Model of invasion of a population by transposable elements presenting an asymmetric effect in gametes

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Model of invasion of a population by transposable

elements presenting an asymmetric effect in gametes

G Morel, R Kalmes, Georges Périquet

To cite this version:

Original

article

Model

of

invasion

of

a

population

by transposable

elements

_presenting

an

asymmetric

effect

in

gametes

G

Morel

R Kalmes

2

G

_Périquet

UFR

Sciences et _{Techniques, D6partement} de _{Math6matiques,} Parc _Grandmont, 37200 Tours ;

2

UFR

Sciences et _Techniques, Institut de _{Bioc6notique Experimentale} des _{Agrosystèmes,} URA CNRS _1298, Parc _Grandmont, 37200 _Tours, France

(Received

2 _April 1992; accepted 9 December

₁₉₉₂₎

Summary - Dynamics of _population invasion _{by transposable} elements is _analyzed and simulated, using a model with a _{very large} number of _{transposition} sites. The _properties of the model are determined in the framework of a conflict between _{transposition} capabilities

of the elements and their harmful effects on the host _genome. Equations are _developed for

the mean and the variance of the number of elements at _equilibrium. We use simulations

to analyze the effects of various _parameters on the _dynamics of the elements, revealing

the _importance of the insertion rate and of the _{self-regulation} properties of the elements. Using values obtained from the P-M system of _{Drosophila melanogaster,} the simulations show that the invasion of these elements is _likely to occur _{in 100 years,} which is an interval

compatible with recent ideas on this invasion. Our _analysis of chained invasions reveals

the possibility of a mean element number gradient occurring, just as has been observed in

European wild _populations.

transposable elements _/ hybrid dysgenesis / model of invasion _/ simulation

Résumé - Modèle d’invasion d’une population par des éléments transposables présentant une action asymétrique selon les _gamètes. L’invasion de _populations par

des éléments _{transposables} est _analysée et simulée en utilisant un modèle à grand nombre

de sites de _{transposition.} Les _propriétés du modèle sont déterminées dans le cadre du conflit

entre les _capacités de transposition des éléments et les _effets délétères qu’ils induisent sur

le _génome hôte. Les équations sont développées pour la moyenne et la variance du

nom-bre d’éléments à l’équilibre. Les simulations permettent d’analyser les effets des _différents paramètres sur la dynamique des éléments et montrent l’importance du taux d’insertion

et des _{propriétés d’autorégulation} des éléments. En utilisant les valeurs obtenues pour le

*

système PM de Drosophila melanogaster, les simulations montrent _que l’invasion de tels éléments est _susceptible de se _produire en une centaine d’années, intervalle _compatible

avec les données récentes sur cette invasion. Une _analyse d’invasions en chaîne met en

évidence la _possibilité d’obtenir un gradient de _fréquence des éléments dans les populations,

similaire à celui actuellement observé dans les

_populations

naturelles _{européennes.}

éléments transposables / dysgenèse hybride / modèle d’invasion _/ simulation

INTRODUCTION

About 15% of the

_eukaryote

_{genome consists of} a

family

of

repeated

and

dispersed

DNA _sequences.

_Many

of these sequences have been described

before,

and some of

them have been found

_capable

of

_mobility

_(review

in

_Berg

and

Howe,

1989).

Several models have been

_proposed

to characterize the distribution laws of these

_transposable

elements in

_populations

as a function of different variables

such as their

_{transposition}

and excision rate, and the selective values _given to

carrier individuals

_(reviewed

in

Charlesworth,

1985, and

Brookfield,

1986 and

_1991).

Generally speaking,

in all sexed

_organisms,

these models have shown that a

_family

of elements could be

kept

in stable

_{equilibrium by}

the

_opposed

effects of

replicative

transposition

and selection

_against

the harmful carriers.

However,

much

_experimental

research has

_proved

the existence of

_{self-regulation}

mechanisms

_by

which the

_probability

of

_{transposition}

of an element decreases as a

function of the number of elements of the same

_family

_present

in the host _genome

(reviewed

in

_Berg

and

Howe,

1989).

Different models have shown that such

self-regulation

could also lead to a state of stable

_equilibrium

for the distribution law of a

_given

family

of elements

(Charlesworth

and

Charlesworth,

1983:

Langley

et

al,

1983,

Charlesworth and

_Langley,

_1986;

_Langley

et

_al,

_1988;

Rio,

_1990).

In

Drosophila

melanogaster,

the research on

hybrid dysgenesis

induced

by

families of

I,

P and hobo elements

_(reviewed

in

_Berg

and

Howe,

1989; and in

_Berg

and

_Spradling,

₁₉₉₁₎

has

_generated

a set of data

_by

which more

_specific

basic models can be conceived. Such models have been

proposed

for

describing

the evolution of

such _systems, in consideration of some of their

_{characteristics,}

but either

by

dealing

only

with the case of a

single

_{transposition}

site

(Ginzburg

et

al,

1984;

Uyenoyama,

1985)

or with an infinite number of

_sites,

and

analyzing

the selective values at the

individual level

_{(Brookfield, 1991).}

In the present

article,

we

analyze

a model for a

family

of

transposable

elements

whose

_{transposition}

and excision rates are functions of the copy

number,

and whose

dysgenic

effects

_depend

on the _type of

_crossing.

The invasion conditions of these

elements in a

_population

are determined

analytically.

When a state of internal

equilibrium

exists, the mean and the variance of the distributions are found.

Simulations are used to

verify

the

equations,

and as a basis for

discussing

invasion

DESCRIPTION OF THE MODEL

The model

_developed

here is based on the

_opposing

actions of a _transposase and of a _repressor, whose

reciprocal

_{concentrations,}

and

thereby

the

effets,

depend

on the

element copy number. The

equilibrium

or

_{disequilibrium}

_existing

between these 2

components

depends

on the direction of

_crossing,

and is established in the _zygote. Adults have a selective value linked to the

_{possible dysgenetic}

effects

_affecting

the

zygote

they

come from.

In the model the number of sites

_(T)

is

_{supposed large enough}

to allow any

transposition

into an _empty site. The _gametes are characterized

_by

the number

(ranging

from 0 to

_T)

of active elements

_they

contain. For the ovum, this number

is taken as an index of the concentration of _repressor, as we assume the rate

of

_{transposition}

is

_simply

controlled

_by

the _{repressor present} in this _gamete.

Considering

T as _very

_large,

the

_frequency

of

_occupied

sites does _{not appear} as a

_pertinent

_parameter and we address here

only

the distribution of _copy number

per gamete.

The element copy number distributions in the _spermatozoa and in the ova of

gen-eration t are considered to be identical. We define them

_by

_{(pt(0), pt(1), ... pt(T)),}

with:

The _zygote obtained

_by

_crossing

an ovum

_containing

i elements and a sperma-tozoon

_containing

_j,

is denoted

_{(i, j).}

First case

The spermatozoa contain no

elements,

so

that j

= 0. We

suppose that the

transposable

elements have no effect on the

(i, 0)

_zygote because the

_equilibrium

between the repressor concentration and the number of elements in _{the egg is} not

disturbed.

The selective value of these _zygotes is taken as

_reference,

and is therefore set

equal

to

_unity,

_{w(i, 0)}

= 1.

Finally,

_we let

(1 - G(i))

be the

frequency

of the

gametes without elements in the set of _gametes

_{produced by}

the

_{(i, 0)}

_{type zygotes.} When the sites of the elements are on a

_single

_pair

of

_chromosomes,

and when there

is no recombination

_possible,

we have

_G(i)

=

1/2

for i > 0.

Second case

The ovum has no _repressor

_(i

=

0)

and the _spermatozoon has elements

(j

_>

0).

In this

_{configuration}

there is a

_high

level of element

_activity,

and we let

A(j)

be

the mean increase in the number of elements for the _type

_{(0, j). A(j)}

is the mean

number of elements

_created,

less the mean number of elements lost.

W(j)

= 1-

S(j)

_represents the selective value of these

zygotes.

[1

-

B(j)]

is the

frequency

of _gametes without elements in the set of _gametes

_produced

from _type

(O

J

)

zygotes.

Third case

i > 0 and j > 0.

We define

_a(j),

_w(j)

= 1 —

s(j)

and

b(i,j).

The values a and w are

_supposed

to

depend only

on

_j,

which induces the

disequilibrium

between the number of elements

and the repressor concentration.

As in the first _case, the repressor concentration of the ovum is assumed to balance its element copy number. In the zygote, the

_{disequilibrium}

therefore

_depends

_only

on the number of

_{elements j}

introduced

_by

the _{spermatozoon.} The values

_a(j)

and

w(j)

= 1 —

s(j)

_correspond

_to the _{mean increase in} the number of elements and _to

the selective value of the zygotes

(i,j).

In these _zygotes, the _presence of repressor

limits the

_activity

of

_{the j}

elements

introduced,

which means

a(j)

<

_A(j)

and

w(j)

>

_W(j).

_Finally,

and as

before,

we define

(1 &mdash; b(i,j))

to be the

_frequency

of

ANALYSIS OF THE MODEL

Analysis

of initial element

_propagation

conditions

If we assume

_panmixia

in an infinite

population,

we _get,

_considering

the fertile

individuals,

p

t+

1 (0)

=

D

o/

D,

with:

T

By replacing

p

t

(0)

with

_{(1- LP}

_t

_(i)),

_Pt+

_{I (0)}

_{becomes a function of !pt(1), ... , pt(T)!.}

i = l

In order for the element

_frequency

to be able to _increase, the function

_{pt+1 (!)}

-T

p

t(O)

=

pt+1 (0) - (1 &mdash;

L

P

t(i))

must be

strictly

negative

in a

neighborhood

of i=i 1

(Pt(1) = 0,... ,pt(T)

=

0)

.

As

_p

_t+i

₍₀₎

is differentiable in

_{(0, ... ,}

_0),

we have to _compute the

_partial

deriva-tives in

_{(0, ... , 0).}

We _get for 1 ! k $ T

_{(Appendix 1):}

A sufficient condition for an increase of elements

starting

from a small initial

number of _gametes

_possessing

elements is therefore

_G(k)

+

_W(k)B(k)

> 1 for

1 ! k ! T. When the gametes introduced have few

elements,

it is sufficient for the first

_inequalities

alone to be satisfied..

These

_inequalities

are _{easy to}

_interpret

because

G(k)

[respectively W(k)B(k)]

is the

_probability

that a

(k, 0)

_{type zygote}

[resp

(0,

k)]

that would be viable and

nonsterile will

_produce

a _gamete

_containing

at least one element.

It will be seen that this model

generalizes

that of

Ginzburg

et al

(1984),

which assumes a

single

insertion site, or that the number of elements has no

effect,

and on a

single

pair

of chromosomes. In this case, the notation is

G(i)

=

1/2

for 1 ! i !

T;

S(j)

=

S;

s(j)

=

_0;

B(j) =

!3+

1/2(1 - /3)

=

1/2 + /3/2

for

1 ! j ! T;

b(i,j)

= 1

for 1 ! i ! T and

_{1 ! j !}

T

_(!3

_being

the

_probability

that the maternal genome be

contaminated

by

transposable

elements in a

(0, j )

_type

_mating.

The T

inequalities

are identical with

G(k)

=

1/2, W(k)

= 1 - S and

B(k)

=

1/2

₊

/3/2.

So we once

again

find the _necessary and sufficient condition of

_expansion

which

they

reach in

their

_special

case, ie

fl

>

_{S/(1 -}

_S).

In the _present

model,

B(k)

is not

_fixed,

but rather

_depends

on the

transposition

process. In the case of

only

one chromosomal

pair

and

assuming

that the k elements

of the

_paternal

chromosome are not

_excised,

that the increase in the number of elements is

_A(k)

for any

(0, k)

zygote, and that any new element is inserted

randomly

in 1 of the 2

_chromosomes,

we _get

B(k)

= 1 -

(1/2)!!)+!.

The kth

_inequality

is then written

Each

_{inequality yields}

a relation between

S(k)

and

A(k)

for

determining

the

conditions under which the element _copy nomber will increase.

The hatched area of

figure

1

corresponds

to the values of

S(k)

and

A(k)

verifying

this

_inequality.

The harmful effect of the

_transposable

elements can increase as the

Analysis

of the

_positions

of

_equilibrium

Analysis

of the mean

Pt

=

(

Pt

(0) , ... ,

,pt(T))

_{is an}

equilibrium point

if p

t+

i(i)

=

p

t

(i)

for 0 ! i _! T.

The

_modelling

described here cannot be used to determine the

_p

_t+1

values as a function of

P

t

for i > 0. This is

possible only

if we

_know,

for each _type of

zygote, the distribution of the _gametes

_produced

as a function of the number of

elements

_they

contain. Each of these distributions

_requires

T _parameters

_(the

sum

of them

_being

less than or

_equal

to

_unity)

in order to be defined. It should be

possible

to reduce this excess of _parameters

by adopting

_{assumptions concerning}

the mode of action of the

_transposable

element. This

_problem

is not addressed in the _present paper.

Instead,

we _attempt to obtain the

_equations

for the mean and

the variance of the distributions of elements at

_equilibrium,

when it exists. Such

equilibria

have been found for the

_{corresponding}

model of

_Ginzburg

et al

_(1984).

The mean and variance

depend

on the parameters

previously

defined. This way we

get

Pt+

i (0)

=

Do!D

(see

Analysis of initial

element

_propagation

_conditions)

and the

mean

E(X

t+1

)

=

E(Y

t+1

)

for the variables

_X

_t+i

(resp Y

t+

,),

number of elements

in the ova

(resp

in the

_{spermatozoa),}

of the

_(t

+

_1)th

generation

(see

Appendix

2).

At a

_point

of

_equilibrium

we have:

For a _point of

_equilibrium

(p, pl, ... , pT)

other than

(1, 0, ... , 0),

_equation

[1]

can

be written:

In the case considered before of a

_species

_{having only}

one

_pair

of

_chromosomes,

and

_supposing

that the elements are not

excised,

the

_{(i, j)}

_type

_yields

no _gametes

without

elements;

so we have

_{b(i, j)}

= 1 for i _> 0

and j

_>

0,

and therefore

e =

_0,

which

corresponds

to an

_equilibrium

where all of the _gametes would _possess

elements. _p = 0 is then a solution of the

_equation

_[1].

It is the

_{only equilibrium}

possible

if d >

_0,

because c >

_0,

_(w(j)

>

_W(j)).

This situation occurs in

particular

when the

_inequalities

related to the element _copy number

_growth

conditions are

verified.

When there is more than 1

_pair

of

chromosomes,

or when the element can be

excised,

b(i,j)

may be other than

unity;

but it

_approaches

it _very

_quickly

as i and

j

increase. It is therefore not

_surprising

to find

_populations

in

_equilibrium

in which

p(0)

can be considered zero. If it _is,

_equation

_[2]

is reduced too, and the mean

number

_E(X’)

of elements per gamete satisfies:

If

_a(.)

and

_w(.)

are linear functions

(a(j)

=

a.j.w(j)

= i -

(s.j),

this

_equation

is

written:

in which

_Var(X’)

_designates

the variance in the number of elements _{per gamete.}

E(X’)

therefore does not

_{depend only}

on the mean increase and the selective

value,

but

_through

the variance of X’ it also

_depends

on the

dispersion

of the

insertion-excision _process. This variance therefore deserves

_{being analyzed.}

Analysis

of the variance

Let

_{Var(i, j)}

be the variance of the number of elements of _gametes

_{produced by}

_type

(i, j)

zygotes. Even with a deterministic model of the number of

transpositions

and

excisions in these _zygotes,

_{Var(i, j)}

is not zero.

Var(i, j)

is

_analyzed

in

_{Appendix 3}

for the case of a

single

_pair

of chromo-somes. These new _parameters are introduced in order to calculate the variances

Var(X

t+d

=

Var(Y

t+i

)

of the number of elements in the

_gametes

of

generation

(t + 1).

At a

_point

of

equilibrium

we have

)

t+1

Var(X

=

Var(X

t

),

which

provides

a third

condition

_[3]

of

_equilibrium

_(see

_Appendix

_!,).

This condition

_depends

on the third

Even when

_p(0)

= 0 and the functions

a(.), w(.)

and

Var(.,.)

are

simple,

the

simulations

_{(see below)}

have shown the

_importance

of the third _moment, which has in no case been found to be close to zero.

Equations

[2]

and

[3]

cannot therefore be used to find

_E(X’)

and

_Var(X’).

As the invasion

_dynamics

of the elements are

_just

as

_interesting

as their mean

and variance at

_equilibrium,

we chose to simulate the _process rather than

_simplify

the

_equations

_by

approximation. However,

the mean increase in the number element

per type of _zygote is not sufficient and we must take into account the way a

_{(i, j)}

zygote

produces

new elements.

SIMULATION AND NUMERICAL ANALYSES Evolution simulation _program

To reduce the simulation _program run time and have a first

_approach

to the

process, the program computes the case of a

single

_pair

of chromosomes without

recombination. This has its effect on the numerical results

_by

_way of

G(.), B(.),

b(.,.)

and

_{Var(., .),}

but does not

_change

the mean value. Moreover it will allow an

introduction to the

_general

features of the

_phenomena.

The functions allowed are of the form:

The mean increases are therefore the result of a

(U, u)

transpose and a

(V, v)

excision _process

_{(Charlesworth}

and

Charlesworth,

1983).

To obtain the

_p

_t

_(n)

_frequencies

at the tth

_generation,

we have to determine for each

_(i,j)

_{type zygote} the _gametes it will

_produce.

_However,

the

_knowledge

of

_A(j)

and

_a(j)

are not

_sufficient,

and the distribution of the

_transposed

and excised elements around these means is _necessary. The _program allows the user to

choose between a distribution

_ranging

between the two

_integers

to either side of the

mean, or a Poisson distribution. From such a distribution the final

_composition

of

gametic

types is

_{determined, giving}

to each chromosome

produced

its number of

new elements.

The simulation stops at the tth

_generation

when the

_frequencies

_p

_t

_(n)

and

p

t

-i

(n)

are within

10-

6

of each other

_{(0 fi n fi T).}

The

_stability

of the mean and the variance of the tth

_generation

is verified

_by

computing

the mean and the variance of the

(t+1)th

_generation

from the formulae

that led to

_equations

_[2]

and

_[3].

Examinations of a few

_special

cases

The

_examples

considered here are based on linear functions

(D

= E = d = e =

F =

f

=

1),

and the increase of elements is distributed _over 2 consecutive

_integers.

Using

the notation of the

model,

we get for the average increases in the number of elements:

and for the selective values of the _zygotes:

Using

the available

_experimental

data

_(from

_Bingham

et

al, 1982;

Engels,

1988;

Berg

and

_Spradling,

₁₉₉₁₎

for the P-M system

of Drosophila melanogaster,

orders of

magnitude

were defined

along

with rates of

_insertion,

excision and selective values. A first series of

_simulations,

carried out as a

check,

shows as

expected that,

when there is no deleterious effect

(S

= s =

0)

the mean of the number of elements

increases

_{indefinitely,}

at a rate that

_depends

on the insertion and excision rates.

In a second series of

simulations,

the relations between the harmful effects of the

elements and their

regulation

capacities

were examined.

Variations with counterselection and

_{self-regulation}

of elements

0.25 _per element in

_dysgenic

_mating

_{((0, j) zygote),}

and

_considering

an excision

rate 100 times smaller than the insertion rate

_{(Engels, 1988),}

we _get U = 0.252

and V = 0.002. If the

self-regulation phenomena

did not exist in the

_{(i, j)}

_zygotes, the _{parameters u,} v and s would be

equal

to

_U,

V and

_{S, respectively.}

Under these conditions

_{(fig 2),}

the simulations show that the

_transposable

elements may invade

the

_population

rather

_quickly

_{(250 generations)}

when the deleterious effect is not too _great

_(S

=

0.05),

but _can

only

set in once S reaches a threshold

value,

which

is S = 0.11 here. For intermediate values of

S(S

=

0.08),

the invasion time will be

greater

(500 generations)

and

_only

a _part of the gametes would have

_transposable

elements.

The

_{self-regulation}

effect in the

_{(i, j)}

_zygotes can then be

_{analyzed by}

_assigning

values 10 times smaller to u and _v, or u =

U/10

and v =

_V/10,

and

_choosing

a low deleterious effect of s = 0.012. The simulation _{is initialized with 1}

gamete

among 1000

carrying

a

_{single element,}

or

_p

_o

₍₁₎

=

10-

3

.

The _curves obtained for

different values of S

_(0.05

and

_0.08)

are

_given

in

_figure

3.

_Here,

the elements can

totally

invade the

_population,

while the selection coefficient S

_against

the

_dysgenic

zygotes is < 0.11. The invasion is slower than before and the

_population

reaches a

stable

_equilibrium

in 1 500 to 2 000

_generations,

but with a

_higher

mean number

of elements

_(13.4

on _{the average, and with} a SD of

_4.8).

On the other

_hand,

It will be noted that the invasion is not

_necessarily

assisted if the rare founder

gametes carry a

_large

number of elements. This is due to the fact that the

_majority

of the _zygotes in the first

_generations

will be

_dysgenic

and _very

_highly

counterselected. When the spermatozoa introduced carry more than

1/S

elements,

the

(0, j)

_zygotes

provided

will be all

_dysgenetic

_(W(j)

=

sup(0,1 -

S.j)

=

0)

and there will be

no invasion.

However,

the invasion

_might

occur for lesser values. Such a case is

illustrated in

_figure

3 with

_p

_o

₍₁₁₎

= 10-3 and S = 0.08. We then observe that

the evolution

_proceeds

in a manner similar to the

_previous

cases, but a little less

rapidly.

Effects of variations in the insertion-excision and

_{self regulation}

rates

As the excision rates under

_dysgenic

conditions are known

_{only approximately}

_(to

within a factor of

_10;

see

Engels,

1988),

a set of simulations were run to

_get

an idea

of its role in the case of our model.

_Using

the conditions of the

_{preceding example,}

of a mean increase of

_0.25,

we chose to increase the excision rate

_10-fold,

or:

and s = 0.012.

The results obtained

_{(fig 4)}

show that the evolutions are

_quite

similar to those obtained

_previously,

with

_equilibrium

achieved in 1500 to 2 000

_generations

with a mean number of elements of 12.6 and a SD of 4.7. In

fact,

when the insertion rate

In the same _way, the role of element

_mobility

_regulation

and the

_dysgenic

effects observed in

_{(i, j)}

_{type zygotes} can be evaluated

_{by modifying}

the values of the

corresponding

parameters. We examined the case of

_relatively

ineffective

_regulation

in this way,

by

increasing

the values of u and _v, and

_generally

the simulations show

that the invasion time is shorter.

In the

_examples

_given

in

_figure

5

_(with

U = 0.252, V =

0.002,

u =

U/2,

v =

V/2

and s =

S/2),

the invasion process is

possible only

for S _<

0.11,

and the

equilibria

are achieved at between 250 and 500

_generations.

In the

_limiting

case, this situation

tends toward the one

_presented

before

(fig 2),

in which

self-regulation

did not exist in the

_{(i, j )}

_zygotes.

Finally,

a last set of simulations was

undertaken,

to estimate the effect of very

high

transposition

rates that could lead to a mean increase A of one element. The

values chosen

_{(fig 6)}

of U = _1.5, V = _{0.5, u} =

U/10,

_v =

V/10

and _{s =} 0.03 when

the invasion takes

_place

_(S

E

_{[0, 0.24))}

showed that the invasion is faster

_(from

200

to 800

_generations

for the 3

_examples

_considered)

but leads to

_equilibrium

values very similar to the

_previous

ones

(average

of 13.7 and a SD of

4.9).

Examination of an invasion in a _{sequence of}

_stages

The model

_proposed

can also be used to

_study

an invasion

_occurring

in successive

waves. An

_{original population}

A is invaded

_{by transposable}

elements and

_then,

af-ter a certain number of

_generations,

a _part of its individuals

_emigrate

into a fresh

distri-bution of elements in the _gametes introduced into

_population

B will be more or

less close to the

_equilibrium

distribution of

_population

A,

but

_usually

very different

from the initial distribution introduced into this same

_population

A.

The simulations were carried out to create mixed

_populations

from _{90% gametes} without elements and 10% gametes

originating

from a _parent

population

in

equi-librium. In all cases, the mixed

populations

evolved toward the

_equilibrium

state

of the _parent

_population

while the _parameters remained

_unchanged.

However,

the

_analysis

of several families of

_transposable

elements has revealed the formation of deleted elements in the course of the

_generations,

which

might

play a

role in the

_dynamic

_regulation

of the invasion

_(Black

et

_al,

_1987;

Jackson et

al, 1988;

P6riquet

et

_al,

_1990;

_Raymond

et

_al,

_1991).

The _parameters then have to be

modified,

as the

regulatory

process decreases the

_mobility

of the elements and

_thereby

their

_dysgenic

effects.

_Considering

the

importance

of the insertion _rate, a series of simulations was made with

only

this

parameter

modified,

in order to _represent the invasion of a series of 3

populations.

The _parent

_population

is defined as U _{= 0.27,} V = 0.02, S = 0.05, u =

U/10,

v =

V110

and _s = _0.012.

’

At

_equilibrium,

10% of the _gametes are introduced into a

population

with no

elements,

which then evolves with U = 0.22. The _same

process will be

repeated

in

a third

_population

for which U = 0.15.

The results

_given

in

_figure

7 show that each

_population

then reaches a

specific

state of

_equilibrium,

with lower and lower average numbers of elements:

12.6,

6.5 and then 3.0. It is also seen that these states of

_equilibrium

are achieved more and

more

_quickly.

Such a _process of successive colonizations can therefore lead to the establishment of a

geographical

differentiation in the distribution of the number

of elements. It should be

_pointed

out,

though,

that

_considering

the formation of deleted

_elements,

each

_population

will contain a mixture of different elements. The

values obtained

_by

our simulations concern

_{only complete}

and active elements.

DISCUSSION

The model and

_dynamic

simulations of

_transposable

elements

_presented

here are

based on a

_genetic

_approach

to the

_phenomena

of

_hybrid

_dysgenesis,

described

mainly

for D

_{melanogaster.}

The model leads to a

generalization

of the model of

Ginzburg

et al

₍₁₉₈₄₎

for a

_large

number of

_{transposition}

sites.

The

_equations

for the mean and the variance of the number of elements at

equilibrium

depend

on the third _moment,

_however,

which cannot be

_neglected.

The

_validity

of the model has been confirmed

_by

simulations,

mainly by

examining

the

population

invasion

_dynamics.

The main results of these simulations can now

be discussed and

_compared

with

_{knowledge acquired}

on P and hobo elements in

D

_{rnelanogaster, although}

the model is an

oversimplified

version for the

_complex

mechanisms of

_regulation

known for these elements.

The parameter values used in our simulations come from the insertion and

excision rates observed in

_dysgenic

_matings

of the P-M _system.

_Although

these values are

_only

orders of

_magnitude,

_they

do show the

_impact

of the various model

parameters. The element invasion conditions is

_{given by}

a _system of

_inequalities

the net increase in the element copy number A. When it _occurs, the invasion is slower with

_increasing

S. With the values

_used,

the critical threshold of S is of the order of 0.10. The conflict between the deleterious effect of the elements and their invasion can be decided either

_{by increasing}

the

_{transposition}

rate or

_{by decreasing}

their deleterious

_effect,

or

by

_acquiring

a

self-regulation

_process.

In the case of P and hobo

dysgenic

_systems, the harmful effect due to

_sterility

is smaller at the _temperatures

_usually

encountered

_by

the

_individuals,

and a

_complex

self-regulation

system was

developed.

The _present model does not account for

cytoplasmic

type

repression effects,

but

_clearly

shows the role of chromosomal

regulation

which,

while

_diminishing

the harmful effects when the egg has elements

(s

<

_S),

favors the survival of the carriers and allows the effective invasion of the

population

by

a

large

number of elements.

It is

_{particularly interesting}

to note

_that,

for a

_species

like D

melanogaster,

the

values used lead to a state of

_equilibrium

in 1 500

_generations,

for a

_population

that

had one _gamete to

_begin

with out of 1000

_carrying

a

_single

element.

_Considering

the

species,

such an invasion _process would be of the order of some 100 years, which is

compatible

with the

_hypothesis

of recent invasion

_by

P and hobo elements

_(Kidwell,

1983;

Anxolab6h!re et

_{al, 1988 ;}

Pascual and

_P6riquet,

_1991).

So the

_importance

of

_setting

up

self-regulation

mechanims seems to be

_primordial

This would be the case of the KP elements and other deleterious elements in the P-M

system, as well as the Th element in the hobo _system. A model has been

_developed

recently

that includes the effect of this _type of element

_{(Brookfield, 1991).}

The author shows that deleterious elements can indeed be favored

by

selection if

they

are a favored substrate for

_{transposition,}

and that the

populations

then consist of a

combination of

_complete

and deleted elements. In wild P

_strains,

the total number of element is ! 50. For those

analyzed,

some one-third are

complete,

or ! 15 of

these

elements,

which

_corresponds

to the values obtained in our simulations.

Finally,

the

_impact

of the deleted elements on the

dynamics

of the whole also

appears to be

_important

in the case of invasions in series. In our

_simulations,

we saw that invasion was not facilitated

_simply

when the gametes of the

_original

population

contained more than one element. This _suggests that the diffusion of

elements

_starting

from individuals that are _part of a

_population

that has

_already

been invaded would be based rather on those individuals

containing

the fewest

complete

elements.

Moreover,

the presence of

incomplete elements, limiting

the

activity

of active ones, leads the

receiving population

to a state of

_equilibrium

at a lower number

of

_complete

elements. When the process is

repeated

from

population

to

_population

with a low elements transfer

flow,

it can lead to the onset of a

decreasing gradient

of the element copy number in all of the

populations.

This process is thus consistent with the invasion of

_European

strains

_by

American P ones and their dilution into M

cytotype,

leading

to the

_gradual

variation

_currently

observed. Here

_again,

the values obtained in our

simulations,

from 12.6 to 3.0

_complete

elements,

are

compatible

with data observed in wild

_populations:

from 35 elements in all for the French

populations,

7 or 8 for those of central Asia

(P6riquet

et

al,

1989).

It will be noted

_that,

if the element transfer flow continued in the course of

_generations,

all of the

_populations

would then tend to

_homogenize,

which is the case of the I

and Hobo

_elements,

for which there is no evidence of

_geographical

differentiation

(reviewed

in

_Bregliano

and

_Kidwell,

_1983;

Pascual and

_P6riquet,

_1991).

For the P-M _system, for which the invasion in D

_melanogaster

appears to be the most recent,

the

_{homogenization}

_process

would be under _way and recent observations on the

French

_population

do in fact show a trend in this direction over the last decade

(Fleuriet

et

_al,

_1992).

As the model

_presented

here _suggests, this _process could extend to all

_European

populations,

but this would

require

a

relatively long

time. We will be able to

answer these

_questions

by analyzing wild

populations

and

_comparing

the results

with an extended version of the _present

_model,

_introducing

recombination and

segregation

between more than one

_pair

of chromosomes and

_taking

into account some _aspects of the mechanisms of

regulation

of these elements. Such a program is under

_{investigation.}

’

ACKNOWLEDGMENTS

The authors would like to thank J _Danger and JC Landré for their technical assistance.

This work was supported by _grants from the Minist6re de

1’Education

Nationale

(DRED:

APPENDIX 1

Initial element

_propagation

conditions

APPENDIX 2

Analysis

of the means

E(X

t+1

)

and

E(Yt+i)

We will now rewrite these

_{equations using}

the variables

Xt

and

5q’,

restrictions

of the variables

_X

_t

and

_Y

_t

_{on the}

_gametes

_containing

elements. This is

_obviously

possible

when

_{7! 7!}

_{(1, 0, ... ,}

_0),

but this is a trivial

_point

of

equilibrium

for which we studied the

_instability

conditions in

_{Analysis of}

initial element

_propagation

conditions. To find the nontrivial

_points

of

_equilibrium,

we notice

that,

when

they

exist,

X’

and

Yt

have the same

_{distribution,}

which is defined

_by:

Giving

this distribution and

_p

_t

_(O)

is

_equivalent

to

_giving

P

t

.

If we

simplify

the

notation

_{by letting}

_p

_t

_(O)

=

p,

Xt

= X’

and Yt

=

Y’,

p

t+1

(0)

and

E(X

t+

i)

_are

written:

A

_point

of

_equilibrium

_{(p, p}

_l

_{, ... ,}

_pT)

other than

_{(1, 0, ... , 0)}

must therefore

_verify

the 2

_following

conditions:

APPENDIX 3

Analysis

of

_Var(ij)

in the case of a

_{single pair}

of chromosomes

When

_{i = j}

=

0,

we have

Var(0,0)=0.

Let us

_{begin by finding}

Var(i, j)

in the case where the increase in the number of

elements is the result of the loss of nd

elements

_(nd

<

_i+j)

and of the

transposition

of na elements

(the

increase is thus assumed to be constant for the

(i,j) strain).

We moreover assume that each element has the same

probability

of

disappearing

and that any new element has 1 chance out of 2 of

meeting

the chromosome from

the egg. The distribution of the number of elements lost

by

the chromosome from the egg then follows a

_{hypergeometric}

distribution with _parameters N = i +

j,

n = nd and

N

I

= i. The distribution of the number of new elements is binomial

with _{parameters n} = _na and

p =

1/2.

In the

_{(i, j)}

strains,

the distribution of the number of elements of the

_gametes

obtained from the maternal chromosome has the mean

(i - nd. (i/i + j ) + na/2)

and

the

_{variance ( (nd.i. j. (i}

_{+ j -}

_{nd) / (i}

_{+ j )}

₂

_{. (i}

_{+ j -1 ) )}

+

_na/4).

We conclude that the

distribution

of the number of elements of

_gametes

obtained from the

_{(i, j)}

strains has mean

(1 /2). (i

+ j &mdash; nd

+

_na),

and variance:

(If

i

_{+ j}

=

_1,

then i = 0

or j

= 0 and the first term must be considered

zero).

In the

_{(i, j)}

_strains, we no

longer

assume that the number of elements lost and

the number of new elements are _constant;

they

have distributions with means md

and ma,

respectively,

variances Var d and Var a, in which the difference

(ma &mdash;

md)

is

_equal

to

_A(j)

if i = 0 and to

_a(j)

if i

_!

_0).

Using

the

_previous

case, we deduce that the distribution of the number of

and variance:

Note : When the distribution of the number of elements lost

_{(resp created)}

is concentrated on the 2

_integers

ndi and ndi +

_{1 (resp}

nai and nai

_{+ 1 )}

to either side of md

_{(resp ma),}

we have:

APPENDIX 4

Analysis

of the variances

_{Var(Xt+i )}

and

_Var(Y

_t+1

₎

At

_equilibrium,

we should have

Var(X

t+1

)

=

Var(X

t

),

which

provides

_a third

condition of

_equilibrium.

When

_{Pt !}

_{( 1, 0, ... , 0),}

the notation of

_A

_P

_{pendix 2}

leads

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