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The Mathematical Basis of MendelianGeneticsB.A.C. Dudley aa Centre for Science Education , Chelsea College, Universityof London , EnglandPublished online: 09 Jul 2006.

To cite this article: B.A.C. Dudley (1973) The Mathematical Basis of Mendelian Genetics ,International Journal of Mathematical Education in Science and Technology, 4:2, 193-204, DOI:10.1080/0020739730040215

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Int. J. Math. Educ. Sci. Technol., VOL. 4, 193-204 (1973)

The Mathematical Basis of Mendelian Genetics*

B. A. C. Dudley†

Centre for Science Education, Chelsea College, University of London, England

SummaryIn this paper Mendelian genetics is reinterpreted in terms of mathematics. The gene in manyliving things -- but in by no means all of them -- is here viewed as an ordered pair of elements(called alleles), and each offspring as an element of a Cartesian product. The paper illustrateshow this view provides a means of presenting both instances of classical Mendelian inheritanceand also special cases of inheritance.

Introduction

Many biology (and mathematics) teachers readily acknowledge that Mendelian geneticsis one of the more mathematical topics in biology but have not studied the mathematics(or biology) that is involved to see if it can contribute to the teaching of this topic. It isthis which has been attempted in the present paper, itself a development and extension ofmaterial first presented elsewhere.1

One important exercise in biology is the classification of information (that is, theidentification of sets and subsets). This can be seen most readily by considering living things.For example insects, crabs, spiders, centipedes and the like all belong to one phylum, theArthropoda (literally, animals with jointed legs); within this is the class of insects, theInsecta, of which one order is the Hymenoptera (wasps, ants and bees). Other examplesof sets include the genes of an individual, the set of parents and the set of all possibletypes of fertilized egg (zygote) which can arise from them. In taxonomy, which is thescience of classifying animals and plants, the 'elements' of a set are whole organisms (aman, an oak tree) but in genetics, while some sets may also be made up of individualorganisms, others are composed of individual genes, individual zygotes and so forth.

The identification of sets, subsets and their elements and the relations between themis also of course part of mathematics. Sets can be represented by letters and relations andoperations by symbols. This method of notation has been used to portray classificationin biology2 and blood groupings3 and can also be used to represent sets in genetics. Forexample, the set of parents, P, of an individual is made up of a male parent, c, and afemale parent, d, as follows

P = {c,d)

Any gene, h, of a set of genes H of an individual is itself a set composed of twoelements—m, contributed by the male parent and / , contributed by the female parent.It is these elements, m and/, which biologists call 'alleles'.

* Reproduced by kind permission of the Institute of Biology. This article was first published in theApril issue (Vol. 6, No.2) of the Journal of Biological Education.

† Present address: Department of Education, University of Keele, Keele, Staffordshire, England.

Received 9 March 1972

© 1973 by The Institute of Biology, London.

193

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An individual is made of many cells all of which arise from the one fertilized egg, orzygote. With the important exception of one special type of cell—the gamete—all cellsof an individual have two elements (alleles) to each gene (see Figure 1). The number ofelements (alleles) of each gene of a gamete is reduced to one.

Individual*

Cell

Chromo-somefrom oneporent

Gene IH

Chromo-some fromsecondporent

Figure 1. Diagrammatic representation of the genetic composition of an individual

When a male gamete (a sperm in the case of animals) fuses with a female gamete (anovum or unfertilized egg) the fertilized egg or zygote results, in which each gene is againmade up of two alleles. The individual grows from this zygote (Figure 2). It is because

I

N>Male \ ^C Female

/ Fertilization -

Gamete1

/

Figure 2. The cycle of the inheritance of genetic material

the maternal and paternal chromosomes do not fuse but remain discrete within thefertilized egg that each gene of an individual is made up of a pair of elements (alleles),

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MATHEMATICAL BASIS OF MENDELIAN GENETICS 195

one in each of two chromosomes. As a consequence a gene, h, of the set of genes H,consists of a pair of alleles

h = {m,f}

where m is an allele of gene h that has been inherited from the male parent and/is anallele that has been inherited from the female parent. Every gene is made up of selectedpairs of elements, one from each of two sets, namely the set of alleles (M) of the maleparent and the set of alleles (F) of the female parent. To sum up,

1. A fertilized egg, and hence the individual arising from it, originates from one malegamete fusing with one female gamete.

2. Each gamete contains only a single allele (element) of each gene.3. Each individual (organism) contains a pair of alleles (elements) for each gene.

The number of chromosomes per cell varies between species, e.g. for humans it is 46,for mulberry trees 308,4 while for one particular fern it is approximately l,020.5

The Inheritance of a Single CharacterBiologists work from the premise that each character or feature of the body is controlledby a single gene. The study of the inheritance of such characters in isolation is called'monohybrid inheritance' though there is more than one form to the character taken forstudy. For instance, the one gene for eye colour may result in the eyes being either brownor blue, the alternative colours resulting from the different alleles of the one gene. Oneof these alleles (suppose it to be for brown eyes in this instance) is found to be expressedwhenever it is present while the other (for blue eyes) is expressed only when the first typeis absent. This first type therefore is said to be dominant and is represented by a 'capital'letter while the other allele is said to be recessive and is represented by the correspondingly'small' letter.

Consider an arbitrary gene A. When an individual has both alleles of this gene domi-nant then its genetic composition (its genotype) is represented by AA. When an individualhas both alleles of this gene recessive its genotype is represented by aa. When one parenthas the genotype AA and the other aa, the types of offspring possible can be shown bythe co-ordinates of a lattice, thus

Gamete types ofsecond parent(aa) a

(A, a)

(A, a)

(A, a)

(A, a)

Gamete types offirst parent (AA)

All the offspring in this example are of the genotype (A, a) and have the appearance(that is, the phenotype) corresponding to dominant allele A.

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196 B. DUDLEY

When one parent has the genotype Aa and the other aa, the types of offspring that arepossible are shown to be as follows:

Gamete types ofsecond parent{Aa) a

{a, A)

{a, a)

(a, A)

(a, a)

Gamete types offirst parent (aa)

Here two kinds of product can form, namely (a, A) and (a, a). On this occasion thesubsets of the set of gametes produced by the recessive parent, (aa), may be united sincethey are identical though those of the other parent may not be united. The simplifiedlattice then appears as

AGamete types ofsecond parent(Aa) a

(a, A)

(a, a)

Gamete types offirst parent (aa)

Experiments that have been performed with parents of this type (namely Aa x aa)show not only that these two types do occur amongst the offspring but also that theyoccur with approximately equal frequency. This leads to the conclusion that within eachparent the types of gamete themselves also occur with equal frequency. The lattice nowbecomes

Gamete types ofsecond parent(Aa)

(a, A)1

(a, A)

i

a(1)

Gamete types offirst parent (aa)

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MATHEMATICAL BASIS OF MENDELIAN GENETICS 197

It is also possible for both parents to have the genotype Aa, when the possible geno-types of the offspring are as follows

AGamete types ofsecond parent(Aa) a

(A, A)

(A, a)

(a, A)

(a, a)

Gamete types offirst parent (Aa)

This can also be written as follows:Set of allele types of first parent is M = {A, a}.Set of allele types of second parent is F = {A, a}.Set of possible types of offspring is M x F = {(A, a), (A, A), (a, A), (a, a)}.Here MxFh the 'Cartesian product' of the sets M,F.

Four kinds of genotype result amongst these offspring, though in genetics (A, a) isalways functionally indistinguishable from that of (a, A). In three of the possible types,the dominant allele, A, is present. In only one kind are there recessive alleles alone,i.e. (a, a). Thus individuals will have one of two appearances (phenotypes), some havingthe phenotype of the dominant allele and some that of the recessive allele.

Mendel, with his work on peas, showed that the dominant and recessive phenotypesoccur in the ratio of 3 : 1 and this is explained by assuming once again that the gametictypes of each parent are equally frequent. The frequency of the subset of offspringidentified by having allele A present is thus | and the frequency of the subset of offspringidentified as having only the recessive allele will be £, giving the classical 3 : 1 ratio.

The Inheritance of Two CharactersThe study of the inheritance of two genes simultaneously is called dihybrid inheritance.In such studies it is first necessary to identify the types of gametes produced by eachparent and this is a matter of confusion amongst biology pupils because the model whichrepresents such experiments is incomplete.

One classical experimental situation is where one parent possesses only dominantalleles of the genes being considered, while the other parent possesses only recessivealleles. If the two genes under consideration are given the symbols A and B, then such anexperiment is traditionally represented by the notation AABBxaabb. This possiblyconfusing notation simply means that the gamete from the first parent has one allele Aand one allele B, and the gamete from the second parent, one of type a and one oftype b. So

motcheswith either (i)©G)

or(ii)0

and the possibilities for the offspring's genotypes are (i) (A,a),(B,b); (ii) (A,b),(B,a).The offspring must be genetically identical to each other and of the type AaBb.

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198 B. DUDLEY

In appearance (phenotype) they will be of the type represented by A and B and willall appear similar to the dominant parent.

When these offspring, namely with the genotype AaBb, themselves function as bothparents each produces a gamete of type

As these alleles can be in the reverse order there are four possibilities within eachgamete as is shown by the following lattice

B<4,B)

(A,b)

(a,B)

(a,b)

Gamete typesofgeneO&>)

A aGamete types ofgene (Aa)

Each parent manufactures all four kinds.The lattice showing the kinds of offspring resulting from these is

A,B

A,bTypes ofpaternalgamete a, B

a,b

(a,b,A,B)

(a,b,A,b)

(a,b,a,B)

(a, b, a, b)

(a,B,A,B)

(a,B,A,b)

(a,B,a,B)

(a,B,a,b)

(A,b,A,B)

(A,b,A,b)

(A,b,a,B)

(A,b,a,b)

(A,B,A,B)

(A,B,A,b)

i.A,B,a,B)

(A,B,a,b)

a,b a,B A,bTypes of maternal gamete

A,B

Here, for example, (a,B,A,b) is a genotype; this means that that particular offspringhas these alleles (which will probably, in fact, all be on different chromosomes).

Four phenotypes can be identified as subsets of this lattice, namely (A, B), {A, b),(a, B) and (a, b), when there are found to be nine containing A and B, three containing Aand b, three containing a and B and one containing a and b. In classical experiments, suchas those performed by Mendel with round or wrinkled, yellow or green peas, these fourphenotypes occur amongst the offspring with a frequency of 9 : 3 : 3 : 1 and once againthe unavoidable conclusion is that each of the gamete types of each parent occurs withequal frequency.

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MATHEMATICAL BASIS OF MENDELIAN GENETICS 199

The genotypes of possible offspring of parents can be represented in this fashion, andthis presentation becomes useful when a study is being made of those special cases ofinheritance, such as multiple alleles, sex linkage and linkage, that are part of the secondaryschool biology course.

Multiple AllelesA multiple allele system is one in which an allele (e.g. A) has more than two expressions,say A, a, a instead of just two possibilities A,a. The best known example is that of theABO system of human blood groups, though others exist, such as the system whichincludes the condition of rabbit's fur known as chinchilla.

With regard to the ABO blood groups in humans, the maternal and paternal gametesof a population as a whole must be elements of the set S = {A, B, O}. Each gametecarries only a single allele for this character. Furthermore, any individual of the popula-tion can carry at most only two kinds of allele for this character. Under such a systemthe types of offspring that arise are shown in the following lattice, where the kinds ofoffspring that result are identified by the co-ordinates.

B

Paternal alleles(gamete types in Othe population)

(A,B)

(A,O)

(A, A)

(O,B)

(O,O)

(O,A)

(B,B)

(B,O)

(B,A)

o B

Maternal alleles(gamete types in the population)

(A, B) for example, represents the gene

There are nine possible elements of the Cartesian product SxS but as far as geneticsis concerned some are equivalent (for example (A, O) and (O, A)). Six genotypes occurbecause alleles A and B are expressed in the phenotype whenever they are present but Ois expressed only when A and B are absent. These six genotypes result in there being fourdifferent blood groups (phenotypes), namely groups A, B, O and AB. Thus at the levelof the phenotype, genotypes AO, OA and AA for instance are indistinguishable one fromthe other.

This information is used in cases of disputed parenthood. Once the blood group of thechild and the mother are known then the necessary blood group of the father can bededuced. This is used in law as a means of eliminating from suspicion a man accused ofbeing the father of that child. The complete set of possibilities are given in the latin squarein Table I which in fact illustrates a mapping of SxS^~S, where S is the set of ABOblood groups.

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200 B. DUDLEY

Table I. The identification of the blood group of the father in disputed paternal cases(the entries in the table give the possible blood groups of the father)

ABO groupof the

A

B

O

AB

A

AABOB

AAB

AAB

ABABO

ABO group

B

BAB

ABABO

BAB

ABABO

of the

O

ABO

ABO

O

0

child

AB

BAB

AAB

0

ABAB

Sex Linkage

An allele is part of a chromosome. Two chromosomes, the X and the Y, are concerned^with determining the sex of the individual. In most animals, including man, an individualwith one Z a n d one Yin each cell of the body is male while one with two X's is female.However, these sex chromosomes also carry alleles of genes which are involved indetermining other features of the organism. For instance, red-green colour deficiency(though not colour blindness) and haemophilia are abnormalities in humans, the allelesfor which occur on the X chromosome. Other alleles determining the more normalcondition of sight and blood clotting exist and, since it is these which are expressed when-ever present, are dominant to the alleles causing the defects mentioned. Alleles for suchsex-linked characteristics, whether dominant or recessive, are on the part of the Xchromosome for which there is no counterpart on the Y chromosome, that is a part of theso-called differential part of the X chromosome; their inheritance therefore shows anunusual pattern, different from the Mendel's ratios associated with alleles on the'ordinary' chromosomes (which are called 'autosomes').

However, their inheritance is no more than a special case of dihybrid inheritanceinvolving the empty set, as can be illustrated with the traditional example, namely theeye colour of the fruit-fly, Drosophila melanogaster, where white eye colour (r) is recessiveto that for red eye (R).

A Cross Between a Red-Eyed Male and a White-Eyed Female (rr)

The male is X Y and, as far as the genes on the differential part of the X chromosomeare concerned, the corresponding part of the Y chromosome consists of the empty set.

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MATHEMATICAL BASIS OF MENDELIAN GENETICS 201

The Cartesian product of such a cross (red-eyed male, white-eyed female) can berepresented as follows

(X)RMalegametes

(7)0(r,0)

(r,R)

(r,0)

r r(X) (X)

Female gametes

Ordered pairs including the empty set will be male and since the only possible malesare (r, 0 ) they all will have white eyes. Products without the empty set will be female, allof which in this particular cross are (r, R) and so all will have red eyes. Hence the offspringresulting from this cross will be such that all the males have white eyes and all the femalesred eyes.

A Cross Between a White-Eyed Male (r0) and a Heterogametic Female (Rr)

The offspring of such a cross are shown in the following diagram.

(X)rMalegametes

(7)0

(R,r)

(R,0)

(r,r)

(r,0)

(X)(X)R r

Female gametes

Some males will have red eyes (R,0) and some white eyes (r ,0). Similarly, somefemales will have red eyes (R, r) and some white eyes (r, r). Since in experiments thesefour types occur with equal frequency then the relative frequency of the two types ofmale gamete is a half, as is the relative frequency of the two types of female gamete,showing again that in each parent the types of gametes occur with equal frequency.

A Cross Between a White-Eyed Male (r0) and a Pure Bred Red-Eyed Female (RR)

The offspring of such a cross are shown in the following diagram.

(X)rMalegametes

(7)0

(R,r)

(R,0)

(R,r)

(R,0)

R R

(X) (X)Female gametes

All the offspring, male and female, will have red eyes.

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202 B. DUDLEY

On this occasion, because the female is homogametic it is possible to unite the sets offemale gametes to give

fflrMalegametes

(Y)0

(R,r)

(R,0)

R

Female gamete

A Cross Between a Red-Eyed Male (R, 0) and a Heterogametic Female (R, r)In this case the only white-eyed individuals amongst the offspring are male (r,0); all

the female offspring and half of the males have red eyes. This is shown to be the casefrom the following lattice.

(X)RMalegametes

(100

(R,R)

(R,0)

(T,R)

(r,0)

R

Female gametes

LinkageThere is another kind of cross involving dihybrid inheritance which is of special interest.This is the cross in which one parent has the genotype AaBb (and is said to be hetero-gametic for both genes) and the other has the genotype aabb (and is said to be homo-gametic and recessive for both genes). This can be represented by the notation AaBb xaabb and it follows that the types of gametes that can be formed in the first parent are(A,B), (A,b), (a,B) and (a,b) while those of the second parent are all (a,b). This kindof cross and the types of offspring that can arise from it are shown in the lattice (below)

Types of OffspringGenotype Phenotype

A,B

Types of gamete A, bproduced by theheterogameticparent {AaBb) a, B

a,b

(a,b,A,B)

(a,b,A,b)

(a,b,a,B)

(a,b,a,b)

A,B

A,b

•a,B

•a,b

a,bTypes of gamete produced bythe homogametic parent (aabb)

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MATHEMATICAL BASIS OF MENDELIAN GENETICS 203

and its special interest lies in the fact that the phenotype of each offspring is governed bythe alleles inherited from the heterogametic parent. As a consequence, amongst theoffspring, the frequency of each phenotype is governed by the frequency with which thecorresponding gamete was produced in the heterogametic parent. When all four pheno-types occur with equal frequency the cross is said to be one involving classical Mendelianinheritance—the types of gamete occur with equal frequency in the heterogametic parentand the alleles of each gene combine in a random fashion when the gametes are beingformed, as indicated in the tree diagram

-AB flu)

Sometimes this result is not obtained, and the first of these exceptions6 is consideredhere (Table II).

Table II. Results obtained by Morgan6 on Drosophila, an exception to those expected ofclassical Mendelian inheritance

Cross

PercentageNumbers incidence ofobtained phenotypes

Frequency ofgamete types

(heterogameticparent)

A,B

A,b

a,B

a,b

(a,b,A,B)-

(a,b,A,b)

(a, b, a, B)

(a,b, a, b)

586

111

106

465

46-21

8-75

8-36

36-67

a,b

It was argued that since the total frequency (P) of the alleles is such that P(A) ^P(a)and P(B) ~P(b), then in forming the gametes, the alleles must behave in the followingfashion

46-21 percent 0\)

8-75 percent (<V4)

8-36 percent (<W

3 5 67 percent (>'/4)

so that (A and B) and {a and b) respectively are not being inherited independently, buttend to stay together.

Morgan proposed that they remain associated (linked) by being on the same chromo-some and this was confirmed in subsequent experiments. The existence of the two leastfrequent classes is due to a phenomenon whereby the chromosome sometimes breaks andin rejoining, parts are exchanged (recombined). If it is assumed that the breaks occur atrandom throughout the length of the chromosome, the incidence of these two classes is

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used as a measure of the length of chromosome which lies between the two genes inquestion. By studying the inheritance of three genes simultaneously (a three-point cross)the linear sequence and their relative distances can be established which form the basisof a map. It is in this way that chromosome maps are built up at least for animals andfor plants. (See, for example, Reference 7, page 188.)

AcknowledgementsThe author's ideas for the work in this paper developed after conversations with SenorPatricio Orduz and it is a pleasure to acknowledge his help in this respect; also that ofProfessor G. Matthews (Shell Professor of Mathematical Education, Centre for ScienceEducation) whose invaluable comments helped form the final result.

REFERENCES1. B. A. C. Dudley, Mathematics and Biology, M.Ed. dissertation, University of London, 1971.2. D. Reid and P. Booth, Biology for the Individual. Book 1. Sorting Animals and Plants into Groups,

Heinemann, London, 1970.3. K. R. Rebman, 'Blood types as sets', in Some Mathematical Models in Biology (Ed. R. M. Thrall,

J. A. Mortimer, K. R. Rebman and R. F. Baum), University of Michigan, 1967.4. M. J. D. White, The Chromosomes, Methuen, London, 1942.5. I. Manton and W. A. Sledge, 'Observations on the cytology and taxonomy of the pteridophyte

flora of Ceylon', Phil. Trans. R. Soc. B. 238, 127-185 (1955).6. T. H. Morgan, 'Sex limited inheritance in Drosophila', Science, N.Y. 32, 120-122 (1910).7. E. J. Gardener, Principles of Genetics, Wiley, New York, 1968.

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