FROM MENDELEEV TO MENDEL VIA DR. SEUSS

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2 FROM MENDELEEV TO MENDEL VIA DR. SEUSS M. J. Orlove A periodic table of the Digits allows us to fit a curve to a single sample point if a "magic x-value is chosen, Y is quoted to enough places, and the coefficients are positive or negative multiples of one half. There is a kind of function called an "Exponential Function". It can describe many phenomena in nature. Three examples of such can be presented as sequences that represent the y-values when x = 1, 2, 3, 4, 5, etc. : The first could describe the number of bacteria in a bowl of soup at hourly intervals when each bacterium becomes two bacteria at one hour of age. The second represents the temperature of a cup of water at regular intervals when room temperature is 1/3. The third is just to show the generality and scope of the formula, and all of them could represent distances from the hive entrance of a canary waterer filled with sugar water during successive training sessions. If the human trainer places the feeder at distances corresponding to the first 3 numbers during the first 3 sessions, then the bees will be waiting for it at the location corresponding to the 4th number, provided all 4 locations are in the same direction from the hive (James Gould, 1976). Gould didn't necessarily use these examples, but used other examples of the Exponential Function. If we call the first number in the sequence, P1, the second, P2, the third, P3, and the infinite-th one, P8 (because an infinity sign looks like an "8" on its side), then the Ith number in the sequence, which we will call PI, is given by the formula: PI = P8 - (P8 - P1)R^(i -1), where R is the deviation of PI this generation from P8 divided by the similar deviation last generation. R = (P3 - P2)/(P2 - P1), and, P8 = (P2^2 - P3P1)/(2P2 -P3 -P1) Consider a thought experiment with a population of beetles on an island. One locus controls their color (Either purple or green). Another locus their behavior (either altruistic or selfish).

3 The beetles are either blind, color blind, or nocturnal, with the consequence that each other's coloration has no effect on choices made by beetles, and the color differences are posited to assist human investigators in tracking events only. Some thinkers have questioned the category of thought experiments upon which we are about to embark - notably Nowak, Tarnita, and Wilson, This should be no deterrent concerning the current exercise. Appropriate discussion of Nowak et al's onslaught is given in Appendix C. Altruistic beetles give up rearing some or all of their own offspring and rear their nieces and nephews instead. According to Hamilton's (1964) inequality (It's not an equation), an organism is predicted to be the product of natural selection, and as such willing to sacrifice its life to save more than 8 first cousins, but not less than 8 of them. For exactly 8 it would be predicted to be ambivalent or indifferent about the deed. If memes (ideas, beliefs) as well as genes were an important part of the organism's programming, then 2 distantly related, but ideologically similar, individuals might treat each other like close relatives, however we are talking beetles here! By this same kind of Hamiltonian reasoning, a beetle would be indifferent, or ambivalent, about the choice between raising one of its offspring and two of its nieces or nephews. If a heterozygote at the altruism-selfishness locus spends half of its time and other resources in offspring production and half in niece and nephew production, and the altruism homozygote spends it all on niece and nephew production, and the selfishness homozygote spends it all on offspring production, and the cost of raising 2 nieces and/or nephews is raising one less own offspring, then the altruism allele will stay at a constant frequency. If the island was colonized by pure purple altruists on its west shore and pure green selfish ones on its east, and further the initial colonization consisted of a small number of individuals, then the initial total correlation of color and behavior can be easily seen as a likely, but still coincidental, event A glacier separates the Easterners from the Westerners until both become so numerous that we can simulate the goings on with an infinite population, i.e., the larger the population, the closer the parameters will be to the ideal values found in an infinite population. Hamilton, himself, used such infinite populations. This makes the model a deterministic one rather than a stochastic one. Any of these unlikely constraints can be relaxed somewhat later without damage to the theory, but for training ourselves they are helpful. Say, the recombination rate between the behavior locus and the color locus is 0.1. and we number the years as "years since the glacier melted, and the beetles started interbreeding, and there's one generation a year. -One last constraint is that the altruism doesn't start until 2 generations after the melt to allow the 2 populations to mix.

4 The reason for giving 2 generations to allow the population to become mixed is that it takes that long for relatedness coefficients between siblings to go from 1 to 1/2 and any further waiting before doing any experiments doesn't change the distribution of different kinds of pairs of interacting individuals. Thus we turn on crossing over and selection the third generation after the glacier melts. When selection is turned off, every beetle has the same number of kids and rears them. Once selection is turned on, they rear kids for themselves and/or sibs as already delineated based on the genes of the would-be helper. -As we run the simulation the altruism allele breaks even and stays at 0.6. Of course the selfishness allele stays at 0.4. Purple increases and green decreases. The frequency of the purple allele increases according to the exponential function. The a/s locus does the work but doesn't budge, but purple rises (in frequency) and green descends. It is reminiscent of an elevator's motor not rising but sending the car up and the counterweight down. The phenomenon is called "The Elevator Effect" (Orlove, 1981). The frequency of purple goes to the sequence: The rate of approach is 0.9, and the asymptote is: In other words P1 = 0.6, P2 = and P3 = When we run our exponential function program, we find out P8 = and R = 0.9. If we plug in P2, P3 and P4, in place of P1, P2, and P3, we get the same values for R and P8, thus demonstrating that the sequence of numbers generated by The Elevator effect Thought Experiment really is an exponential function. I chose 0.6 as the initial frequency of purple altruists and switched on selection and crossing over simultaneously in order to get numbers that would be ratios of small whole numbers and, except for zeroes, not repeating decimals, and ran it on several brands of computers under 40 decimal places in order to convince any doubting Thomases among you we are not seeing an artifact of rounding error here. Hamilton's inequality is stated as: Altruism is selected for if and only if: K > 1/r. The Elevator Effect is can be stated as:

5 A marker, coupled with altruism, will be selected of, if and only if,: K > (1 - p) ^ n /r, where K = benefit to beneficiary / cost to altruist, r = the Coefficient of relationship between altruist and beneficiary, p = the recombination rate between the markers locus and the altruism/selfishness locus, n = the number of generations since the common ancestor between the potential altruist and the potential beneficiary. The second inequality is a generalization of the first which turns into the first when p = 0. One can use the exponential fitting program to get R and then we can subtract R from 1 to get p and from p we can find out how many Morgans are between the altruism-selfishness locus and the purple-green locus. We can call them the behavior locus and the marker's locus. The use of "selected of" instead of "selection for" is to satisfy the followers of Eliot Sober who makes the distinction that the allele being favored isn't causing the events that select it but is only correlated with them. I have some problems with this, but that's for another paper. To put it tersely, ultimately all selection would have to be selection of, ultimately. In a nutshell, the case can be defended that all selection is selection of. This is because the genes expressed in the somatoplasm won't be the same individual ones replicated in the germ plasm, merely replicas of them. The Elevator effect could be used to locate a/s loci, and other loci on chromosome maps, and noninvasively at that. It could be enlisted to solve the riddle as to whether the helpless young of a duck-billed dinosaur, Maasauria, and injured adolescent Tyrannosaurus individuals, were fed exclusively by their parents, or did siblings, cousins, aunts, and uncles chip in, as often happens in avian dinosaurs. As just demonstrated, when the initial frequency of purple altruists is 0.6, the asymptote is other examples are: p1 P

6 Most of these look mysterious, at first sight, but the first one looks very promising. It is interesting to note we can get the point , and even get In "On Beyond Zebra", Dr. Seuss (1955) used our familiar alphabet to spell the names of new letters in order to describe new (imaginary) creatures in his poem, e.g., "Quan is for Quandry who lives on a shelf.". In a similar way we can learn to think in (say) base 100: Sometimes examples of normally difficult mathematical operations exist which can be calculated by simply looking at them. When it happens it is called "Solving by Inspection". An example of solving by inspection is that multiplying by 10 can be accomplished by moving the decimal point one place to the right, and dividing by ten by moving the point one place to the left. An example of translating from one base to another is that of translating from base 10 to base 100. Instead of inventing new characters to get 100 symbols for the digits in base 100, we will use the 100 possible two digit numbers in base 10 to represent the 100 possible single digits in base 100. To translate from base 10 to base 100 by inspection, we simply put a comma after every other digit in our base 10 numeral. As a result of which the two digit numbers between the commas will be single digits in base 100. We can similarly translate from base 10 to base 1000 by putting a comma after every third digit in our base 10 number and so on. In these sample points we can see by inspection that the jth digit of the p8 coordinate is a linear function of 1/p1, when these digits are from the representation of P8 in base 1/P1. For all 3 examples: The first digit = 1. The linear equation has a slope of 0 and an intercept of 1. When j = 2, the slope is 1/2 and the intercept is -1. When j =3 the slope and intercept are both 0." when j = 4 the slope is 1 and the intercept is -1. Thereafter, odd values of j will have slopes and intercepts of 0, and even values of j will have slopes of 1 and intercepts of -1.

7 We can get the ith coefficient of the formula for P8 as a function of P1 by adding the ith intercept to the (i + 1)th slope. This gives us the equation: P8 = 1.5P1 - P1^2 + P1^3 - P1^4 +P1^5 - P1^6 + P1^7.... The slope when j = 2 is the slope of the line through the points (10, 4) and (100, 49). The intercept for j = 2 is the intercept of that line. The formula for this slope is: M2 = (49-4)/(100-10), and the intercept is given by: B2 = M2. On a computer which inputs and outputs in base 8, we might have produced this table: The 2nd point to determine the linear equation for j = 2, could just as well be (8, 3), instead of (100, 49). We don't need a base 8 computer display to know this. We can use our proportion rule: digit1/base1 roughly equals digit2/base2. So we go: 4/10 = x/8 and solve for x. Which is: x = 4 times 8/10. which gives x = 3.2 Rounding this gives 3, which does for base 8, what 4 does for base 10, and 49 does for base 100. We can calculate the slope as: (3-4)/(8-10), and the intercept as: 3-8 (3-4)/(8-10). If base1 and base2 differ by 2, the equation we get will be a natural law.

8 If they don't, you'll get an equation that is represented by a curve that goes through the sample point but it won't be a natural law. In the Periodic Table of the Elements, elements in the same column, but different rows tend to have similar chemical properties. Thus lithium, sodium, calcium, aluminium, titanium, and strontium share a column and can substitute for each other in similar, or the same chemical reactions. As the rows get lower on the Periodic table, there are more elements in them. To get around the apparent paradox, great gaps exist in the middles of the higher rows. The different number bases are like rows in a periodic table. 9 would be in the same column in the base 10 row as 7 in the base 8 row. But instead of having one gap in the middle, we have 2 gaps at 1/4 and 3/4 along the interval. With the elements, we bung the entries toward the ends. With numbers we bung them toward the ends or middle, whichever is nearer. Thus a simple periodic table of the numerals would go: To show that 4 is to base 10 what 49 is to base 100, we could use the proportion rule to go from base 10 to base 12 and similarly from base 12 to base 14 and on through bases 16, 18, 20, 22, and so on. But we can get all we want with base 10 and base 12. Infinite series can be dealt with by multiplying the y coordinate of the sample point by (1- x coordinate ^ M) and putting a denominator in the derived formula of (1 - x^m), where M is a moderately sized even number. If our civilization hadn't abandoned Roman numerals, it is tempting to speculate that the method just described here would have been discovered a long time ago. MJO References: Fisher, R.A. (1930) The Genetical Theory of Natural Selection, Clarendon Press, Oxford Gould, J.L. (1976) The Dance-Language Controversy, The Quarterly Review of Biology, vol 51, issue 2, pp Haldane, J.B.S. (1919) The combination of linkage values, and the calculation of distances between

9 the loci of linked factors, J. Genet. v8, pp Hamilton, W. D. (1964) The Genetical Evolution of Social Behaviour I and II, J. Theor. Biol. v7, pp 1-16, and Kosambi, D.D. (1944) The estimation of map distance from recombination values, Annals of Eugenics, Vol.12, pp , 1944 Michod, R.E. & Hamilton, W.D. (1980) Coefficients of relatedness in sociobiology, Nature 288, (18 December 1980) Nowak, M.A., Tarnita, C.E. & Wilson, E.O. (2010) The evolution of eusociality, Nature 466, (26 August 2010) Orlove, M. J. (1975) A Model of Kin Selection not Invoking Coefficients of Relationship, J. Theor. Biol. v49 pp Orlove, M.J. (1979) A Reconciliation of Inclusive Fitness and Personal Fitness Approaches: a Proposed Correcting Term for the Inclusive Fitness Formula, J. Theor. Biol. v81 pp Orlove, M. J. (1981) The Elevator Effect or a Lift in a Lift: How a Locus in Neutral Equilibrium Can Provide a Free Ride for a Neutral Allele at Another Locus, J. Theor. Biol. v90 pp Orlove, M.J. & Wood, C. L. (1978) Coefficients of relationship and coefficients of relatedness in kin selection: A covariance form for the RHO formula, Journal of Theoretical Biology, Volume 73, Issue 4, 21 August 1978, Pages Skinner, B.F. (1947) 'Superstition' in the Pigeon, Journal of Experimental Psychology #38, pp Seuss, Dr. (1955) On Beyond Zebra!, Random House, New York Trivers, R.L. & Hare, H. (1976) Haplodiploidy and the evolution of the social insects, Science 191, West-Eberhard, M.J. (1975) The evolution of social behavior by kin selection, Quart. Rev. Biol. 50(1), JSTOR APPENDIX A The relationship between distance and recombination probability can be expressed with Haldane's (1919) Mapping Functions: D = - LOG(2 (1/2 - r))/2 r = 1/2 - EXP( -2 * D) / 2 Some folks would say that Haldane's formula for the relationship between recombination rate and map distance (1919) should be replaced by that of Kosambi(1944). This is because the occurrance

10 of a crossover might inhibit another one occurring nearby. APPENDIX B The Elevator Effect is most easily introduced by generalizing Hamilton's Inequality: K > 1/r to: K > (1 - p)^n/r In order to convince a most-likely incredulous contingent of any new audience to take seriously something as seemingly counterintuitive as the Elevator Effect it helps to slightly reformulate the 2nd inequality as: K > 1/r* where: r* = r/(1 - p)^n. The 2 formulae are equivalent but the latter is easier to se intuitively: If the world has existed for m generations and the potential altruist (worker) and potential beneficiary (queen) had their most recent common ancestor n generations ago, then r* has a numerator and denominator. The numerator tells how efficiently the altruism gene can pump purpleness genes into next generation by raising the queen's kids. The denominator tells how efficiently it can by raising the workers kids. Both numerator and denominator get multiplied by a factor of (1 p) for every generation since the beginning till n generations ago, after which each generation contributed a factor of 1/4 to the numerator, but continued delivering a factor of (1 - p) to the denominator. (Siblings are related by 1/2 because they are really each others half-siblings (r = 1/4) twice (making the total r = 1/2).) Whenever p is greater than 0, r* is greater than r. Thus when altruism is breaking even (k=1/r), k will be greater than 1/r*, at which time the neutral marker, which was initially coupled with altruism, will ride the elevator going up. APPENDIX C The Nowak et al 2010 paper claims that inclusive fitness theory, AKA kin theory, kin selection theory, and therefore Hamilton's Inequality ( K > 1/r is the necessary and sufficient condition to select for altruism) are irrelevant, that their time in the sun has come and gone, and it gives three reasons: 1) You can do anything with a personal fitness approach that you can do with an inclusive fitness approach

11 2) Hamilton's Inequality can be expanded to: a) K > 1/"something" is the necessary and sufficient condition to select for altruism b) "something" = r (where "something" is a variable name, not just the word) It argues that "something" may exist, but that "something" might not always equal r. 3) If we define r = "something", in areas of our conceptual space where r is undefined, then the approach is allegedly ad hoc, and consequently poor science. For point 1, a personal fitness approach gives you high predictive power, an inclusive fitness approach gives you high explanatory power. The prediction by Trivers and Hare (1976) that a queen ant strives toward the reproductive swarm being 1/2 female by weight, and the workers strive toward it being 3/4 female by weight, and in normal ants the workers win, and in slave making ants the queen wins, could not have been made without inclusive fitness theory. They subsequently confirmed their prediction with observations on many live ant species (reported in the same paper). Also, the Elevator Effect, Orlove (1981), and further discussed in this paper, could not have been explained without inclusive fitness theory. Hamilton (unpub pers com) was able to duplicate the model (Orlove 1975) with a purely inclusive fitness approach. The only variables describing the population were the gene frequency in the females, the gene frequency in the male, and the relatedness coefficient between members of pairs of ova destined to become each others sisters. The Nowak et al 2010 paper's saying that retaining both inclusive and personal fitness approaches is unnecessarily redundant, is like saying that retaining both fractions and decimals is unnecessarily redundant, or retaining both Cartesian and polar coordinates is unnecessarily redundant. For point 2, the following segment, though by the current author, is adapted from the talk page (aka discussion page) belonging to the wikipedia article on Inclusive Fitness. Since wikipedia material is potentially quite ephemeral, paraphrasing it rather than citing it appears appropriate: One glaring error in the Nowak et al 2010 paper is the unnumbered equation right after inequality (1) on page The following is a scan of Page 4 from BU-344-M showing how the unnumbered equation that follows Hamilton's inequality in Nowak et al is derived. This is for clarification (of both Nowak et al and my derivations) purposes. The West referred to here is Mary Jane West Eberhard, not Stuart West. (BU-344-M is the companion text for the 344th lecture in a series for the Biometry unit at Cornell University. The companion texts for this series are currently in the process of being posted on the web, in reverse chronological order, and as of this writing 344 has yet to be posted.)

12 There are two mistakes in that it wants to have "something" as the left hand side, as opposed to r, lest it gets caught in circular reasoning because it defines Q as the relatedness between the potential altruist and the potential beneficiary and Q bar as the relatedness between one of them and an average member of the population, and since r and relatedness are the same thing according to the paper, that is a circular definition. But for this obvious oversight, the paper would have said that "something" equals the right hand side of the equation and then it challenged the claim that "something" always equals r. If it had used "genes alike in state" instead of relatedness in

13 its definitions of Q and Q bar, then the equation would have been accurate. If the formula is evaluated twice, using "genes alike in state" the first time and "relatedness" the second time, when calculating Q and Q bar, the circularity is eliminated and the r values returned both times will be identical, thus proving "something" always equals r. For point 3, if r is defined such that Hamilton's Rule is preserved, as done by Orlove (1975), Orlove and Wood (1978), and in nine other papers, all of which were reviewed by Michod and Hamilton (1980), then the approach is by definition ad hoc. However, if r is defined in addition such that two propositions about the rate of selection, namely "the total amount of corrected inclusive fitness in the world equals the total amount of personal fitness in the world" and "if everyone has a doppelganger on a possible world and the personal fitness of one's doppelganger equals one's own corrected inclusive fitness (provided that the initial conditions are the same in both worlds) then the plot of gene frequency as a function of time will be identical in both worlds", are preserved then the two definitions of r coincide, which greatly reduces or eliminates the ad hoc nature of Hamilton's Rule[9]: where corrected inclusive fitness equals inclusive fitness in the original sense, plus C, where C is a constant the same for everyone in the population at any instant in time and it equals the total number of stranger equivalents reared in the current generation divided by the population size (e.g., if r equals 1/8 between cousins and I rear 8 offspring from my cousin, I have reared one of my offspring equivalents and 7 stranger equivalents.) Evenly dividing the stranger equivalents between all members of the population is a little white lie equivalent to saying I reared one of my offspring and seven strangers. However, it does preserve the proposition about rates of selection. Hamilton's 1964 not using C was not an oversight because he wanted his definition of fitness to be incapable of decreasing as time goes by, as R. A. Fisher (1930) predicted for Classical fitness in his fundamental theorem of natural selection. Novak et al. cite the fact that both inclusive fitness and personal fitness can be used "to do the same thing" as evidence for the superfluous nature of inclusive fitness. However, Orlove (1979) and Orlove (1975) use it as a way of finding philosophical implications, notably seeing inclusive fitness theory as isomorphic with the much older karma theory, C being equivalent to group karma. The similarity between kin theory and karma theory stems from natural selection working on correlated events just as well as when they are or are not causally linked, much as in B. F. Skinner (1947) finding a similar phenomenon in operant conditioning which he called "superstition".