda). N U s U/s quiet% mean load var load n0 quiet N exp(-l) ratio ------------------------------------------------------------------------------ 500 0.10 0.05 2.0 10.9 2.0907 2.0892 62.0839 67.6676 0.9175 1000 0.10 0.05 2.0 69.8 2.0766 2.0580 124.1577 135.3353 0.9174 2000 0.10 0.05 2.0 99.1 2.0378 2.0281 259.1247 270.6706 0.9573 2000 0.20 0.05 4.0 0.7 4.0224 4.0762 37.8689 36.6313 1.0338 100 0.10 0.10 1.0 14.0 1.1015 1.0851 32.9257 36.7879 0.8950 200 0.10 0.10 1.0 87.0 1.0513 1.0421 69.5410 73.5759 0.9452 500 0.10 0.10 1.0 100.0 1.0191 1.0160 180.1454 183.9397 0.9794 1000 0.10 0.10 1.0 100.0 1.0082 1.0069 364.6455 367.8794 0.9912 2000 0.10 0.10 1.0 100.0 1.0043 1.0033 732.1679 735.7589 0.9951 200 0.20 0.10 2.0 0.7 2.0630 2.0591 25.2565 27.0671 0.9331 500 0.20 0.10 2.0 55.0 2.0725 2.0540 62.3038 67.6676 0.9207 1000 0.20 0.10 2.0 99.7 2.0324 2.0257 130.5769 135.3353 0.9648 2000 0.20 0.10 2.0 100.0 2.0155 2.0119 266.0129 270.6706 0.9828 2000 0.40 0.10 4.0 1.6 4.0755 4.0513 33.4477 36.6313 0.9131 ------------------------------------------------------------------------------ Cells with enough quiet data: 14 mean load / lambda : mean 1.0288, sd 0.0258, range 1.0043 to 1.1015 var load / lambda : mean 1.0249, sd 0.0215, range 1.0033 to 1.0851 class zero / N exp(-l) : mean 0.9533, sd 0.0397, range 0.8950 to 1.0338 restricted to n0 >= 8 (14 cells): 0.9533 +/- 0.0106 And the same class-zero size measured WITHOUT that restriction, i.e. the least-loaded class in the moving frame, which is what an observer of a real population would actually see: n0 pred band cells median n0 seen/pred range ------------------------------------------------------------------------------ 0 - 0.1 24 1.36e+05 59.9 to 5.47e+15 0.1 - 1 7 18.65 11.22 to 43.23 1 - 3 3 4.903 3.567 to 11.28 3 - 8 4 4.368 2.453 to 6.414 8 - 30 9 1.336 0.9513 to 2.973 >= 30 13 0.9649 0.9137 to 1.161 ------------------------------------------------------------------------------ ============================================================================== PART 1b. WHERE THE CLASSICAL RATE PREDICTION FAILS. ============================================================================== n0 pred band cells median meas/pred min max ------------------------------------------------------------------------------ 0 - 0.1 24 0.1853 0.04386 0.4707 0.1 - 1 7 0.09526 0.07241 0.15 1 - 3 3 0.2975 0.1079 0.4129 3 - 8 4 1.702 0.5539 11.69 8 - 30 9 1.133e+04 181.1 4.11e+09 >= 30 9 4.30e+26 1.42e+13 2.13e+112 ------------------------------------------------------------------------------ The prediction is wrong in both directions. Below n0 = 1 it is too fast, because the least-loaded class in a clicking population is not at its deterministic size. Above n0 = 3 it is far too slow, because the class is not required to die in a single draw. ============================================================================== PART 1c. IS THE CLICK RATE A FUNCTION OF n0 ALONE? Three cells built to share n0 = 5 and lambda = 4, with the selection coefficient varying by a factor of five. ============================================================================== N U s n0 pred n0 seen rate SE clicks ------------------------------------------------------------------------------ 273 0.080 0.02 5 22.03 0.01788 3.02e-04 2145 273 0.200 0.05 5 14.26 0.02656 4.32e-04 2125 273 0.400 0.10 5 10.53 0.03544 5.36e-04 2835 ------------------------------------------------------------------------------ s = 0.02 rate divided by s = 0.10 rate: 0.5044 Difference -1.7563e-02 +/- 6.1532e-04, which is 28.5 standard errors from zero. Fitting rate proportional to s^p over the three cells gives p = 0.425. n0 on its own does not determine the click rate. Raising s at fixed n0 and fixed lambda makes the ratchet turn FASTER, which is the direction Stephan, Chao and Smale (1993) predicted for intermediate N and s. ============================================================================== PART 1d. n0 SWEEP at fixed s = 0.05 and U = 0.20 (lambda = 4), with the population size tuned so that n0 walks from below one to thirty. ============================================================================== N n0 pred n0 seen rate SE exp(-n0) meas/pred clicks ------------------------------------------------------------------------------ 27 0.4945 5.189 0.08651 8.73e-04 0.60986 0.142 6921 55 1.007 7.036 0.06155 6.31e-04 0.36518 0.169 4924 82 1.502 8.275 0.05133 6.35e-04 0.22271 0.23 4106 109 1.996 9.319 0.04404 6.10e-04 0.13582 0.324 3523 164 3.004 11.07 0.03687 5.35e-04 0.04960 0.743 2950 218 3.993 12.62 0.03099 4.73e-04 0.01845 1.68 2479 328 6.008 15.13 0.02389 4.15e-04 0.00246 9.71 2389 437 8.004 17.67 0.02050 3.44e-04 0.00033 61.4 2050 546 10 19.67 0.01770 2.94e-04 4.54e-05 390 2124 655 12 22.2 0.01562 2.55e-04 6.16e-06 2.53e+03 2187 874 16.01 25.56 0.01292 2.42e-04 1.12e-07 1.2e+05 2068 1092 20 29.04 0.01051 1.64e-04 2.06e-09 5.1e+06 2103 1365 25 33.38 0.00835 1.50e-04 1.39e-11 6.0e+08 2172 1638 30 37.61 0.00692 1.10e-04 9.35e-14 7.4e+10 2075 ------------------------------------------------------------------------------ Two fits over the 13 sweep cells with n0 >= 1 and more than 100 clicks: exponential ln(rate) = -3.1031 -0.0707 n0 rms residual 0.1957 power law ln(rate) = -2.6635 -0.6293 ln(n0) rms residual 0.0819 the classical exp(-n0) estimate would be a = 0, b = -1. Measured exponential decay constant: 0.0707, i.e. 14.1 times weaker than the classical one. Over this range of n0 the power law is the better description (smaller residual), and the ratchet in this window therefore slows roughly as n0 to the power -0.63, not exponentially. ============================================================================== PART 1e. A PREDICTION, WRITTEN DOWN BEFORE THE CELLS WERE RUN, AND THEN TESTED. Stephan, Chao and Smale (1993) said n0 is not enough and that s matters separately. Jain (2008) put the interclick time in a scaling form involving the least-loaded class size and the selection coefficient together. Neher and Shraiman (2012) name the controlling combination as N s x0, which is s times n0. If they are right, then the click rate DIVIDED BY s should be a function of s times n0 alone: populations matched on s*n0 should agree even when s and n0 each differ five-fold, while populations matched on n0 alone (Part 1c) should not. A designed grid, all at lambda = U/s = 4. Five values of s*n0 crossed with three values of s, plus the rows needed to match on n0 instead. Each cell is 200 fresh replicate populations. ============================================================================== Designed cells: 18 N U s n0 s x n0 rate SE rate / s clicks ------------------------------------------------------------------------------ 109 0.080 0.02 1.996 0.03993 0.02744 5.10e-04 1.3719 2195 273 0.080 0.02 5 0.1 0.01780 3.22e-04 0.8900 2136 682 0.080 0.02 12.49 0.2498 0.01068 1.77e-04 0.5338 2135 1092 0.080 0.02 20 0.4 0.00813 1.39e-04 0.4067 2115 1365 0.080 0.02 25 0.5 0.00709 1.20e-04 0.3545 2127 2730 0.080 0.02 50 1 0.00416 7.57e-05 0.2082 2082 5460 0.080 0.02 100 2 0.00198 4.87e-05 0.0990 1188 109 0.200 0.05 1.996 0.09982 0.04420 6.24e-04 0.8840 3536 273 0.200 0.05 5 0.25 0.02701 4.30e-04 0.5403 2161 546 0.200 0.05 10 0.5 0.01727 3.01e-04 0.3453 2072 1092 0.200 0.05 20 1 0.01039 1.81e-04 0.2077 2077 2184 0.200 0.05 40 2 0.00531 9.31e-05 0.1062 2017 55 0.400 0.10 1.007 0.1007 0.09038 8.48e-04 0.9038 7230 109 0.400 0.10 1.996 0.1996 0.06250 6.65e-04 0.6250 5000 136 0.400 0.10 2.491 0.2491 0.05601 5.81e-04 0.5601 4481 273 0.400 0.10 5 0.5 0.03584 4.63e-04 0.3584 2867 546 0.400 0.10 10 1 0.02083 3.62e-04 0.2083 2083 1092 0.400 0.10 20 2 0.01047 1.89e-04 0.1046 2093 ------------------------------------------------------------------------------ TEST 1. Matched on s x n0. Does rate/s agree? s x n0 rate/s at s = 0.02, 0.05, 0.10 spread as % of mean as SE ------------------------------------------------------------------------------ 0.10 0.8900 0.8840 0.9038 0.0198 2.2 1.3 0.25 0.5338 0.5403 0.5601 0.0264 4.8 2.5 0.50 0.3545 0.3453 0.3584 0.0130 3.7 1.7 1.00 0.2082 0.2077 0.2083 0.0006 0.3 0.1 2.00 0.0990 0.1062 0.1046 0.0072 6.9 2.3 ------------------------------------------------------------------------------ Largest disagreement anywhere in TEST 1: 6.9% TEST 2. Matched on n0 alone. Does the rate agree? n0 rate at s = 0.02, 0.05, 0.10 spread as % of mean as SE ------------------------------------------------------------------------------ 2.00 0.02744 0.04420 0.06250 0.03506 78.4 41.8 5.00 0.01780 0.02701 0.03584 0.01804 67.1 32.0 20.00 0.00813 0.01039 0.01047 0.00233 24.1 9.9 ------------------------------------------------------------------------------ Largest disagreement anywhere in TEST 2: 78.4% SUMMARY OF THE TWO TESTS matched on s x n0 : disagreements of 0.3% to 6.9%, median 3.7% matched on n0 : disagreements of 24.1% to 78.4%, median 67.1% the collapse variable is tighter by a factor of 18.1 on the medians. The prediction holds. s times n0 is the number that governs this ratchet and n0 by itself is not, which is why no constant in front of exp(-n0) could ever have rescued the classical estimate. EVERY SUFFICIENTLY CLICKING CELL IN THE STUDY, IN COLLAPSE COORDINATES, sorted by s x n0. The lambda column is what the collapse does NOT absorb. # N s lambda n0 s x n0 rate/s ------------------------------------------------------------------------------ 1 100 0.02 5.0 0.6738 0.01348 1.8456 2 1000 0.05 8.0 0.3355 0.01677 1.2132 3 27 0.05 4.0 0.4945 0.02473 1.7303 4 200 0.02 5.0 1.348 0.02695 1.4019 5 2000 0.05 8.0 0.6709 0.03355 0.9740 6 1000 0.10 8.0 0.3355 0.03355 0.9904 7 109 0.02 4.0 1.996 0.03993 1.3719 8 55 0.05 4.0 1.007 0.05037 1.2310 9 2000 0.10 8.0 0.6709 0.06709 0.7670 10 500 0.02 5.0 3.369 0.06738 0.9533 11 82 0.05 4.0 1.502 0.07509 1.0265 12 100 0.05 4.0 1.832 0.09158 0.9530 13 109 0.05 4.0 1.996 0.09982 0.8807 14 109 0.05 4.0 1.996 0.09982 0.8840 15 273 0.02 4.0 5 0.1 0.8938 16 273 0.02 4.0 5 0.1 0.8900 17 55 0.10 4.0 1.007 0.1007 0.9038 18 1000 0.02 5.0 6.738 0.1348 0.6925 19 164 0.05 4.0 3.004 0.1502 0.7375 20 200 0.05 4.0 3.663 0.1832 0.6278 21 100 0.10 4.0 1.832 0.1832 0.6613 22 218 0.05 4.0 3.993 0.1996 0.6198 23 109 0.10 4.0 1.996 0.1996 0.6250 24 136 0.10 4.0 2.491 0.2491 0.5601 25 682 0.02 4.0 12.49 0.2498 0.5338 26 273 0.05 4.0 5 0.25 0.5312 27 273 0.05 4.0 5 0.25 0.5403 28 2000 0.02 5.0 13.48 0.2695 0.4757 29 328 0.05 4.0 6.008 0.3004 0.4778 30 200 0.10 4.0 3.663 0.3663 0.4365 31 1092 0.02 4.0 20 0.4 0.4067 32 437 0.05 4.0 8.004 0.4002 0.4100 33 500 0.05 4.0 9.158 0.4579 0.3818 34 546 0.05 4.0 10 0.5 0.3540 35 273 0.10 4.0 5 0.5 0.3544 36 1365 0.02 4.0 25 0.5 0.3545 37 546 0.05 4.0 10 0.5 0.3453 38 273 0.10 4.0 5 0.5 0.3584 39 655 0.05 4.0 12 0.5998 0.3124 40 100 0.05 2.0 13.53 0.6767 0.2876 41 874 0.05 4.0 16.01 0.8004 0.2585 42 1000 0.05 4.0 18.32 0.9158 0.2270 43 500 0.10 4.0 9.158 0.9158 0.2309 44 1092 0.05 4.0 20 1 0.2103 45 2730 0.02 4.0 50 1 0.2082 46 1092 0.05 4.0 20 1 0.2077 47 546 0.10 4.0 10 1 0.2083 48 1365 0.05 4.0 25 1.25 0.1671 49 200 0.05 2.0 27.07 1.353 0.1444 50 100 0.10 2.0 13.53 1.353 0.1501 51 1638 0.05 4.0 30 1.5 0.1383 52 2000 0.05 4.0 36.63 1.832 0.1177 53 1000 0.10 4.0 18.32 1.832 0.1199 54 5460 0.02 4.0 100 2 0.0990 55 2184 0.05 4.0 40 2 0.1062 56 1092 0.10 4.0 20 2 0.1046 57 200 0.10 2.0 27.07 2.707 0.0549 58 500 0.05 2.0 67.67 3.383 0.0352 59 2000 0.10 4.0 36.63 3.663 0.0451 60 100 0.10 1.0 36.79 3.679 0.0150 61 500 0.10 2.0 67.67 6.767 0.0039 ------------------------------------------------------------------------------ 61 cells in the table, spanning 202-fold in N, 5-fold in s and 8-fold in lambda. Restricting to the 45 cells at lambda = 4 and fitting one smooth curve in log-log (a quadratic, three free constants): ln(rate/s) against ln(s n0) : rms residual 0.0527 (5.3%) ln(rate) against ln(n0) : rms residual 0.1880 (18.8%) collapse coordinates are tighter by a factor of 3.6. fitted curve: ln(rate/s) = -1.6150 -0.8639 L -0.0854 L^2, with L = ln(s n0) Across the whole table lambda still matters. At a given s x n0 the cells with lambda = 8 click more slowly than the cells with lambda = 1, by up to a factor of a few at the edges of the grid. The collapse is good, not exact, and we are not claiming a law. ============================================================================== PART 2. CONVERGENCE. The running estimate as replicate populations accumulate, for three cells spanning the grid. ============================================================================== N = 100 U = 0.20 s = 0.05 n0 pred = 1.832 generations = 400 clicks = 3812 replicates running rate running SE 1 0.06000 0 2 0.05125 0.00875 5 0.04400 0.00562 10 0.04975 0.00407 25 0.04730 0.00228 50 0.04770 0.00140 100 0.04727 9.38e-04 150 0.04795 7.38e-04 200 0.04765 6.28e-04 final: 0.04765 +/- 6.28e-04 (relative SE 1.32%) N = 500 U = 0.20 s = 0.05 n0 pred = 9.158 generations = 600 clicks = 2291 replicates running rate running SE 1 0.01000 0 2 0.01583 0.00583 5 0.01800 0.00220 10 0.01950 0.00129 25 0.01953 6.70e-04 50 0.01923 5.45e-04 100 0.01912 3.80e-04 150 0.01926 3.42e-04 200 0.01909 3.00e-04 final: 0.01909 +/- 3.00e-04 (relative SE 1.57%) N = 1000 U = 0.20 s = 0.05 n0 pred = 18.32 generations = 900 clicks = 2043 replicates running rate running SE 1 0.00778 0 2 0.01167 0.00389 5 0.01089 0.00147 10 0.01133 0.00100 25 0.01080 4.69e-04 50 0.01102 3.90e-04 100 0.01136 2.70e-04 150 0.01135 2.15e-04 200 0.01135 1.89e-04 final: 0.01135 +/- 1.89e-04 (relative SE 1.67%) CONVERGENCE DATA FOR THE FIGURE (replicates:running rate:running SE) cell N=100 U=0.20 s=0.05 1:0.06:0 2:0.05125:0.00875 3:0.05:0.00520416 4:0.04875:0.00388641 5:0.044:0.00562361 6:0.04625:0.0051133 7:0.0478571:0.00461069 8:0.0478125:0.00399323 9:0.0505556:0.00446393 10:0.04975:0.00407312 15:0.0493333:0.00347154 20:0.047875:0.00274626 25:0.0473:0.00228181 30:0.0471667:0.00207867 35:0.0473571:0.00181017 40:0.047125:0.00159514 45:0.0478889:0.00152205 50:0.0477:0.00139759 55:0.0472273:0.00129697 60:0.0471667:0.00128203 65:0.0473846:0.00119649 70:0.0466429:0.00116105 75:0.0465667:0.00109952 80:0.0468125:0.00106082 85:0.0468235:0.00103034 90:0.0471111:0.00100242 95:0.0470789:0.000966729 100:0.047275:0.000938174 105:0.047619:0.000911855 110:0.0475455:0.000872371 115:0.047587:0.000836529 120:0.0473333:0.000810111 125:0.04756:0.000796901 130:0.0474231:0.000773623 135:0.047537:0.000755935 140:0.0477679:0.000751709 145:0.0479138:0.000746966 150:0.04795:0.000737714 155:0.0479194:0.000723223 160:0.0480156:0.000715224 165:0.0481364:0.000701616 170:0.0480294:0.000685864 175:0.0477714:0.000681526 180:0.0478472:0.000665841 185:0.0477703:0.000662691 190:0.0477632:0.000649247 195:0.0477949:0.000639362 200:0.04765:0.00062798 cell N=500 U=0.20 s=0.05 1:0.01:0 2:0.0158333:0.00583333 3:0.0172222:0.00364302 4:0.0170833:0.00257975 5:0.018:0.00219848 6:0.0180556:0.00179591 7:0.0185714:0.0016031 8:0.0189583:0.00144123 9:0.0188889:0.00127294 10:0.0195:0.00129219 15:0.0198889:0.000882916 20:0.0198333:0.000714307 25:0.0195333:0.000670268 30:0.0199444:0.000600571 35:0.02:0.000546613 40:0.0195833:0.000519389 45:0.0194074:0.000537994 50:0.0192333:0.000545462 55:0.0194848:0.000515152 60:0.0191111:0.000506497 65:0.0191795:0.000494525 70:0.0193571:0.000467844 75:0.0191556:0.000451195 80:0.0191042:0.000444674 85:0.0190196:0.000426056 90:0.0190926:0.00040728 95:0.019193:0.000391892 100:0.0191167:0.000380099 105:0.0191746:0.000374125 110:0.0190606:0.000380538 115:0.019029:0.00037281 120:0.0190972:0.000361495 125:0.01916:0.000358023 130:0.0191923:0.00035724 135:0.0190864:0.000349488 140:0.019131:0.000341902 145:0.0192414:0.000339808 150:0.0192556:0.00034236 155:0.0192151:0.000342599 160:0.0191667:0.000334053 165:0.0191616:0.000325362 170:0.0190882:0.000329676 175:0.0191048:0.000322979 180:0.0191481:0.000319085 185:0.0191441:0.000312173 190:0.0191579:0.000305368 195:0.0191368:0.000302558 200:0.0190917:0.000300009 cell N=1000 U=0.20 s=0.05 1:0.00777778:0 2:0.0116667:0.00388889 3:0.0114815:0.00225288 4:0.0116667:0.00160375 5:0.0108889:0.00146566 6:0.0107407:0.00120584 7:0.0111111:0.00108433 8:0.0118056:0.00116794 9:0.011358:0.00112305 10:0.0113333:0.00100479 15:0.0108148:0.000714893 20:0.0106667:0.000543136 25:0.0108:0.000469305 30:0.0111481:0.00046894 35:0.0109524:0.000425221 40:0.0108611:0.000416287 45:0.0110123:0.000394535 50:0.0110222:0.000389895 55:0.0110101:0.000358128 60:0.0111111:0.000333019 65:0.0111795:0.00031472 70:0.0112222:0.000300933 75:0.0112:0.000294358 80:0.0112778:0.000285135 85:0.0112549:0.000280665 90:0.011284:0.00027364 95:0.0112749:0.000268106 100:0.0113556:0.000269671 105:0.0112169:0.000266254 110:0.0111818:0.000255613 115:0.0111884:0.00025183 120:0.0111944:0.000244652 125:0.0112978:0.000244977 130:0.0112821:0.00023668 135:0.0112675:0.000230472 140:0.011254:0.000226084 145:0.0113027:0.000221087 150:0.0113481:0.000214787 155:0.0112975:0.000211592 160:0.0113125:0.000205445 165:0.0113468:0.000201458 170:0.0113464:0.000197135 175:0.0112952:0.000195034 180:0.0113025:0.000193152 185:0.0112673:0.000194619 190:0.0113041:0.000193338 195:0.0113675:0.000190722 200:0.01135:0.000189015 ============================================================================== PART 3. MEAN FITNESS DECLINE, and the accounting identity d(ln Wbar)/dt = (click rate) x ln(1 - s). ============================================================================== regime clicks/gen d lnW/dt meas rate x ln(1-s) ratio ------------------------------------------------------------------------------ N=500 U=0.40 s=0.05 0.07765 -3.9621e-03 -3.9828e-03 0.9948 N=500 U=0.20 s=0.05 0.01842 -9.4622e-04 -9.4499e-04 1.0013 N=100 U=0.20 s=0.05 0.04671 -2.3878e-03 -2.3959e-03 0.9966 N=1000 U=0.40 s=0.05 0.06103 -3.1151e-03 -3.1306e-03 0.9950 N=100 U=0.10 s=0.02 0.03770 -7.5558e-04 -7.6164e-04 0.9920 N=2000 U=0.80 s=0.10 0.07677 -8.0638e-03 -8.0885e-03 0.9969 ------------------------------------------------------------------------------ Worst regime here, N=2000 U=0.80 s=0.10, loses a factor of 5.583e-06 in mean fitness over 1500 generations, and its mean load rises from 29.2 to 144.3. FITNESS TRAJECTORY DATA FOR THE FIGURE (generation:ln Wbar:mean kmin) N=500 U=0.40 s=0.05 1:-1.80344:30.030 49:-1.99327:33.880 97:-2.19918:37.790 145:-2.38882:41.395 193:-2.58028:45.250 241:-2.77566:49.020 289:-2.96686:52.775 337:-3.15529:56.410 385:-3.34641:60.135 433:-3.54046:63.925 481:-3.72927:67.600 529:-3.91975:71.340 577:-4.10998:75.005 625:-4.30268:78.845 673:-4.49616:82.545 721:-4.68311:86.200 769:-4.87391:89.935 817:-5.06112:93.470 865:-5.25647:97.290 913:-5.44028:100.965 961:-5.62602:104.605 1009:-5.81390:108.265 1057:-6.00998:112.095 1105:-6.20172:115.865 1153:-6.39864:119.670 1201:-6.57969:123.270 1249:-6.76545:126.790 1297:-6.95335:130.445 1345:-7.15601:134.510 1393:-7.34886:138.125 1441:-7.53802:141.960 1489:-7.72459:145.545 N=500 U=0.20 s=0.05 1:-0.52042:6.695 49:-0.56299:7.535 97:-0.60671:8.395 145:-0.65117:9.290 193:-0.69559:10.150 241:-0.74063:11.035 289:-0.78734:11.955 337:-0.83327:12.885 385:-0.88026:13.795 433:-0.92651:14.615 481:-0.97163:15.560 529:-1.01755:16.470 577:-1.06560:17.345 625:-1.10982:18.245 673:-1.15805:19.175 721:-1.20276:20.005 769:-1.24680:20.875 817:-1.29403:21.810 865:-1.33654:22.665 913:-1.38448:23.560 961:-1.43100:24.505 1009:-1.47401:25.315 1057:-1.51987:26.170 1105:-1.56552:27.140 1153:-1.61075:27.995 1201:-1.65650:28.860 1249:-1.69966:29.690 1297:-1.75123:30.720 1345:-1.79566:31.595 1393:-1.83619:32.390 1441:-1.88345:33.295 1489:-1.92691:34.105 N=100 U=0.20 s=0.05 1:-1.07673:18.260 49:-1.19522:20.605 97:-1.31492:22.950 145:-1.44123:25.345 193:-1.55641:27.610 241:-1.67243:29.825 289:-1.78061:31.960 337:-1.88915:34.130 385:-2.00160:36.335 433:-2.11017:38.455 481:-2.22543:40.710 529:-2.34174:42.920 577:-2.45706:45.295 625:-2.57633:47.555 673:-2.69054:49.815 721:-2.80914:52.095 769:-2.93069:54.365 817:-3.04483:56.640 865:-3.15699:58.935 913:-3.27656:61.170 961:-3.38573:63.300 1009:-3.49715:65.465 1057:-3.61355:67.695 1105:-3.72614:69.945 1153:-3.84300:72.185 1201:-3.95826:74.410 1249:-4.06746:76.550 1297:-4.18130:78.840 1345:-4.29547:81.000 1393:-4.41491:83.325 1441:-4.51998:85.475 1489:-4.63951:87.750 ============================================================================== PART 4. RECOMBINATION RESCUE, on the explicit-genome engine. R is the fraction of offspring made by recombining two parents: R = 0 is the asexual control, R = 1 an obligate sexual. Base cell N = 200, U = 0.30, s = 0.05 (lambda = 6.0, n0 pred = 0.4958), 1024 sites per genome, 200 replicates, 800 generations after a 150 generation burn-in. ============================================================================== R up/gen SE net kmin/gen SE vs R=0 load end down ------------------------------------------------------------------------------ 0.000 0.06158 4.59e-04 0.06158 4.59e-04 1.00000 62.01 0 0.005 0.07412 6.86e-04 0.05478 4.28e-04 0.88958 55.87 3094 0.010 0.08336 8.39e-04 0.04930 4.27e-04 0.80057 51.24 5450 0.020 0.09920 9.68e-04 0.03931 3.32e-04 0.63828 42.15 9583 0.050 0.11985 0.00112 0.01839 3.42e-04 0.29859 23.68 16234 0.100 0.13514 0.00102 0.00431 1.98e-04 0.07003 10.87 20933 0.150 0.14171 9.63e-04 6.56e-04 1.28e-04 0.01066 7.47 22569 0.200 0.14811 9.92e-04 1.62e-04 1.08e-04 0.00264 6.87 23671 0.300 0.16416 9.26e-04 -1.69e-04 9.77e-05 -0.00274 6.40 26292 1.000 0.26876 9.72e-04 1.06e-04 8.21e-05 0.00173 6.29 42985 ------------------------------------------------------------------------------ Mutations landing on an already-mutated site (the finite-sites leak): R=0.000 3.23%, R=0.005 2.91%, R=0.010 2.71%, R=0.020 2.27%, R=0.050 1.42%, R=0.100 0.82%, R=0.150 0.66%, R=0.200 0.62%, R=0.300 0.59%, R=1.000 0.58% Asexual control, net kmin drift : 0.06158 per generation R at which the ratchet runs at 50% of the asexual control : R = 0.0268, one offspring in 37 R at which the ratchet runs at 10% of the asexual control : R = 0.0843, one offspring in 12 R at which the ratchet runs at 1% of the asexual control : R = 0.1520, one offspring in 7 First grid R below 1% of the control : R = 0.200 First grid R whose net drift is within two standard errors of zero : R = 0.200 Net drift at R = 1 (obligate sex) : 1.06e-04 +/- 8.21e-05, which is 1.3 standard errors from zero, so the ratchet is stopped, not merely slowed. RECOMBINATION FITNESS TRAJECTORIES FOR THE FIGURE (generation:ln Wbar:mean load) R=0.000 0:-0.65343:12.857 24:-0.73603:14.466 48:-0.81369:15.980 72:-0.89217:17.507 96:-0.97207:19.066 120:-1.05973:20.774 144:-1.13923:22.326 168:-1.21570:23.814 192:-1.29071:25.281 216:-1.37155:26.854 240:-1.45376:28.457 264:-1.53398:30.022 288:-1.61121:31.527 312:-1.68938:33.050 336:-1.76767:34.576 360:-1.83696:35.928 384:-1.91232:37.396 408:-1.98441:38.802 432:-2.05988:40.278 456:-2.13307:41.701 480:-2.21262:43.252 504:-2.28594:44.681 528:-2.35544:46.039 552:-2.42695:47.427 576:-2.49807:48.814 600:-2.57275:50.272 624:-2.64728:51.720 648:-2.72668:53.274 672:-2.79745:54.655 696:-2.87104:56.091 720:-2.94329:57.495 744:-3.01542:58.899 768:-3.08537:60.267 792:-3.15035:61.533 R=0.005 0:-0.61716:12.154 24:-0.69245:13.618 48:-0.76503:15.037 72:-0.83382:16.374 96:-0.90253:17.718 120:-0.96518:18.940 144:-1.03016:20.206 168:-1.09402:21.449 192:-1.16489:22.830 216:-1.23544:24.203 240:-1.30447:25.552 264:-1.37505:26.927 288:-1.44953:28.379 312:-1.51616:29.677 336:-1.58068:30.933 360:-1.65270:32.338 384:-1.71560:33.568 408:-1.78353:34.887 432:-1.85347:36.253 456:-1.92764:37.702 480:-1.98930:38.899 504:-2.06182:40.317 528:-2.12193:41.487 552:-2.18946:42.806 576:-2.25174:44.019 600:-2.31949:45.337 624:-2.38276:46.571 648:-2.44971:47.878 672:-2.51377:49.128 696:-2.58413:50.494 720:-2.64612:51.705 744:-2.70208:52.800 768:-2.76830:54.087 792:-2.83561:55.394 R=0.020 0:-0.54234:10.706 24:-0.58622:11.554 48:-0.64032:12.613 72:-0.68517:13.487 96:-0.72937:14.351 120:-0.78012:15.337 144:-0.83261:16.360 168:-0.88169:17.322 192:-0.93191:18.296 216:-0.98439:19.319 240:-1.03240:20.259 264:-1.07754:21.137 288:-1.13215:22.196 312:-1.18128:23.157 336:-1.23171:24.140 360:-1.28026:25.085 384:-1.32945:26.047 408:-1.37714:26.976 432:-1.42887:27.984 456:-1.47716:28.927 480:-1.52316:29.824 504:-1.56227:30.586 528:-1.61520:31.612 552:-1.65768:32.444 576:-1.70389:33.346 600:-1.75918:34.425 624:-1.80130:35.243 648:-1.85073:36.203 672:-1.91021:37.368 696:-1.95415:38.223 720:-2.00229:39.160 744:-2.05244:40.137 768:-2.09841:41.040 792:-2.13908:41.827 R=0.100 0:-0.37284:7.416 24:-0.38344:7.622 48:-0.39318:7.814 72:-0.39633:7.880 96:-0.39894:7.920 120:-0.41008:8.143 144:-0.41494:8.240 168:-0.41713:8.280 192:-0.42306:8.398 216:-0.43334:8.599 240:-0.43277:8.588 264:-0.44040:8.736 288:-0.44408:8.803 312:-0.45154:8.950 336:-0.45421:9.005 360:-0.45939:9.101 384:-0.46788:9.271 408:-0.47402:9.392 432:-0.47828:9.474 456:-0.48160:9.533 480:-0.47878:9.482 504:-0.48761:9.654 528:-0.49361:9.772 552:-0.49523:9.801 576:-0.50579:10.011 600:-0.50809:10.052 624:-0.50590:10.010 648:-0.51361:10.164 672:-0.51581:10.203 696:-0.52177:10.321 720:-0.52261:10.339 744:-0.53248:10.525 768:-0.53505:10.574 792:-0.54892:10.851 R=1.000 0:-0.31802:6.350 24:-0.31762:6.343 48:-0.31521:6.293 72:-0.31791:6.347 96:-0.32132:6.415 120:-0.31907:6.370 144:-0.32073:6.405 168:-0.32208:6.433 192:-0.31997:6.388 216:-0.31556:6.300 240:-0.31698:6.332 264:-0.31535:6.297 288:-0.32013:6.392 312:-0.32259:6.441 336:-0.31866:6.362 360:-0.32092:6.407 384:-0.32053:6.401 408:-0.32273:6.442 432:-0.31657:6.319 456:-0.32010:6.393 480:-0.31983:6.387 504:-0.31901:6.368 528:-0.32059:6.401 552:-0.31700:6.331 576:-0.31826:6.357 600:-0.31615:6.313 624:-0.31702:6.329 648:-0.31595:6.309 672:-0.31343:6.257 696:-0.31661:6.323 720:-0.31369:6.264 744:-0.31904:6.370 768:-0.31979:6.386 792:-0.31710:6.333 ============================================================================== PART 5. CROSS-CHECKS. Three independent implementations of the same asexual model, asked for the same number. ============================================================================== cell load-class individual packed genomes ------------------------------------------------------------------------------ N=100 U=0.20 s=0.05 0.04757 +/- 6.14e-04 0.04540 +/- 0.00133 0.04422 +/- 8.50e-04 N=200 U=0.30 s=0.05 0.06480 +/- 5.85e-04 0.06753 +/- 0.00125 0.06289 +/- 8.87e-04 ------------------------------------------------------------------------------ N = 100 load-class minus individual-by-individual : +0.00217 (1.5 standard errors) load-class minus packed-genome : +0.00335 (3.2 standard errors) the packed-genome run lost 1.84% of its mutations to sites already carrying one, so its realised mutation rate is U = 0.1963, not 0.20. load-class at that realised U : 0.04610 +/- 5.25e-04 difference after the correction : +0.00188 (1.9 standard errors) N = 200 load-class minus individual-by-individual : -0.00273 (2.0 standard errors) load-class minus packed-genome : +0.00191 (1.8 standard errors) the packed-genome run lost 2.57% of its mutations to sites already carrying one, so its realised mutation rate is U = 0.2923, not 0.30. load-class at that realised U : 0.06276 +/- 6.57e-04 difference after the correction : -0.00013 (0.1 standard errors) The packed-genome engine clicks a little more slowly than the other two, and the finite-sites leak accounts for it. Correct the mutation rate for the leak and the three implementations agree. ============================================================================== PART 6. SUMMARY OF THE VALIDATION. Club value beside the accepted or analytic value, with the difference. ============================================================================== quantity club accepted difference ------------------------------------------------------------------------------ clicks with U = 0, 6 cells x 200 reps 0 0 0 mean fitness with U = 0 1.000000000 1.000000000 0 mean load / lambda, 14 quiet cells 1.0288 1.0000 +0.0288 variance of load / lambda, same cells 1.0249 1.0000 +0.0249 class zero / N exp(-U/s), same cells 0.9533 1.0000 -0.0467 rate / exp(-n0), median over 70 cells 0.3978 1.0 x 0.3978 worst single cell 1.40e+55 1.0 x 1.40e+55 d lnW/dt over rate x ln(1-s), 6 regimes 0.9961 1.0000 -0.0039 sd over those regimes 0.0031 - - click rate N=100, load-class vs packed genome 0.04757 0.04422 +0.00335 click rate N=200, load-class vs packed genome 0.06480 0.06289 +0.00191 net kmin drift at R = 1 (obligate sex) 1.06e-04 0 +1.06e-04 ------------------------------------------------------------------------------ THE DISAGREEMENT THAT SURVIVES The classical one-generation estimate exp(-n0) is not a usable prediction of the click rate, and no amount of debugging made it one. It fails in both directions, so here are the worst cell on each side. CLASSICAL TOO FAST, worst cell: N = 1000, U = 0.20, s = 0.02, n0 pred = 0.0454 club value : 0.04651 +/- 4.85e-04 clicks per generation classical : 0.95562 difference : -0.909103, standard error 0.0004853 the club value sits 1873 standard errors below the classical one, a ratio of 0.04867. CLASSICAL TOO SLOW, worst cell: N = 1000, U = 0.10, s = 0.05, n0 pred = 135.3 club value : 2.34e-04 +/- 2.18e-05 clicks per generation classical : 1.68e-59 difference : +0.000234, standard error 2.183e-05 the club value sits 11 standard errors above the classical one, a ratio of 1.40e+55. The explanation is not a coding error, and PART 5 is the evidence: three separately written implementations give the same click rate. The classical argument assumes the least-loaded class sits at its deterministic size and then has to be wiped out in one sampling draw. In the runs it wanders for many generations first, and it is those excursions, not one unlucky draw, that end it. What survives intact is Haigh's equilibrium itself, wherever the ratchet is quiet enough for an equilibrium to be defined, and the fitness accounting identity, which holds in every regime we tested. ============================================================================== Wall clock at the start of each part, seconds: PART 1. 2, PART 1a. 48, PART 1b. 48, PART 1c. 48, PART 1d. 49, PART 1e. 56, PART 2. 68, PART 3. 68, PART 4. 74, PART 5. 165, PART 6. 170, end 170 Total wall clock: 169.7 s Every number above came from this file, seed 20250921, in one run. ==============================================================================