Computer

ScienceUniversity

of

Manchester

The

Evolution

of

Polygenic

Sex

Determination

with

Potential

for

Environmental

Manipulation

M.J.

Hatcher

C.

Tofts

Technical

Report

The

Evolution

of

Polygenic

Sex

Determination

with

Potential

for

Environmental

Manipulation?y

M.J.

Hatcher

C.

Tofts

Department

of

Pure

and

Applied

Biology

Department

of

Computer

Science

University

of

Leeds

University

of

Manchester

Leeds

Oxford

Road

Manchester

UK

cmnt@cs.man.ac.uk

Copyright

c

All

rights

reserved

Reproduction

of

all

or

part

of

this

work

is

permitted

for

educational

or

research

purposes

on

condition

that

-LRB-

-RRB-

this

copyright

notice

is

included

-LRB-

-RRB-

proper

attribution

to

the

author

or

authors

is

made

and

-LRB-

-RRB-

no

commercial

gain

is

involved

Recent

technical

reports

issued

by

the

Department

of

Computer

Science

Manchester

University

are

available

by

anonymous

ftp

from

ftp.cs.man.ac.uk

in

the

directory

pub/TR

The

files

are

stored

as

PostScript

in

compressed

form

with

the

report

number

as

filename

They

can

also

be

obtained

on

WWW

via

URL

Alternatively

all

reports

are

available

by

post

from

The

Computer

Library

Department

of

Computer

Science

The

University

Oxford

Road

Manchester

UK

yM.J

Hatcher

is

supported

by

the

NERC

C.

Tofts

is

supported

by

an

EPSRC

Advanced

Fellowship

Abstract

We

present

a

formal

account

of

a

genetic

mechanism

for

Environmental

Sex

Determination

-LRB-

ESD

-RRB-

using

a

probabilistic

process

algebra

-LRB-

WSCCS

-RRB-

We

exploit

the

process

algebra

to

demonstrate

that

a

single

chromosome

pair

based

polyfactorial

system

is

intrinsically

unstable

The

system

evolves

with

a

deterministic

relationship

between

an

animal

's

sex

and

its

polyfactor

count

to

three

possible

forms

of

genetic

mechanism

pure

ESD

pure

genetic

sex

determination

-LRB-

GSD

-RRB-

-LRB-

both

XY

and

ZW

being

possible

outcomes

-RRB-

or

a

mixed

solution

with

a

division

between

ESD

and

GSD

With

a

probabilistic

dependence

between

gene

count

and

sex

we

observe

the

first

and

last

of

the

above

solutions

-LRB-

no

pure

GSD

-RRB-

but

with

population

gene

frequencies

that

can

be

far

from

Hardy-Weinberg

equilibrium

A

model

of

the

systemic

behaviour

in

the

presence

of

recombination

is

presented

and

the

effect

of

the

relative

recombination

and

evolution

rates

is

discussed

Our

process

algebraic

model

is

extended

to

account

for

the

polyfactors

being

distributed

over

many

possible

chromosome

pairs

and

demonstrates

that

only

one

essentially

new

stable

solution

is

reachable

We

demonstrate

that

within

a

multichromosome

system

there

will

almost

always

be

a

leading

chromosome

pair

which

determines

sex

other

chromosome

pairs

will

have

identical

sex

determining

information

upon

both

members

of

the

pair

but

not

necessarily

CR

categories

F.

I.

G.

Introduction

In

this

paper

we

attempt

to

provide

a

genetic

basis

for

Environmental

Sex

Determination

-LRB-

ESD

-RRB-

Rather

than

assuming

that

genetic

make

up

forces

gender

we

shall

assume

that

the

underlying

genotype

of

an

animal

gives

a

predisposition

towards

a

particular

sex

but

that

other

factors

may

affect

the

individual

's

final

choice

of

sex

To

that

end

we

will

describe

sex

as

being

determined

by

a

polygene

whose

intensity

defines

the

prediposition

towards

a

particular

sex

-LRB-

male

-RRB-

Of

particular

interest

to

us

are

two

separate

populations

of

Gammarus

duebeni

one

of

which

shows

conventional

genetic

sex

determination

and

the

other

ESD

-LSB-

-RSB-

The

allocation

of

resources

to

each

sex

within

a

population

is

a

major

evolutionary

pressure

in

sexual

species

-LRB-

-LSB-

-RSB-

and

many

others

-RRB-

In

many

diploid

organisms

sex

is

determined

with

respect

to

a

single

pair

of

sex

chromosomes

which

may

be

structurally

distinct

from

the

autosomes

-LRB-

non-sex

chromosomes

-RRB-

and

from

each

other

-LRB-

termed

heteromorphic

e.g.

-LSB-

-RSB-

-RRB-

Where

cytology

has

not

identified

heteromorphism

gender

appears

to

be

determined

by

one

or

more

major

sex

factors

and

in

some

cases

by

the

actions

of

many

minor

sex

factors

-LRB-

the

terms

are

relative

reviewed

in

-LSB-

-RSB-

-RRB-

In

this

paper

we

develop

models

relating

to

major

and

minor

factor

control

systems

which

we

term

multifactorial

or

polygenic

Factors

are

distributed

over

a

number

of

sites

-LRB-

multiple

loci

-RRB-

on

one

or

more

chromosome

pairs.We

shall

refer

to

the

total

sum

of

factors

as

a

polygene

and

to

the

factors

collectively

as

polyfactors

The

dynamics

of

multilocus

or

polygenic

factors

for

determining

organism

phenotype

have

been

widely

studied

-LSB-

-RSB-

and

the

effect

of

using

such

a

system

for

sex

determination

has

been

studied

in

-LSB-

-RSB-

There

are

many

examples

of

animals

in

which

it

is

believed

that

gender

is

determined

by

some

underlying

multifactor

system

-LSB-

-RSB-

Necessarily

many

of

these

studies

have

concentrated

on

sex

determined

by

genotype

at

conception

although

for

many

organisms

there

may

be

a

selective

advantage

to

determination

of

sex

with

respect

to

environmental

cues

-LSB-

-RSB-

This

advantage

may

arise

whenever

the

environment

is

heterogenous

in

time

or

space

and

the

relationship

between

environment

quality

and

adult

reproductive

success

differs

between

the

two

sexes

Under

these

circumstances

selection

favours

those

individuals

which

produce

relatively

more

of

the

sex

with

the

greater

relative

fitness

for

that

patch

-LSB-

-RSB-

When

patch

quality

can

not

reliably

be

predicted

by

the

parent

but

can

be

predicted

from

environmental

cues

experienced

by

the

offspring

ESD

is

favoured

-LSB-

-RSB-

Among

others

ESD

occurs

in

plants

-LSB-

-RSB-

nematodes

-LSB-

-RSB-

crustaceans

-LSB-

-RSB-

fish

-LSB-

-RSB-

and

reptiles

-LSB-

-RSB-

It

is

usually

supposed

that

ESD

has

multifactorial

genetic

underpinnings

as

-LRB-

-RRB-

adaptive

interpretations

of

ESD

imply

selection

which

requires

that

some

aspects

of

ESD

are

heritable

and

-LRB-

-RRB-

it

is

difficult

to

envisage

how

ESD

could

be

encoded

under

simple

chromosomal

sex

determination

-LSB-

-RSB-

Unlike

the

analysis

of

-LSB-

-RSB-

we

shall

not

assume

that

the

polyfactors

are

normally

distributed

and

then

look

for

a

single

control

point

which

induces

stability

We

shall

analyse

two

forms

of

decision

deterministic

system

if

the

total

factor

count

of

the

polygene

is

below

a

particular

value

then

the

animal

will

become

female

if

the

loading

is

above

a

-LRB-

possibly

different

-RRB-

control

value

the

animal

will

become

male

otherwise

it

will

decide

its

sex

in

an

unbiased

random

fashion

-LRB-

-RRB-

nondeterministic

system

we

assume

that

a

particular

polygenic

load

leads

to

a

particular

probability

of

becoming

female

for

simplicity

this

distribution

will

initally

be

linear

For

our

purposes

the

random

region

represents

that

area

in

which

the

environment

can

potentially

have

an

effect

on

the

outcome

of

sex

determination

although

we

do

not

model

ESD

explicitly

The

model

methodology

that

we

adopt

is

essentially

a

formal

individual

based

techinique

Process

Algebra

Process

algebra

-LSB-

-RSB-

is

particularly

appropriate

to

the

modelling

of

evolutionary

systems

as

it

permits

us

to

track

precisely

the

genetic

make

up

of

every

individual

within

a

population

To

express

the

behaviours

of

our

system

of

organisms

we

need

to

model

the

effects

of

time

probability

and

priority

Hence

our

process

algebra

will

need

to

be

able

to

express

all

of

these

behaviours

The

Weighted

Synchronous

Calculus

of

Communicating

Systems

-LRB-

WSCCS

-RRB-

-LSB-

-RSB-

is

capable

of

expressing

all

of

these

properties

and

we

shall

use

it

to

derive

the

properties

of

our

evolutionary

system

An

application

of

these

methodologies

to

the

behaviour

of

social

insects

can

be

found

in

-LSB-

-RSB-

and

their

biological

consequences

in

-LSB-

-RSB-

Evolutionary

systems

are

increasingly

of

interest

to

computer

scientists

as

the

methods

are

often

used

for

exploration

of

complex

systems

via

genetic

algorithms

-LSB-

-RSB-

The

behaviour

of

genetic

algorithms

is

highly

complex

and

providing

appropriate

formal

analysis

techniques

for

such

computational

methodologies

is

difficult

From

a

computing

prespective

an

evolutionary

system

is

a

large

scale

parallel

device

which

optimises

its

use

of

resources

The

analysis

of

such

natural

parallel

systems

may

help

in

the

design

and

analysis

of

similar

artificial

computing

systems

In

this

initial

study

of

polygenic

sex

determination

we

shall

concentrate

on

the

stability

of

the

gene

pool

That

is

to

say

we

shall

examine

what

the

stable

distribution

of

genes

within

the

population

will

be

after

many

generations

In

particular

we

shall

perform

our

analysis

in

the

absence

of

any

selective

pressures

That

is

we

are

studying

the

structural

stability

of

our

genetic

model

of

ESD

The

analysis

of

polygenes

for

other

behaviours

-LSB-

-RSB-

suggests

that

it

is

highly

unlikely

that

the

distribution

of

polyfactors

will

be

Guassian

as

assumed

by

-LSB-

-RSB-

Thus

it

is

of

great

importance

to

know

what

the

underlying

distribution

of

our

factors

should

be

in

the

absence

of

an

enviromental

bias

giving

advantage

to

particular

individuals

In

the

succeeding

section

we

describe

our

basic

process

algebraic

description

of

the

polygenic

system

and

analyse

its

stability

given

sex

selection

systems

of

both

types

stated

above

This

initial

analysis

is

undertaken

under

the

assumption

that

all

of

the

factors

contributing

to

a

polygene

are

resident

on

a

single

chromosome

pair

We

consider

the

effect

upon

our

stable

states

of

recombination

between

the

genes

In

Section

we

analyse

the

effect

of

distributing

the

polygene

over

many

chromosome

pairs

The

main

observation

is

that

under

finite

population

assumptions

the

deterministic

system

evolves

to

one

of

three

basic

forms

of

stability

monogenic

purely

random

sex

determination

-LRB-

all

individuals

genetically

identical

-RRB-

two

factor

pure

genetic

sex

determination

two

factor

mixed

sex

determination

with

each

showing

a

sex

ratio

as

we

should

expect

-LSB-

-RSB-

in

our

sexually

unbiased

system

The

non-deterministic

system

evolves

into

one

of

two

solution

types

either

or

of

above

When

the

factors

are

distributed

over

many

chromosomes

then

we

see

essentially

the

same

behaviour

with

one

pair

becoming

leading

in

sex

determination

and

all

of

the

other

pairs

being

monogenic

with

respect

to

our

polyfactor

Female

Male

Number

of

sex

factors

Low

High

Definitely

Definitely

Figure

Sex

allocation

function

with

deterministic

regions

Sex

Allocation

with

Deterministic

Regions

We

consider

the

case

of

stability

for

a

polygenic

sex

determination

system

when

the

sex

determining

function

has

the

following

form

there

is

some

lower

limit

of

factors

-LRB-

Low

-RRB-

below

which

all

animals

are

female

there

is

an

upper

limit

of

factors

-LRB-

High

-RRB-

above

which

all

animals

become

male

in

between

these

limits

sex

is

allocated

at

random

on

a

basis

Pictorially

such

a

selection

function

can

be

viewed

as

given

in

Figure

We

are

modelling

a

system

in

which

each

is

of

unitary

value

contributing

to

the

genotype

A

formal

description

of

the

behaviour

of

our

shrimp

would

be

the

following

collection

of

WSCCS

processes

The

following

process

describes

those

animals

which

are

dead

Dead

def

p

Dead

Since

we

are

using

a

synchronous

description

system

the

above

process

is

the

identity

with

respect

to

parallel

composition

as

a

result

adding

any

such

processes

to

a

population

will

have

no

effect

on

the

system

's

behaviours

sexes

occur

in

pairs

are

many

different

versions

of

the

gene

for

each

character

are

many

places

on

the

chromosomes

at

which

the

alleles

may

occur

genetic

factor

which

may

specify

or

contribute

to

the

specification

of

the

animal

-LRB-

phenotype

-RRB-

opposed

to

more

complicated

relationship

In

some

senses

these

can

be

coded

up

in

the

response

function

from

number

to

sex

and

so

we

do

not

need

to

consider

more

complicated

allelic

behaviour

A

male

process

with

genotype

x

y

which

represent

the

number

of

polyfactors

on

each

of

the

two

chromosomes

tries

to

export

its

genes

to

a

female

process

and

then

dies

Male

-LRB-

x

y

-RRB-

def

mate

-LRB-

x

y

-RRB-

Dead

:p

Dead

Obviously

the

male

animal

will

prefer

to

mate

if

it

possibly

can

and

hence

priority

is

given

to

achieving

that

end

Since

we

are

modelling

the

system

as

having

nonoverlapping

generations

the

male

will

die

unmated

if

it

can

not

mate

at

this

point

in

time

A

female

process

takes

in

the

genes

specified

by

the

male

to

produce

a

process

with

their

combined

genetic

information

Female

-LRB-

u

v

-RRB-

def

mate

-LRB-

x

y

-RRB-

P

air

-LRB-

x

y

u

v

-RRB-

:p

Dead

This

may

seem

an

unnatural

representation

of

the

actual

process

of

mating

where

the

males

genetic

information

is

divided

into

sperm

on

a

per

chromosome

basis

and

then

passed

onto

the

female

From

our

point

of

view

as

long

as

the

division

of

genetic

information

occurs

in

an

unbiased

manner

then

the

point

in

time

at

which

it

occurs

is

irrelevant

provided

it

occurs

before

the

final

allocation

of

sex

to

the

young

In

the

above

the

factors

u

and

v

represent

the

number

of

polyfactors

on

the

two

chromosomes

of

the

female

Again

female

animals

prefer

to

mate

rather

than

not

doing

so

and

if

they

can

not

then

they

will

not

influence

future

generations

The

pair

now

produces

some

number

of

offspring

which

will

later

acquire

their

genetic

type

so

at

this

point

we

retain

the

information

on

all

four

parental

P

air

-LRB-

x

y

u

v

-RRB-

def

offi

:p

Y

oung

-LRB-

x

y

u

v

-RRB-

i

In

the

above

offi

represents

the

probability

of

having

i

offspring

and

Offspring

the

set

of

permissible

values

of

number

of

young

It

is

simpler

to

model

the

production

of

a

number

of

young

with

the

full

parental

genetic

information

and

assign

an

appropriate

division

of

that

genetic

information

at

a

subsequent

point

The

young

now

acquire

their

genotype

in

the

correct

proportions

Y

oung

-LRB-

x

y

u

v

-RRB-

def

:p

Animal

-LRB-

x

u

-RRB-

:p

Animal

-LRB-

x

v

-RRB-

:p

Animal

-LRB-

y

u

-RRB-

:p

Animal

-LRB-

y

v

-RRB-

This

is

the

standard

pattern

of

inheritance

in

a

diploid

system

following

the

laws

of

Mendelian

inheritance

That

is

to

say

that

the

members

of

a

homologous

pair

assort

at

random

independently

By

splitting

mating

offspring

production

and

genetic

assignment

into

separate

stages

we

permit

the

introduction

of

genetic

based

biases

in

each

of

these

life

stages

and

therefore

could

permit

selection

to

occur

upon

any

of

them

Hence

the

process

theoretic

some

sense

we

are

performing

meiosis

on

both

parents

in

the

young

but

formally

this

is

irrelevant

abbreviation

P

i

def

P

P

P

i

times

framework

we

have

provided

can

be

exploited

for

more

general

problems

of

evolution

without

great

alteration

although

in

this

simple

system

there

is

no

selection

based

upon

genotype

Now

the

animals

have

to

eat

in

order

to

survive

Animal

-LRB-

x

y

-RRB-

def

eat

Ani

-LRB-

x

y

-RRB-

:p

Dead

This

stage

ensures

that

the

population

of

adults

remains

finite

limited

by

the

availability

of

the

food

resource

Again

we

could

model

selective

advantage

of

a

particular

genotype

in

foraging

at

this

point

but

choose

not

to

do

so

for

simplicity

in

this

analysis

Notice

that

as

a

result

of

the

priority

if

there

is

a

sufficient

population

to

take

up

all

of

the

available

resource

then

this

will

happen

Alternatively

we

have

bounded

the

population

to

the

carrying

capacity

of

its

environment

Finally

the

animal

's

sex

is

determined

Ani

-LRB-

x

y

-RRB-

def

p

Male

-LRB-

x

y

-RRB-

if

x

y

High

p

Female

-LRB-

x

y

-RRB-

if

x

y

Low

:p

Female

-LRB-

x

y

-RRB-

:p

Male

-LRB-

x

y

-RRB-

otherwise

which

allocates

animals

a

sex

on

the

basis

of

their

genotype

given

the

decision

procedure

we

outlined

above

A

process

to

provide

the

animals

with

food

F

ood

def

-LSB-

N

-RSB-

eati

F

ood

The

above

process

limits

the

total

number

of

breeding

animals

in

the

population

to

N

it

is

possible

to

have

many

more

young

in

the

system

but

their

numbers

are

reduced

by

the

limited

availability

of

this

resource

Note

that

is

p.

Letting

xi

yi

zj

wj

with

i

I

and

j

J

arbitrary

index

sets

range

over

some

intial

sex

factor

distribution

the

complete

system

is

given

by

P

oly

def

-LRB-

-LRB-

Qi

Male

-LRB-

xi

yi

-RRB-

Qj

Female

-LRB-

zj

wj

-RRB-

F

ood

-RRB-

dfpg

-RRB-

The

above

process

describes

the

intial

state

of

a

number

of

animals

under

polygenic

sex

determination

given

by

the

sex

allocation

function

we

described

earlier

It

should

be

noted

that

sex

allocation

is

the

only

point

in

the

system

at

which

the

genetic

make

up

of

the

individual

is

used

to

make

a

decision

about

its

future

All

other

decisions

survival

number

of

offspring

and

mating

success

are

unbiased

with

respect

to

genotype

Thus

the

system

must

be

selectively

neutral

other

than

with

respect

to

an

animal

's

sex

Solutions

with

Deterministic

Allocation

Function

With

the

above

allocation

function

the

system

P

oly

will

stabilise

in

one

of

the

following

three

states

Number

of

sex

factors

Low

High

Number

of

animals

Figure

Location

of

type

solution

Type

In

this

case

there

is

no

genetic

sexual

differentiation

all

of

the

animals

have

the

same

number

of

sex

factors

on

both

chromosomes

The

animals

are

-LRB-

a

a

-RRB-

with

Low

a

a

High

In

terms

of

the

allocation

function

all

of

the

animals

are

in

the

undetermined

region

see

Figure

If

there

is

an

ESD

modifier

in

this

case

then

it

will

affect

of

the

animals

Notice

that

this

achieves

-LRB-

on

average

-RRB-

a

sex

ratio

Type

In

this

case

the

sex

of

the

animals

is

entirely

determined

by

genotype

Depending

upon

the

precise

values

of

Low

and

High

and

the

maximum

number

of

copies

on

either

chromosome

the

system

becomes

either

an

XY

or

a

ZW

genetic

sex

determination

-LRB-

male

or

female

heterogamety

-RRB-

In

the

former

case

females

are

-LRB-

a

a

-RRB-

and

males

are

-LRB-

a

b

-RRB-

with

a

a

Low

High

a

b

shown

pictorially

in

Figure

Alternatively

the

system

can

evolve

into

a

ZW

state

In

this

case

female

animals

are

-LRB-

a

b

-RRB-

and

male

animals

are

-LRB-

b

b

-RRB-

with

a

b

Low

High

b

b.

Pictorially

the

distribution

of

animals

is

given

in

Figure

In

either

case

the

number

of

animals

of

each

sex

is

half

of

the

total

population

Type

In

this

case

there

is

a

mixture

between

animals

whose

sex

is

determined

genetically

and

those

whose

sex

is

determined

randomly

In

this

case

females

have

genotypes

-LRB-

a

a

-RRB-

or

-LRB-

a

b

-RRB-

and

males

have

genotype

-LRB-

a

b

-RRB-

or

-LRB-

b

b

-RRB-

with

a

a

Low

a

b

High

b

b.

Pictorially

this

genotype

distribution

is

given

in

Figure

Number

of

sex

factors

Low

High

Number

of

animals

a

b

Figure

Location

of

type

solution

for

XY

Number

of

sex

factors

Low

High

Number

of

animals

a

b

Figure

Location

of

type

solution

for

ZW

Number

of

sex

factors

Low

High

Number

of

animals

a

b

Figure

Location

of

type

solutions

Again

this

achieves

a

sex

ratio

However

the

populations

of

each

genotype

are

not

in

the

standard

ratio

see

Section

for

details

There

are

several

observations

that

can

be

made

about

special

cases

for

the

limits

Low

and

High

and

their

effects

on

the

solution

classes

possible

If

Low

High

-LRB-

for

example

Low

and

High

see

also

-LSB-

-RSB-

-RRB-

that

is

the

boundaries

are

adjacent

then

there

is

no

space

for

type

or

solutions

to

occur

and

we

have

only

XY

and

ZW

type

solutions

occuring

Similarly

if

Low

High

and

Low

is

an

odd

number

-LRB-

for

example

Low

and

High

-RRB-

then

there

can

be

no

Type

solutions

Assuming

that

there

is

some

maximum

number

of

copies

-LRB-

max

-RRB-

of

the

polygene

possible

on

either

chromosome

in

the

system

and

that

High

-LRB-

Low

-RRB-

max

-LRB-

for

example

max

Low

and

High

-RRB-

then

there

can

be

no

Type

XY

solutions

Since

there

is

no

choice

for

the

b

value

chromosome

since

b

with

b

a

How

stability

arises

-LRB-

Informally

-RRB-

Of

note

is

the

longstanding

observation

that

sexual

animals

should

allocate

the

sex

of

their

offspring

on

a

basis

-LSB-

-RSB-

This

results

from

the

observation

that

if

the

population

has

a

biased

ratio

in

favour

of

either

sex

then

an

animal

can

ensure

a

greater

proportion

of

its

genes

will

be

present

in

the

population

in

generations

time

by

producing

more

of

the

rarer

sex

The

stable

solutions

of

the

system

arise

as

the

result

of

the

interaction

of

two

forces

preciseley

equal

resources

should

be

invested

in

each

sex

ffl

the

rare

female

effect

-LSB-

-RSB-

ffl

probabilistic

symmetry

breaking

The

first

force

is

the

result

of

departure

from

Fisherian

-LSB-

-RSB-

sex

ratio

If

this

occurs

then

the

rarer

sex

will

get

to

breed

more

frequently

thus

increasing

its

numbers

in

the

subsequent

generations

The

second

force

results

from

the

absence

of

selection

Probabilistic

drift

will

cause

asymmetries

in

the

numbers

of

factors

on

particular

chromosomes

within

the

population

to

arise

In

the

absence

of

any

sexual

-LRB-

Fisherian

-RRB-

selection

the

system

can

continue

to

drift

As

a

result

of

feedback

from

culling

small

biases

in

population

numbers

of

particular

gene

counts

tend

to

become

exaggerated

and

eventually

some

gene

counts

will

be

eliminated

Only

if

there

is

some

selective

advantage

in

keeping

a

particular

gene

count

will

it

be

retained

Thus

the

system

attempts

to

reduce

to

the

smallest

number

of

different

counts

Notice

that

for

all

of

our

solutions

if

at

this

point

any

gene

count

's

frequency

is

reduced

then

the

population

's

sex

ratio

will

depart

from

Fisher

and

rarity

effects

will

restore

that

particular

gene

count

's

frequency

in

the

population

The

precise

solution

that

is

reached

depends

on

which

gene

counts

are

drifted

out

in

the

early

period

of

the

system

For

instance

if

gene

count

is

removed

then

the

system

can

not

reach

a

solution

based

upon

Counts

are

removed

at

random

until

some

selective

pressure

is

encountered

at

this

point

those

that

give

a

Fisherian

ratio

are

retained

and

those

that

do

not

are

eventually

lost

Whilst

we

can

predict

that

the

system

eventually

arrives

in

one

of

our

three

solution

types

it

is

extremely

difficult

-LRB-

and

highly

parameter

dependent

-RRB-

to

demonstrate

which

solutions

it

will

arrive

at

with

what

probability

Stability

-LRB-

Formally

-RRB-

In

order

to

demonstrate

that

the

system

will

eventually

reach

one

of

the

stable

states

stated

above

we

have

to

demonstrate

two

facts

-LSB-

-RSB-

the

sets

of

states

given

by

those

situations

are

themselves

closed

-LRB-

stable

-RRB-

from

an

arbitrary

state

one

or

more

of

the

stable

sets

is

reachable

Whilst

this

will

demonstrate

the

eventual

solutions

of

the

system

are

stable

it

can

not

tell

us

with

what

frequency

each

of

the

solution

types

will

arise

We

proceed

by

establishing

the

first

fact

Proposition

The

following

sets

of

states

are

closed

-LRB-

with

i

j

k

l

m

and

n

free

in

the

positive

naturals

-RRB-

With

Low

High

f

-LRB-

-LRB-

Qi

Male

-LRB-

a

a

-RRB-

Qj

Female

-LRB-

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

P

air

-LRB-

a

a

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Y

oung

-LRB-

a

a

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Animal

-LRB-

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Ani

-LRB-

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

g

With

Low

and

High

a

b

f

-LRB-

-LRB-

Qi

Male

-LRB-

b

a

-RRB-

Qj

Female

-LRB-

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

P

air

-LRB-

b

a

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Y

oung

-LRB-

b

a

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Animal

-LRB-

a

a

-RRB-

Qj

Animal

-LRB-

b

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Ani

-LRB-

a

a

-RRB-

Qj

Ani

-LRB-

b

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

g

symmetrically

for

a

b

Low

and

High

With

Low

a

b

High

f

-LRB-

-LRB-

Qi

Female

-LRB-

a

a

-RRB-

Qj

Female

-LRB-

a

b

-RRB-

Qk

Male

-LRB-

a

b

-RRB-

Ql

Female

-LRB-

b

a

-RRB-

QmMale

-LRB-

b

a

-RRB-

Qn

Male

-LRB-

b

b

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

P

air

-LRB-

b

b

a

a

-RRB-

Qj

P

air

-LRB-

b

b

a

b

-RRB-

Qk

P

air

-LRB-

b

a

a

a

-RRB-

Ql

P

air

-LRB-

b

a

a

b

-RRB-

Qm

P

air

-LRB-

a

b

a

a

-RRB-

Qn

P

air

-LRB-

a

b

b

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Y

oung

-LRB-

b

b

a

a

-RRB-

Qj

Y

oung

-LRB-

b

b

a

b

-RRB-

Qk

Y

oung

-LRB-

b

a

a

a

-RRB-

Ql

Y

oung

-LRB-

b

a

a

b

-RRB-

Qm

Y

oung

-LRB-

a

b

a

a

-RRB-

Qm

Y

oung

-LRB-

a

b

b

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Ani

-LRB-

a

a

-RRB-

Qj

Ani

-LRB-

a

b

-RRB-

Qk

Ani

-LRB-

b

a

-RRB-

Ql

Ani

-LRB-

b

b

-RRB-

F

ood

-RRB-

dfpg

-RRB-

g

Proof

For

the

case

above

it

is

straightforward

to

demonstrate

closure

from

the

definition

of

the

processes

For

the

second

and

third

cases

there

is

a

problem

as

to

why

they

do

not

become

a

system

of

type

It

is

certainly

possible

for

the

systems

to

reach

at

least

one

of

the

following

states

-LRB-

-LRB-

Qi

Ani

-LRB-

a

a

-RRB-

F

ood

-RRB-

dfpg

-RRB-

-LRB-

-LRB-

Qi

Ani

-LRB-

b

b

-RRB-

F

ood

-RRB-

dfpg

-RRB-

In

which

case

the

system

is

now

apparently

of

type

It

should

be

noted

that

if

either

of

these

cases

occur

then

there

will

be

no

more

males

-LRB-

females

-RRB-

to

pair

with

and

the

population

will

go

extinct

The

probability

of

this

event

is

comparable

to

that

for

spontaneous

as

a

result

of

all

offspring

being

one

sex

-LRB-

a

possibility

for

all

of

our

three

systems

-RRB-

Thus

it

is

not

the

case

that

the

system

can

lose

either

of

the

polygenes

without

the

population

going

extinct

the

process

becoming

equivalent

to

our

Dead

process

Hence

we

can

see

that

the

second

and

third

types

are

stable

in

the

same

sense

as

the

first

as

long

as

spontaneous

extinction

as

a

result

of

monosexuality

is

avoided

It

is

this

observation

that

helps

in

resolving

the

issues

of

reachability

To

demonstrate

the

inevitability

of

reaching

one

of

our

three

stable

states

we

observe

from

the

proof

above

that

particular

gene

counts

will

only

be

retained

if

they

are

necessary

some

senses

we

are

making

a

distinction

between

very

long

lived

transients

and

short

lived

ones

There

is

an

ultimate

closed

set

of

the

singleton

Dead

but

the

time

scale

to

reach

it

is

many

orders

of

magnitude

greater

than

evolutionary

time

Since

it

results

from

all

surviving

offspring

in

a

generation

guessing

their

sex

incorrectly

it

is

a

very

low

probability

event

Thus

we

distinguish

between

stable

sets

which

are

persistant

on

this

time

scale

from

others

for

the

maintenance

of

the

presence

of

both

sexes

Hence

in

a

complex

system

where

many

gene

counts

are

present

there

is

a

non-zero

probability

that

any

particular

count

can

be

lost

either

through

the

parent

failing

to

mate

or

its

offspring

having

the

count

failing

to

survive

this

process

will

continue

until

one

of

the

three

states

above

is

reached

It

is

sufficient

to

demonstrate

the

effect

of

adding

a

single

alternative

count

into

the

above

three

systems

and

consider

its

consequences

for

the

presences

of

both

males

and

females

to

observe

the

non-existence

of

any

other

stable

solutions

Proposition

The

addition

of

animals

with

a

new

factor

count

to

any

of

the

stable

states

is

not

a

stable

state

and

the

system

subsequently

evolves

to

a

stable

state

Proof

We

consider

each

of

the

three

types

in

turn

Consider

a

type

with

Low

High

and

an

extra

factor

c

then

we

have

animals

of

type

a

c

and

there

are

two

cases

to

consider

-LRB-

a

-RRB-

c

a

then

since

a

c

and

fall

into

either

the

female

only

or

the

regions

we

can

lose

the

c

gene

without

affecting

sex

ratio

if

is

in

the

random

region

then

we

can

lose

the

a

gene

but

will

still

have

a

type

solution

but

with

c

in

place

of

a.

Finally

if

both

a

c

and

lie

below

Low

then

we

can

not

lose

the

a

gene

as

it

is

the

only

source

of

males

but

the

system

can

lose

the

c

without

such

problems

-LRB-

b

-RRB-

c

a

as

above

if

High

then

c

can

replace

a

otherwise

c

will

inevitably

be

lost

Hence

the

system

will

always

stay

as

type

it

should

be

noted

that

the

probability

of

losing

a

gene

from

the

system

in

competition

is

many

orders

of

magnitude

greater

than

that

of

spontaneous

extinction

Consider

an

XY

system

with

Low

and

High

a

b

and

a

competitor

gene

c

then

there

are

three

basic

cases

-LRB-

a

-RRB-

c

a

in

this

case

animals

will

always

be

female

if

b

c

High

then

the

a

or

c

genes

are

interchangable

and

the

system

will

become

a

b

or

c

b

XY

failing

that

c

can

be

evolved

out

but

a

and

b

can

not

so

the

system

becomes

a

b

XY

-LRB-

b

-RRB-

a

c

b

if

Low

then

as

above

similarly

High

see

-LRB-

c

-RRB-

below

So

if

Low

High

then

the

system

can

lose

both

a

b

to

become

a

type

system

or

lose

the

c

to

become

type

In

this

case

there

is

no

evolutionary

advantage

and

the

timescale

will

be

that

of

symmetry

breaking

-LRB-

c

-RRB-

b

c

by

an

identical

argument

to

that

for

c

a

and

results

in

either

an

a

c

or

a

b

type

system

and

there

is

no

need

to

retain

all

three

genes

counts

Type

with

Low

a

b

High

and

introducing

a

new

value

c

we

have

three

cases

-LRB-

a

-RRB-

c

a

if

Low

c

b

High

then

we

can

get

c

b

type

otherwise

the

system

must

evolve

to

a

b

Type

-LRB-

b

-RRB-

a

c

b

if

Low

and

Low

c

b

High

then

it

can

evolve

to

either

a

b

or

c

b

type

alternatively

if

c

b

High

then

it

can

become

either

a

b

type

or

c

b

XY

Similarly

if

High

and

Low

a

c

High

then

it

can

evolve

to

a

c

or

a

b

type

or

if

a

c

Low

it

can

become

a

c

ZW

or

a

b

type

If

Low

High

then

we

have

a

competition

between

c

c

type

or

a

b

type

but

the

system

will

eventually

reach

one

of

them

-LRB-

c

-RRB-

b

c

if

Low

a

c

High

then

it

can

evolve

to

a

c

or

a

b

type

otherwise

it

will

become

a

b

type

The

basic

method

of

system

evolution

is

to

lose

particular

gene

counts

until

a

point

is

reached

at

which

selection

for

the

rarer

sex

forces

the

sex

ratio

to

equality

Those

genes

that

do

not

aid

in

the

maintenance

of

equal

sex

ratio

are

evolved

out

otherwise

there

is

a

competition

between

essentially

equivalent

genes

which

will

be

resolved

by

symmetry

breaking

That

is

once

one

of

the

competitors

gets

an

advantage

then

it

is

likely

to

increase

that

advantage

in

the

next

generation

until

it

reaches

fixation

This

process

is

very

fast

in

comparison

with

that

of

spontaneous

extinction

In

a

particular

competitive

circumstance

the

actual

resolution

will

depend

upon

the

initial

frequencies

of

the

various

gene

counts

This

is

of

particular

relevance

when

considering

the

effect

of

mutation

or

recombination

upon

the

stable

states

of

the

system

In

particular

if

mutants

arise

infrequently

or

on

becoming

dominant

would

lead

to

a

non-Fisherian

sex

ratio

then

they

are

highly

unlikely

to

become

common

within

a

stable

population

Note

that

in

the

above

type

solutions

in

competition

will

often

be

changed

into

type

or

and

thus

we

could

expect

to

find

relatively

few

of

these

solutions

Genotype

Frequencies

in

Type

Earlier

we

stated

that

the

distribution

of

genotypes

in

a

type

system

was

not

the

standard

ratio

In

this

section

we

calculate

the

stable

genotype

frequencies

assuming

that

animals

have

to

occur

in

male/female

pairs

to

breed

From

the

type

system

we

have

that

females

are

-LRB-

a

a

-RRB-

or

-LRB-

a

b

-RRB-

and

males

are

-LRB-

a

b

-RRB-

or

-LRB-

b

b

-RRB-

We

can

assume

from

symmetry

that

the

proportions

of

-LRB-

a

a

-RRB-

and

-LRB-

b

b

-RRB-

animals

must

be

the

same

Hence

we

proceed

by

assuming

that

a

proportion

of

males

are

-LRB-

a

b

-RRB-

and

that

of

males

are

-LRB-

b

b

-RRB-

and

for

females

are

-LRB-

a

b

-RRB-

and

that

of

females

are

-LRB-

a

a

-RRB-

tracing

the

breeding

ratios

-LRB-

we

label

the

axes

with

the

frequency

of

males

or

females

of

that

type

-RRB-

Hence

the

gene

frequencies

of

the

offspring

are

as

follows

Number

of

sex

factors

Probability

of

being

male

max

Figure

The

result

of

breeding

two

polygenic

shrimp

females

-LRB-

a

a

-RRB-

-LRB-

a

b

-RRB-

males

-LRB-

a

b

-RRB-

-LRB-

a

a

-RRB-

-LRB-

a

a

-RRB-

-LRB-

a

b

-RRB-

-LRB-

a

b

-RRB-

-LRB-

b

b

-RRB-

-LRB-

b

b

-RRB-

-LRB-

a

b

-RRB-

-LRB-

a

b

-RRB-

-LRB-

b

b

-RRB-

The

above

system

is

stable

when

the

resultant

number

of

a

genotype

remains

the

same

in

subsequent

generations

Solving

for

genotype

-LRB-

a

b

-RRB-

we

obtain

the

equation

which

simplified

with

the

eqaution

gives

us

the

quadratic

equation

which

has

a

solution

in

-LSB-

-RSB-

of

-LRB-

or

approximately

-RRB-

In

the

above

eqautions

setting

and

does

not

give

a

stable

solution

So

the

ratio

of

genotypes

in

type

is

or

approximately

Exactly

Allocation

with

Probabilistic

Determination

In

this

case

we

assume

that

the

probability

of

being

male

is

a

linear

function

of

the

total

number

of

sex

factors

carried

-LRB-

see

Figure

-RRB-

Formally

this

is

represented

by

changing

the

definition

of

the

state

Animal

-LRB-

x

y

-RRB-

to

the

following

Animal

-LRB-

x

y

-RRB-

def

-LRB-

x

y

-RRB-

:p

Male

-LRB-

x

y

-RRB-

-LRB-

-LRB-

x

y

-RRB-

-RRB-

:p

Female

-LRB-

x

y

-RRB-

Whilst

leaving

the

definitions

of

the

other

states

unchanged

In

this

case

we

lose

the

Type

solutions

of

the

deterministic

function

and

only

retain

the

Type

and

solutions

Initially

one

might

assume

that

all

solutions

should

be

of

Type

but

initially

the

sex

bias

selection

pressure

is

very

low

so

the

genes

necessary

for

a

Type

solution

-LRB-

for

example

with

max

then

or

-RRB-

could

have

been

lost

by

probabilistic

drift

In

this

case

the

best

solution

the

system

can

adopt

is

a

Type

with

a

sex

ratio

as

close

to

Fisher

as

possible

In

the

latter

case

as

above

we

do

not

generally

observe

population

ratios

of

Determining

population

makeup

In

the

case

of

the

Type

solution

we

give

some

gene

flow

equations

to

calculate

the

stable

population

Take

the

following

collection

of

variables

ffl

x

the

proportion

of

-LRB-

a

a

-RRB-

animals

in

the

population

ffl

y

the

proportion

of

-LRB-

a

b

-RRB-

animals

in

the

population

ffl

z

the

proportion

of

-LRB-

b

b

-RRB-

animals

in

the

population

ffl

fx

the

probability

of

an

-LRB-

a

a

-RRB-

animal

being

female

ffl

fy

the

probability

of

an

-LRB-

a

b

-RRB-

animal

being

female

ffl

fz

the

probability

of

an

-LRB-

b

b

-RRB-

animal

being

female

We

define

mx

fx

and

my

mz

respectively

Then

define

the

proportion

of

females

f

in

the

population

to

be

f

xfx

yfy

zfz

and

the

proportion

of

males

m

in

the

population

to

be

m

xmx

ymy

zmz

Consider

the

matings

that

lead

to

an

-LRB-

a

a

-RRB-

animal

This

can

be

the

result

of

-LRB-

a

a

-RRB-

-LRB-

a

a

-RRB-

or

-LRB-

a

a

-RRB-

-LRB-

a

b

-RRB-

or

-LRB-

a

b

-RRB-

-LRB-

a

b

-RRB-

matings

So

the

proportion

of

-LRB-

a

a

-RRB-

animals

in

the

next

generation

is

the

probability

of

an

-LRB-

a

a

-RRB-

-LRB-

a

a

-RRB-

mating

times

the

probability

it

results

in

an

-LRB-

a

a

-RRB-

animal

plus

the

probability

of

an

-LRB-

a

a

-RRB-

-LRB-

a

b

-RRB-

mating

times

the

probability

it

results

in

an

-LRB-

a

a

-RRB-

animal

plus

the

probability

of

an

-LRB-

a

b

-RRB-

-LRB-

a

b

-RRB-

mating

times

the

probability

it

results

in

an

-LRB-

a

a

-RRB-

animal

So

a

similar

chart

to

our

earlier

one

for

type

solutions

in

the

determined

systems

would

be

females

-LRB-

a

a

-RRB-

xfxf

-LRB-

a

b

-RRB-

yfyf

-LRB-

b

b

-RRB-

zfzf

males

-LRB-

a

a

-RRB-

xmx

m

-LRB-

a

a

-RRB-

xmx

m

xfxf

-LRB-

a

a

-RRB-

xmx

m

yfyf

-LRB-

a

b

-RRB-

xmx

m

zfzf

-LRB-

a

b

-RRB-

xmx

m

yfyf

-LRB-

a

b

-RRB-

ymy

m

-LRB-

a

a

-RRB-

ymy

m

xfxf

-LRB-

a

a

-RRB-

ymy

m

yfyf

-LRB-

a

b

-RRB-

ymy

m

zfzf

-LRB-

a

b

-RRB-

ymy

m

xfxf

-LRB-

a

b

-RRB-

ymy

m

yfyf

-LRB-

b

b

-RRB-

ymy

m

zfzf

-LRB-

b

b

-RRB-

ymy

m

yfyf

-LRB-

b

b

-RRB-

zmz

m

-LRB-

a

b

-RRB-

zmz

m

xfxf

-LRB-

a

b

-RRB-

zmz

m

yfyf

-LRB-

b

b

-RRB-

zmz

m

zfzf

-LRB-

b

b

-RRB-

zmz

m

yfyf

We

can

generate

three

equations

in

our

variables

x

y

and

z

which

will

hold

when

the

genotype

proportions

are

stable

from

the

above

matrix

x

xmx

m

xfxf

xmx

m

yfyf

ymy

m

xfxf

ymy

m

yfyf

y

xmx

m

zfzf

xmx

m

yfyf

ymy

m

zfzf

ymy

m

xfxf

ymy

m

yfyf

zmz

m

xfxf

zmz

m

yfyf

z

ymy

m

zfzf

ymy

m

yfyf

zmz

m

zfzf

zmz

m

yfyf

Unfortunately

the

above

equations

are

too

complicated

to

solve

in

general

For

illustration

we

provide

some

numerical

solutions

to

these

equations

for

particular

values

of

fx

fy

and

fz

fx

fy

fz

x

y

z

From

the

above

table

it

is

clear

that

populations

with

equilibriums

far

away

from

that

of

Hardy-Weinberg

are

possible

The

Effect

of

Recombination

For

our

purposes

we

take

recombination

as

follows

Given

an

animal

with

genotype

-LRB-

a

b

-RRB-

then

produce

an

animal

with

genotype

-LRB-

c

d

-RRB-

with

c

linearly

distributed

in

the

range

to

a

b

and

d

-LRB-

a

b

-RRB-

c

We

assume

a

variable

rate

at

which

this

process

occurs

in

animals

Since

sex

is

determined

by

the

total

number

of

sex

factors

which

is

left

invariant

by

the

above

we

can

impose

recombination

at

any

stage

in

our

system

as

it

will

be

independent

of

sex

choice

and

gene

selection

Formally

the

presence

of

recombination

in

our

evolving

system

can

be

represented

by

the

following

processes

RMale

-LRB-

x

y

-RRB-

def

mate

-LRB-

x

y

-RRB-

Dead

:p

Dead

RFemale

-LRB-

z

w

-RRB-

def

mate

-LRB-

x

y

-RRB-

RP

air

-LRB-

x

y

z

w

-RRB-

:p

Dead

RP

air

-LRB-

x

y

z

w

-RRB-

def

wi

:p

RY

oung

-LRB-

x

y

z

w

-RRB-

i

RY

oung

-LRB-

x

y

z

w

-RRB-

def

:p

RAnimal

-LRB-

x

z

-RRB-

:p

RAnimal

-LRB-

x

w

-RRB-

:p

RAnimal

-LRB-

y

z

-RRB-

:p

RAnimal

-LRB-

y

w

-RRB-

RAnimal

-LRB-

x

y

-RRB-

def

eat

RAniR

-LRB-

x

y

-RRB-

:p

Dead

RAni

-LRB-

x

y

-RRB-

def

p

RMale

-LRB-

x

y

-RRB-

if

x

y

High

p

RFemale

-LRB-

x

y

-RRB-

if

x

y

Low

:p

RFemale

-LRB-

x

y

-RRB-

:p

RMale

-LRB-

x

y

-RRB-

otherwise

In

the

above

the

process

RaniR

has

yet

to

be

defined

and

we

do

so

below

RAniR

-LRB-

x

y

-RRB-

def

norec

:p

p

RAni

-LRB-

x

y

-RRB-

rec

:p

RecAni

-LRB-

x

y

-RRB-

RecAni

-LRB-

x

y

-RRB-

def

-LSB-

x

y

-RSB-

:p

RAni

-LRB-

i

-LRB-

x

y

-RRB-

i

-RRB-

In

the

above

the

variables

norec

and

rec

express

the

probability

that

recombination

will

occur

in

this

animal

The

process

RecAni

performs

the

recombination

it

is

at

this

point

that

differences

emerge

between

our

polyfactorial

system

and

one

dependent

on

If

the

animal

does

n't

recombine

then

we

need

to

wait

an

extra

tick

to

allow

the

system

to

resynchronise

Recombination

allows

gene

counts

that

have

been

lost

to

be

reintroduced

it

is

a

mechanism

for

avoiding

the

consequences

of

bad

choices

-LRB-

Muller

's

ratchet

-LSB-

-RSB-

-RRB-

Unfortunately

for

an

analysis

of

the

stable

solutions

of

our

system

recombination

is

a

major

challenge

The

behaviour

of

the

system

is

given

by

a

dynamic

competition

between

drift

removing

rare

genotypes

and

recombination

reintroducing

them

These

rates

are

very

difficult

to

estimate

and

are

highly

dependent

on

the

parameter

values

Essentially

the

rate

of

drift

is

a

function

of

effective

population

size

and

the

recombination

rate

is

dependent

on

the

degree

of

linkage

of

the

genes

Traditional

analyses

of

these

complex

systems

assume

that

linkage

is

low

-LSB-

-RSB-

however

other

authors

-LSB-

-RSB-

suggest

that

linkage

between

related

genes

may

be

very

high

However

there

are

some

observations

that

we

can

make

dependent

on

the

relative

speed

of

the

underlying

rates

in

the

system

ffl

Recombination

very

much

slower

than

drift

In

this

case

our

system

will

behave

as

before

It

will

reach

an

essentially

stable

solution

which

will

be

characterised

by

brief

departures

from

it

as

a

result

of

a

recombination

event

It

is

extremely

unlikely

that

the

system

's

nature

will

change

a

multi-factor

system

an

appropriate

recombination

rule

would

be

RecAni

-LRB-

x

y

-RRB-

def

:p

Rani

-LRB-

x

y

-RRB-

:p

Rani

-LRB-

y

x

-RRB-

effectively

swapping

the

genetic

material

from

the

mother

and

father

This

form

of

recombination

would

have

no

effect

on

our

conclusions

ffl

Recombination

and

drift

rates

are

comparable

In

this

case

the

system

will

tend

to

be

constantly

in

flux

between

different

possible

stable

solutions

occasionally

residing

in

one

for

a

period

before

moving

off

to

another

ffl

Recombination

faster

than

drift

The

system

never

stabilises

but

the

observed

distribution

of

polyfactors

is

highly

non-Guassian

-LSB-

-RSB-

The

effect

of

sex

selection

forbids

certain

states

and

the

system

will

tend

to

avoid

them

Essentially

recombination

has

an

effect

as

structure

starts

to

emerge

in

the

system

If

it

is

strong

enough

to

break

down

the

structure

then

it

will

probably

never

emerge

otherwise

the

structures

we

see

can

emerge

despite

the

presence

of

the

recombination

In

simulo

-LSB-

-RSB-

rates

of

recombination

of

much

greater

than

in

a

population

of

animals

caused

the

system

to

start

to

oscillate

and

recombinant

rates

of

the

order

of

cause

a

complete

loss

of

any

structure

Note

the

system

does

not

go

to

a

normal

distribution

it

becomes

highly

variable

exhibiting

almost

no

structural

properties

in

a

sense

it

is

completely

noise

dominated

Multiple

Chromosomes

We

start

be

analysing

the

system

when

the

polygenic

factors

are

distributed

over

two

chromosomes

This

extension

of

our

earlier

system

requires

that

we

track

factor

counts

rather

than

representing

the

counts

on

each

of

the

chromosomes

A

male

process

with

genotype

tries

to

export

its

genes

to

a

female

process

and

then

dies

-LRB-

-RRB-

def

mate

-LRB-

-RRB-

Dead

:p

Dead

We

abbreviate

the

formal

parameter

list

by

p

A

female

process

takes

in

the

genes

specified

by

the

male

to

produce

a

process

with

their

combined

genetic

information

-LRB-

-RRB-

def

mate

-LRB-

-RRB-

air

-LRB-

p

-RRB-

:p

Dead

The

pair

now

produces

some

number

of

offspring

which

will

acquire

their

genetic

type

in

the

correct

proportions

from

their

parents

air

-LRB-

p

-RRB-

def

offi

:p

oung

-LRB-

p

-RRB-

i

In

the

above

offi

represents

the

probability

of

having

i

offspring

and

Offspring

the

set

or

permissible

values

of

number

of

young

The

young

now

acquire

their

genotype

in

the

correct

proportions

oung

-LRB-

p

-RRB-

def

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

:p

-LRB-

-RRB-

This

is

the

standard

pattern

of

inheritance

in

a

diploid

system

Now

the

animals

have

to

eat

in

order

to

survive

again

we

abbreviate

the

formal

parameter

list

by

a

-LRB-

-RRB-

def

eat

-LRB-

a

-RRB-

:p

Dead

Finally

the

animals

sex

is

determined

we

abbreviate

as

oea

-LRB-

a

-RRB-

def

p

-LRB-

a

-RRB-

if

oea

High

p

-LRB-

a

-RRB-

if

oea

Low

:p

-LRB-

a

-RRB-

:p

-LRB-

a

-RRB-

otherwise

Letting

with

i

I

and

j

J

arbitrary

indexing

sets

range

over

some

initial

sex

factor

distribution

the

complete

system

is

given

by

oly

def

-LRB-

-LRB-

Qi

-LRB-

-RRB-

Qj

-LRB-

-RRB-

F

ood

-RRB-

dfpg

-RRB-

The

system

oly

is

the

same

as

the

system

P

oly

in

that

the

sex

decision

process

and

all

other

evolutions

are

identical

excepting

that

the

polygenic

factors

are

now

distributed

over

chromosome

pairs

Stable

solutions

with

Pairs

Observing

how

the

stable

solutions

arose

in

our

earlier

system

we

can

envisage

that

with

chromosomes

present

the

system

will

evolve

by

losing

factors

until

a

point

is

reached

at

which

the

factors

have

to

be

retained

in

order

that

the

system

does

not

go

extinct

The

first

observation

is

that

the

two

chromosome

pairs

evolve

independently

There

is

no

communication

between

them

Hence

there

can

be

no

selection

for

preservation

of

structure

between

the

chromosomes

Therefore

any

stable

solution

for

both

chromosomes

must

be

stable

with

respect

to

each

pair

separately

as

no

exchange

of

factors

between

them

is

possible

That

is

to

say

chromosomes

belonging

to

different

pairs

do

not

recombine

together

So

as

candidate

stable

solutions

we

can

consider

the

systems

we

observed

earlier

noting

that

we

can

perform

the

proof

of

stability

on

each

of

the

chromosomes

separately

as

we

did

before

This

gives

rise

to

possible

combinations

of

pairs

-LRB-

since

order

is

unimportant

-RRB-

Type

Type

in

this

case

we

have

and

on

the

two

chromosomes

and

given

Low

-LRB-

-RRB-

High

the

system

will

be

stable

Type

Type

in

this

case

we

have

and

so

if

-LRB-

-RRB-

Low

and

High

the

is

stable

Type

Type

in

this

case

we

have

and

and

as

long

as

-LRB-

-RRB-

Low

High

-LRB-

-RRB-

the

system

will

be

stable

Type

Type

in

this

case

we

have

and

factors

present

in

the

system

Let

si

and

li

ai

bi

i

then

-LRB-

WLOG

-RRB-

there

are

two

potentially

stable

cases

-LRB-

a

-RRB-

and

less

than

Low

as

well

as

and

greater

than

High

if

becomes

then

we

still

get

both

type

below

and

above

the

limits

respectively

thus

maintaining

the

sex

ratio

So

in

this

case

the

pair

that

does

not

change

the

position

of

the

other

values

with

respect

to

the

limits

can

be

evolved

out

to

a

Type

Type

-LRB-

b

-RRB-

Low

High

now

if

either

of

the

pairs

loses

one

of

the

gene

counts

then

one

of

the

determined

regions

will

be

lost

So

given

that

of

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

can

form

a

type

solution

then

the

above

will

be

stable

This

is

the

only

new

form

of

solution

we

observe

and

gives

rise

to

up

to

genotypes

we

leave

an

analysis

of

the

population

genetics

of

this

system

for

later

as

the

system

is

highly

unlikely

to

evolve

in

practice

Type

Type

as

above

we

have

and

modifying

the

small

medium

and

large

values

in

the

type

solution

-LRB-

and

-RRB-

In

this

case

there

are

two

arrangements

either

there

is

no-overlap

between

the

two

versions

of

the

type

three

and

they

lie

outside

the

boundaries

in

which

case

the

system

will

become

a

Type

Type

of

either

and

or

and

Alternatively

the

two

sets

overlap

then

if

Low

High

in

which

case

either

of

two

Type

Type

will

evolve

Otherwise

the

system

will

evolve

to

Type

Type

may

seem

possible

to

lose

the

if

Low

and

High

but

there

is

no

other

alternative

gene

to

go

on

chromosome

pair

one

so

will

be

maintained

In

the

presence

of

mutation

or

recombination

however

the

information

on

pair

one

could

be

lost

Type

Type

the

system

will

evolve

to

Type

Type

unless

the

following

pattern

occurs

defining

si

mi

ai

bi

and

li

then

if

Low

High

in

which

case

the

differences

in

and

are

irrelevant

and

they

can

be

evolved

out

leading

to

a

Type

Type

There

is

a

symmetric

case

when

pair

dominates

At

this

stage

we

have

identified

essentially

four

types

of

solutions

our

previous

three

solution

types

and

a

new

version

of

the

mixed

type

in

which

there

are

four

defining

factors

on

two

chromosome

pairs

By

the

same

arguments

used

to

demonstrate

that

our

previous

three

solutions

were

the

only

ones

possible

in

a

single

chromosome

system

we

could

demonstrate

that

the

above

are

the

only

stable

solutions

for

chromosome

pairs

More

than

Chromosomes

As

we

in

Section

introducing

an

extra

chromosome

pair

into

a

system

generally

leads

to

the

information

on

one

of

the

pairs

being

lost

If

we

consider

crossing

a

further

chromosome

pair

with

the

solutions

above

then

we

obtain

no

new

stable

solutions

and

this

continues

as

we

consider

crossing

over

as

many

extra

chromosomes

as

we

wish

to

add

Essentially

all

but

one

-LRB-

occasionally

two

-RRB-

of

the

chromosome

pairs

can

be

treated

as

an

additive

constant

which

move

the

limits

appropriately

for

the

lead

sex

determining

chromosome

pair

Thus

if

the

polygenes

are

distributed

over

many

chromosomes

we

would

expect

to

see

one

of

the

four

following

outcomes

ffl

All

but

two

of

the

chromosome

pairs

have

identical

counts

of

the

polygenes

and

the

last

two

are

arranged

so

that

the

totals

over

all

the

pairs

are

as

ffl

All

but

one

of

the

chromosome

pairs

have

identical

numbers

and

one

pair

is

arranged

as

type

or

as

for

a

single

chromosome

pair

Conclusions

We

have

presented

a

polygenic

system

which

can

evolve

in

the

absence

of

selection

into

one

of

three

basic

types

when

present

on

a

single

chromosome

pair

The

first

two

of

these

three

solutions

represent

well

known

forms

of

sex

determination

The

first

is

where

sex

is

determined

with

respect

to

environmental

cues

alone

-LRB-

ESD

-LSB-

-RSB-

-RRB-

The

second

is

where

sex

is

purely

genetically

determined

the

system

evolving

to

one

of

two

states

where

sex

is

determined

according

to

the

genotype

based

on

one

chromosome

pair

-LRB-

sex

chromosomes

-LSB-

-RSB-

-RRB-

These

states

-LRB-

reviewed

in

-LSB-

-RSB-

-RRB-

are

male

heterogamety

-LRB-

male

is

XY

female

is

XX

as

in

mammals

-RRB-

and

female

heterogamety

-LRB-

female

is

ZW

male

is

ZZ

e.g.

birds

-RRB-

The

third

solution

is

essentially

a

combination

of

the

previous

two

Its

existence

in

natural

populations

has

not

been

explicitly

demonstrated

although

there

is

considerable

evidence

of

mixed

ESD/genotypic

systems

which

may

conform

to

the

structure

-LSB-

-RSB-

Whilst

it

is

known

-LSB-

-RSB-

that

polygenic

sex

determination

is

unstable

to

invasion

by

mutants

we

have

demonstrated

that

in

finite

populations

polygenic

systems

of

sex

determination

are

intrinsically

unstable

A

polygenic

sex

determining

system

can

evolve

towards

a

conventional

system

of

genetic

sex

determination

without

the

introduction

of

any

sex

determining

mutants

The

outcome

is

purely

the

result

of

two

processes

that

affect

factor

frequencies

in

the

gene

pool

Random

drift

reduces

genetic

variability

as

a

consequence

of

sampling

error

in

finite

populations

-LSB-

-RSB-

Fisherian

selection

-LSB-

-RSB-

maintains

only

those

combinations

of

factors

that

yield

an

unbiased

population

sex

ratio

this

latter

process

occurs

as

the

rarer

sex

necessarily

makes

a

relatively

greater

genetic

contribution

to

the

following

generation

In

the

presence

of

selective

advantage

for

ESD

we

might

expect

to

see

a

preference

for

type

or

solutions

but

we

leave

that

analysis

for

a

later

date

It

is

clear

from

our

results

that

the

assumption

that

the

stable

distribution

for

polygenic

sex

determining

factors

is

Guasssian

is

false

-LRB-

as

in

-LSB-

-RSB-

-RRB-

if

there

is

a

high

degree

of

linkage

even

if

the

factors

are

distributed

over

many

chromosome

pairs

Of

particular

interest

is

the

observation

that

generally

if

the

factors

are

distributed

over

many

chromosomes

then

one

-LRB-

occasionally

two

-RRB-

pairs

will

end

up

with

absolute

influence

on

sex

determination

Hence

we

see

the

automatic

evolution

of

sex

chromosomes

with

the

gene

count

on

the

autosomes

-LRB-

non

sex

chromosomes

-RRB-

all

being

identical

between

members

of

each

pair

These

latter

observations

concur

with

some

recent

genetic

observations

where

DNA

sequences

homologous

to

known

sex

factors

on

sex

chromosomes

have

been

found

widely

distributed

throughout

the

autosomes

-LSB-

-RSB-

These

factors

appear

to

be

present

in

wide

range

of

species

-LSB-

-RSB-

Indeed

-LSB-

-RSB-

suggest

that

a

possible

model

consistent

with

their

findings

would

be

one

of

additive

sex

factors

From

the

perspective

of

the

original

problem

of

distinct

ESD

and

GSD

Gammarus

populations

the

outcome

is

very

satisfactory

We

can

account

for

the

observation

of

differing

sex

determination

methods

within

the

same

species

from

the

result

that

one

of

the

populations

is

of

solution

type

and

the

other

is

of

type

both

having

diverged

from

a

single

ancestral

polygenic

state

We

may

use

the

same

observation

to

interpret

a

similar

cross-species

pattern

found

in

the

fish

Menidia

menidia

-LSB-

-RSB-

From

the

perspective

of

experimental

design

most

of

the

solutions

give

a

small

number

of

possible

outcomes

for

offspring

sex

ratio

dependent

on

the

particular

female/male

cross

The

populations

with

the

most

complex

structure

are

those

that

arise

as

a

probabilistic

type

In

that

case

there

can

be

a

considerable

variation

in

familial

sex

ratio

There

are

many

questions

of

rate

left

to

study

Often

when

two

viable

solutions

are

competing

the

system

can

remain

stable

in

this

configuration

for

a

very

long

time

Whilst

the

process

of

symmetry

breaking

is

fast

when

compared

with

that

of

spontaneous

extinction

it

can

be

very

slow

-LRB-

in

large

populations

-RRB-

in

comparison

with

the

generational

time

scale

When

the

effective

population

is

many

thousands

it

can

take

thousands

of

generations

for

a

symmetry

break

to

occur

Equally

there

are

questions

of

the

probability

of

particular

solutions

being

reached

both

in

the

presence

and

absence

of

selective

advantage

We

hope

to

address

these

questions

in

a

later

paper

Taking

a

polygenic

-LRB-

multifactorial

-RRB-

system

as

the

genetic

basis

of

sex

determination

we

can

observe

evolution

to

many

of

the

known

forms

of

sex

determination

More

interestingly

we

see

that

in

a

multichromosome

system

generally

one

of

the

pairs

will

take

the

lead

in

determining

the

animal

's

gender

Finally

it

is

possible

within

the

same

basic

genetic

system

to

have

both

environmental

and

genetic

sex

determination

In

a

general

sense

the

process

algebraic

approach

to

the

description

of

evolutionary

systems

has

great

potential

The

question

of

the

effect

of

point

of

gene

expression

on

evolutionary

dynamics

-LSB-

-RSB-

may

be

very

precisely

addressed

in

this

context

Within

our

process

algebraic

description

is

a

very

general

evolutionary

system

which

could

be

exploited

to

investigate

problems

of

evolution

other

than

sex

The

effects

of

linkage

and

finite

population

size

in

evolution

are

known

to

be

very

hard

to

understand

-LSB-

-RSB-

The

process

algebraic

approach

may

give

new

insights

in

this

area

Acknowledgements

Chris

Tofts

is

supported

by

an

EPSRC

Advanced

Fellowship

-LRB-

-RRB-

Melanie

Hatcher

is

supported

by

an

NERC

Research

Grant

-LRB-

-RRB-

The

authors

thank

Alison

Dunn

and

Jonathan

Adams

for

many

useful

discussions

of

this

work

The

final

version

of

this

paper

is

much

improved

as

a

result

of

diligent

reading

of

an

early

draft

by

Alan

Williams

References

-LSB-

-RSB-

Adams

J.

Greenwood

P.

Naylor

C.

Evolutionary

aspects

of

environmental

sex

determination

Int

J.

Inv

Rep.

Dev

-LSB-

-RSB-

Aida

T.

Genetics

-LSB-

-RSB-

Barton

N.H.

The

Maintenance

of

Polygenic

Variation

Through

a

Balance

Between

Mutation

and

Stabilising

Selection

Genet

Res

-LRB-

-RRB-

-LSB-

-RSB-

Blackmore

M.S.

E.L.

Charnov

Adaptive

variation

in

environmental

sex

determination

in

a

nematode

Am

Nat.

-LSB-

-RSB-

Bull

J.J.

Sex

determination

in

reptiles

Q.

Rev.

Biol.

-LSB-

-RSB-

Bull

J.J.

Charnov

E.L.

How

fundamental

are

Fisherian

sex

ratios

Oxf

Surv

Evol

Biol

-LSB-

-RSB-

Bull

J.

Evolution

of

Sex

Determining

Mechanisms

Benjamin/Cummings

California

-LSB-

-RSB-

Bull

J.J.

R.C.

Vogt

M.G.

Bulmer

Heritability

of

sex

ratio

in

turtles

with

environmental

sex

determination

Evolution

-LSB-

-RSB-

Bull

J.

D.

Hillis

and

S.

O'Steen

Mamalian

ZFY

Sequences

Exist

in

Reptiles

Regardless

of

Sex-Determining

Mechanism

Science

-LSB-

-RSB-

Bulmer

M.G.

and

J.J.

Bull

Models

of

Polygenic

Sex

Determination

and

Sex

Ratio

Evolution

Evolution

-LSB-

-RSB-

Belew

R.

and

L.B.

Brooker

Proceedings

of

the

Fourth

-LRB-

-RRB-

International

Conference

on

Genetic

Algorithms

University

of

California

Morgan

Kaufman

-LSB-

-RSB-

Birky

C.

W.

and

J.B.

Walsh

Effects

of

Linkage

on

Rates

of

Molecular

Evolution

Proc

Natl.

Acad

Sci

USA

-LSB-

-RSB-

Baeten

J.C.M.

and

W.P.

Weijland

Process

Algebra

Cambridge

Tracts

in

Theoretical

Computer

Science

Cambridge

University

Press

-LSB-

-RSB-

Charnov

E.L.

The

Theory

of

Sex

Allocation

Princeton

Princeton

University

Press

-LSB-

-RSB-

Charnov

E.L.

J.J.

Bull

When

is

sex

environmentally

determined

Nature

-LSB-

-RSB-

Charnov

E.L.

R.L.

Los-den

Hartogh

W.T.

Jones

J.

van

den

Assem

Sex

ratio

evolution

in

a

variable

environment

Nature

-LSB-

-RSB-

Conover

D.O.

D.A.

Vanvoorhees

M.

Ethisham

Sex

ratio

selection

and

the

evolution

of

environmental

sex

determination

in

laboratory

populations

of

Menidia

menidia

Evolution

-LSB-

-RSB-

Conover

D.O.

B.E.

Kynard

Environmental

sex

determination

Interaction

of

temperature

and

genotype

in

a

fish

Science

-LSB-

-RSB-

Cox

T.F.

and

M.A.A.

Cox

The

Maximum

Non-unit

Eigenvalue

for

the

Wright-Fisher

Model

of

Random

Drift

Incorporating

Selection

Theor

Popul

Biol

-LSB-

-RSB-

Crow

J.F.

and

M.

Kimura

An

introduction

to

Population

Genetics

Theory

Harper

and

Row

New

York

-LSB-

-RSB-

Davis

L.

Genetic

Algorithms

and

Simulated

Annealing

Research

Notes

in

Artificial

Intelligence

Pitman

-LSB-

-RSB-

Davis

L.

Handbook

of

Genetic

Algorithms

Van

Nostrand

Reinhold

-LSB-

-RSB-

Dunn

A.M.

Adams

J.

Smith

J.E.

Transovarial

transmission

and

sex

ratio

distortion

by

a

microsporidian

parasite

in

a

shrimp

J.

Invert

Pathol

-LSB-

-RSB-

Deeming

D.

C.

and

M.W.J.

Ferguson

Environmental

regulation

of

sex

determination

in

reptiles

Phil

Trans

R.

Soc

Lond

B

-LSB-

-RSB-

Eshel

I.

Selection

on

sex

ratio

and

the

evolution

of

sex-determination

Heredity

-LSB-

-RSB-

Felsenstein

J.

The

Evolutionary

Advantage

of

Recombination

Genetics

-LSB-

-RSB-

Fisher

R.A.

The

Genetical

Theory

of

Natural

Selection

Oxford

Clarendon

Press

-LSB-

-RSB-

Fredga

K.

A.

Groop

H.

Winking

and

F.

Frank

Fertile

XX

and

XY

type

females

in

the

wood

lemming

Myopus

schisticolor

Nature

-LSB-

-RSB-

Frank

S.A.

Swingland

J.R.

Sex

ratio

under

conditional

sex

expression

J.

theor

Biol

-LSB-

-RSB-

Goldberg

D.

E.

Genetic

Algorithms

in

Search

Optimisation

and

Machine

Learning

Addison-Wesley

-LSB-

-RSB-

Gordon

M.

Two

Opposing

Sex

Determing

Mechanisms

one

XX-XY

and

the

other

WY-YY

in

Different

Natural

Populations

of

the

Platyfish

Xiphophorus

maculatus

Proc

Int

Cong

Gen.

-LSB-

-RSB-

Gabriel

W.

and

G.

P.

Wagner

Parthenogenetic

Populations

Can

Remain

Stable

in

Spite

of

High

Mutation

Rate

and

Random

Drift

Naturwissenschaften

-LSB-

-RSB-

Hamilton

W.D.

Extraordinary

sex

ratios

Science

-LSB-

-RSB-

Hamilton

W.D.

Altruism

and

related

phenomena

mainly

in

social

insects

Ann

Rev.

Ecol

Syst

-LSB-

-RSB-

M.

Hennessey

Algebraic

theory

of

Processes

M.I.T.

Press

-LSB-

-RSB-

Hoare

C.

A.

R.

Communicating

Sequential

Processes

Prentice-Hall

-LSB-

-RSB-

Howard

H.W.

The

genetics

of

Armadillidium

vulgare

Latrielle

II

Studies

on

the

inheritance

of

monogeny

and

amphogeny

J.

Genetics

-LSB-

-RSB-

Hatcher

M.J.

and

C.

Tofts

The

Effect

of

Point

of

Expression

on

ESS

Sexratio

in

press

J.

Theor

Biol

-LSB-

-RSB-

Hatcher

M.J.

and

C.

Tofts

The

Evolutionary

Dynamics

of

a

Polygenic

Sex

Determination

System

in

preperation

-LSB-

-RSB-

Janzen

F.J.

Heritable

variation

for

sex

ratio

under

environmental

sex

determination

in

the

common

snapping

turtle

-LRB-

Chelydra

serpentina

-RRB-

Genetics

-LSB-

-RSB-

Jones

K.W.

Evolution

of

Sex

Chromosomes

in

Development

in

Mammals

Vol

ed

M.H.

Johnston

Elsevier

New

York

-LSB-

-RSB-

Kimura

M.

The

Neutral

Theory

of

Molecular

Evolution

Cambridge

University

Press

Cambridge

-LSB-

-RSB-

Karlin

S.

and

S.

Lessard

Theoretical

Studies

on

Sex

Ratio

Evolution

Princeton

University

Press

-LSB-

-RSB-

Kosswig

C.

Polygenic

Sex

Determination

Expermintia

-LSB-

-RSB-

Lagomarisino

I.V.

D.O.

Conover

Variation

in

environmental

and

genotypic

sex

determining

mechanisms

across

a

latitudinal

gradient

in

the

fish

Menidia

menidia

Evolution

-LSB-

-RSB-

Lloyd

D.G.

K.S.

Bawa

Modification

of

the

gender

of

seed

plants

in

varying

conditions

Evol

Biol.

-LSB-

-RSB-

Lande

R.

Expected

time

for

random

genetic

drift

of

a

population

between

stable

phenotypic

states

Proc

Natl.

Acad

Sci

USA

-LSB-

-RSB-

R.

Milner

Calculus

of

Communicating

Systems

LNCS

Springer

Verlag

-LSB-

-RSB-

R.

Milner

Calculi

for

Synchrony

and

Asynchrony

Theoretical

Computer

Science

-LRB-

-RRB-

pp

-LSB-

-RSB-

R.

Milner

Communication

and

Concurrency

Prentice

Hall

-LSB-

-RSB-

Moody

M.E.

L.D.

Mueller

and

D.E.

Soltis

Genetic

Variation

and

Random

Drift

in

Autotetraploid

Populations

Genetics

-LSB-

-RSB-

Muller

H.J.

The

Relation

of

Recombination

to

Mutation

Advantage

Mutation

Research

-LSB-

-RSB-

Nagylaki

T.

Models

and

Approximations

for

Random

Drift

Theor

Popul

Biol

-LSB-

-RSB-

Nagylaki

T.

The

Evolution

of

Multilocus

Systems

Under

Weak

Selection

Genetics

-LSB-

-RSB-

Naylor

C.

J.

Adams

P.

Greenwood

Population

dynamics

and

adaptive

sexual

strategies

in

a

brackish

water

crustacean

Gammarus

duebeni

J.

Anim

Ecol.

-LSB-

-RSB-

Naylor

C.

J.

Adams

P.

Greenwood

Variation

in

sex

determination

in

natural

populations

of

a

shrimp

J.

evol

Biol.

-LSB-

-RSB-

Nei

M.

and

F.

Tajima

Genetic

Drift

and

Estimates

of

Effective

Population

Size

Genetics

-LSB-

-RSB-

Nunney

L.

The

Infleunce

of

Mating

System

and

Overlapping

Generations

on

Effective

Population

Size

Evolution

-LRB-

-RRB-

-LSB-

-RSB-

Nur

U.

The

expected

changes

in

the

frequency

of

alleles

affecting

the

sex

ratio

Theor

Popul

Biol

-LSB-

-RSB-

Ohno

S.

Sex

Chromosomes

and

Sex

Linked

Genes

Springer

Verlag

Berline

-LSB-

-RSB-

Ohno

S.

Major

Sex

Determining

Genes

Berlin

Springer-Verlag

-LSB-

-RSB-

Page

D.C.

R.

Mosher

E.

Simpson

E.

Fisher

G.

Mardon

J.

Pollack

B.

McGillivray

A

de

la

Chapelle

and

L.

Brown

The

Sex-Determining

Region

of

the

Human

Y

Chromosome

Encodes

a

Finger

Protein

Cell

-LSB-

-RSB-

Poethke

H.K.

Sex

Ratio

Polymorphism

The

Impact

of

Mutation

and

Drift

on

Evolution

Acta

Biotheretica

-LSB-

-RSB-

Rice

W.R.

On

the

Instability

of

Polygenic

Sex

Determination

the

Effect

of

Sex

Specific

Selection

Evolution

-LSB-

-RSB-

Rigaud

T.

Juchault

P.

Conflict

between

feminizing

sex

ratio

distorters

and

an

autosomal

masculinizing

gene

in

the

terrestrial

isopod

Armadillidium

vulgare

Latr

Genetics

-LSB-

-RSB-

Sinclair

A.H.

J.

Foster

J.

Spencer

D.

Page

M.

Palmer

P.

Goodfellow

and

J.

Marshall

Graves

Sequences

Homologous

to

ZFY

a

candidate

human

sex-determining

gene

are

autosomal

in

marsupials

Nature

-LSB-

-RSB-

Trivers

R.L.

D.E.

Willard

Natural

selection

of

parental

ability

to

vary

the

sex

ratio

of

offspring

Science

-LSB-

-RSB-

Turelli

M.

and

N.H.

Barton

Dynamics

of

Polygenic

Characters

Under

Selection

Theor

Popul

Biol

-LSB-

-RSB-

Tofts

C.

M.J.

Hatcher

N.

Franks

Autosynchronisation

in

Leptothorax

acervorum

Theory

Testability

and

Experiment

Journal

of

Theoretical

Biology

-LSB-

-RSB-

Tofts

C.

A

Synchronous

Calculus

of

Relative

Frequency

CONCUR

Springer

Verlag

LNCS

-LSB-

-RSB-

C.

Tofts

The

Autosynchronisation

of

Leptothorax

acervorum

-LRB-

Fabricius

-RRB-

Described

in

WSCCS

LFCS-Report

Number

Dept.

Of

Computer

Science

Edinburgh

-LSB-

-RSB-

Tofts

C.

Task

Allocation

in

Monomorphic

Ant

Species

LFCS-Report

Number

Dept.

Of

Computer

Science

Edinburgh

-LSB-

-RSB-

Tofts

C.

Using

Process

Algebra

to

Describe

Social

Insect

Behaviour

Transactions

on

Simulation

-LSB-

-RSB-

Tofts

C.

A

Study

of

Task

Alocation

in

Monomorphic

Ant

Species

Theory

Bull

Math

Bio

-LRB-

-RRB-

-LSB-

-RSB-

Tofts

C.

Processes

with

Probabilities

Priority

and

Time

Formal

Aspects

of

Computer

Science

-LRB-

-RRB-

-LSB-

-RSB-

Turelli

M.

Heritable

Genetic

Variation

via

Mutation-Selection

Balance

Lerch

's

Zeta

Meets

the

Abdominal

Bristle

Theor

Popul

Theo

-LSB-

-RSB-

Uyenoyama

M.K.

B.O.

Bengtsson

Towards

a

genetic

theory

for

the

evolution

of

the

sex

ratio

Genetics

-LSB-

-RSB-

Uyenoyama

M.K.

B.O.

Bengtsson

Towards

a

genetic

theory

for

the

evolution

of

the

sex

ratio

II

Haplodiploid

and

diploid

models

with

sibling

and

parental

control

of

the

brood

sex

ratio

and

brood

size

Theor

Popul

Bio

-LSB-

-RSB-

Verner

J.

Selection

for

Sex-ratio

Am

Nat

-LSB-

-RSB-

Watt

P.J.

J.

Adams

Adaptive

variation

in

sex

determination

in

a

crustacean

Gammarus

duebeni

J.

Zool.

-LSB-

-RSB-

Werren

J.H.

E.L.

Charnov

Facultative

sex

ratios

and

population

dynamics

Nature

-LSB-

-RSB-

Werren

J.H.

The

coevolution

of

autosomal

and

cytoplasmic

sex

ratio

factors

J.

theor

Biol

-LSB-

-RSB-

Wagner

G.

P.

and

W.

Gabriel

Quantitative

variation

in

Finite

Parthenogentic

Populations

What

Stops

Muller

's

Ratchet

in

the

Absence

of

Recombination

Evolution

-LRB-

-RRB-

-LSB-

-RSB-

Winge

O

The

experimental

alteration

of

sex

chromosomes

and

vice

versa

as

illustrated

by

Lebistes

Compt

Rend

Lab.

Carlsburg

Ser

Physiol

-LSB-

-RSB-

Wright

S.

The

Distribution

of

Self-Fertility

Alleles

in

Populations

Genetics

-LSB-

-RSB-

Wright

S.

Evolution

and

The

Genetics

of

Populations

I

Genetic

and

Biometric

Foundations

The

University

of

Chicago

Press

-LSB-

-RSB-

Wright

S.

Evolution

and

The

Genetics

of

Populations

II

The

Theory

of

Gene

Frequency

The

University

of

Chicago

Press

-LSB-

-RSB-

Yamamoto

T.

Genetics

-LSB-

-RSB-

Zimmerman

J.K.

Ecological

correlates

of

labile

sex

expression

in

the

orchid

Catasetum

viridiflavum

Ecology

esd
evolution
probability
process algebra
sex determination
simulation