An Algebra for Theoretical Genetics Ph.D. Dissertation, Massachusetts Institute of Technology, 1940.

In this thesis, an algebra is constructed for studying the dynamics of Mendelian populations. The symbols of the algebra represent groups of individuals or populations. The indexed symbol ?^h J ⁱ k, for example, represents a population in which two gene loci are under consideration (the number of loci corresponds to the number of pairs of indices). The number of allelomorphs in each locus is completely arbitrary, as is the recombination value for the loci. The different components of a population symbol, represented by fixing the indices at specific values, are numbers whose values correspond to the fractions of the population with certain genetic formulae. It is convenient in some cases to consider as populations symbols whose components are negative or even complex. Such symbols cannot, of course, represent an actual group of individuals and are called unrealizable populations, but their use sometimes facilitates the solution of problems.

Addition of two population symbols, R?^h j ⁱ k + S?^h j ⁱ k, results in a third population symbol which is defined in such a way as to represent the population obtained by merely combining the original populations in fractional proportions corresponding to the scalar coefficients R and S. Cross multiplication of population symbols ?^h j ⁱ k ? ?^h j ⁱ k gives a population symbol which is defined in such a way as to represent the expected offspring population when the two original populations are crossmated at random. When two gene loci are considered, this is realized by the mathematical definition ?^h j ⁱ k ? ?^h j ⁱ k = 1/2[p0?^h ? ⁱ ? + p1?^h ? ^? i][p0?^j ? ^k ? + p1?^j ? ^? k] + 1/2[p0?^j ? ^k ? + p1?^j ? ^? k][p0?^h ? ⁱ ? + p1?^h ? ^? i] in which p 1 = 1 - p0 is the recombination value for the two loci, and replacing an index by a dot indicates summation of the population symbol on that index. Cross multiplication is defined analogously for n loci. It is shown that this algebra is commutative on addition and multiplication, distributive, and associative on addition but not on multiplication. These laws together with two fundamental manipulation theorems: one, that summation of a population on all indices gives unity and two, that inverting the upper and lower rows of indices of a population leaves it unchanged, form the basic algorithms of the algebra.

A number of the well known theorems of theoretical genetics are easily proved by means of this algebra. In addition, a number of new results are found. Completely general formulae for the nth generation offspring under random intermating of an arbitrary initial population are developed both for the cases of two and of three linked factors. For two linked factors, the formula for the nth generation is ?^h j ⁱ k = [p ^n-1 0(p0?^h ? ⁱ ? + ^p1?^h ? ^? i) + (1 - p ^n-1 0 )?^h ? ^? ??^? ? ⁱ ?) [p ^n-1 0(p0?^j ? ^k ? +p1?^j ? ^? k) + (1 -^n-1 0)?^j ? ^? ??^? ? ^? ? ^k ?] in which ?^h j ⁱ k is the initial population and p1 the recombination value. Incidental to this, it is shown that a recombination value > 1/2 is impossible when there is no interference. Conditions are found for the stability under random intermating of a population when one or more loci are considered. For the case of one locus, three sets of equivalent necessary and sufficient conditions are established.

By means of certain homogeneous ? populations an arbitrary population may be expanded in a finite series displaying its various components. This expansion, together with the multiplication law for the ? populations, displays the elements of this algebra as hypercomplex numbers. It is shown that an arbitrary population may be expanded uniquely as a sum of any n linearly independent populations where n is the number of different possible genetic formulae for the factors considered.

It is possible to write down various types of equations involving known and unknown populations using the operations of addition and cross multiplication. In general, such an equation can be interpreted as a breeding experiment involving one or more unknown populations and resulting in a genetically known population. Methods are developed whereby most such equations can be solved in case a solution exists. Briefly this method of solution may be summarized as follows. By summing on one or more indices of the unknown populations, enough data about them is obtained to uniquely determine any cross products in which they appear. The cross product terms in the original equations then become known and the equations may be solved in exactly the same way as ordinary linear algebraic equations.

In case a selective action exists favoring individuals of a certain genetic constitution, the previous formulae for stability no longer hold. Although this more difficult problem has not been completely solved, a necessary condition for the possible existence of a stable population under an arbitrary given selective action is established, and a formula for this population is developed. This has only been done for the case of a single locus.

A start has been made toward the development of a calculus of populations, i.e., the study of populations which may vary continuously with time. The time derivative of a population is defined. The derivative of a population, although an indexed symbol, is not itself a population. All the ordinary rules of derivation including the Leibnitz rule for the derivative of a cross product of populations are shown to hold true. Also, a population may be expanded in a Taylor series in powers of time, of the same form as the ordinary Taylor series.