A Stochastic Model for Evolution of Altruistic Genes

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A Stochastic Model for Evolution of Altruistic Genes

Roberta Donato

To cite this version:

Roberta Donato. A Stochastic Model for Evolution of Altruistic Genes. Journal de Physique I, EDP

Sciences, 1996, 6 (4), pp.445-453. �10.1051/jp1:1996223�. �jpa-00247196�

A Stochastic Model for Evolution of Altruistic Genes

Roberta Donato

Department

of

Physics, University

"La

Sapienza",

Piazzale Aldo Moro 2, 00185 Roma,

Italy

(Received

5 October 1995,

accepted

5

January 1996)

PACS.87.10.+e

General,

theoretical, and mathematical

biophysics

PACS.87.90. +y ^Other

topics

in

biophysics

and medical

physics

Abstract. We

study numerically

_a stochastic mortel for evolution and maintenance of _a

population

which

reproduces asexually

under selective _pressure and is divided into smaller groups of variable _size. An altruistic trait is defined _as

Iowering

the fitness of the carrier, but the

sqrvival

probability

of aII the members of

a group with

a

large enough

number of altruists is enhanced.

Numerical results show that there is _a transition in the average proportion of altruists _~ersus the relative

davantage

conferred

by

the _presence of altrmsts _to aII the members of the _group.

At the transition the distribution of altruistic

frequence

in the groups is not trivial, and average deme size reaches _a minimum. We found also _an error-threshold in mutation rate

analogous

to the

quasi-species

mortel of M.

Eigen.

1. Introduction

Natural selection is an

egoistic

_process: individual organisms compete ^for

representation

of their genes in the next

generation,

and

only

those _genomes better able to survive and

reproduce

are

likely

to be maintained in the

evolutionary

_process. In this framework it is diflicult _to

explain

how is

possible

the evolution and the maintenance of "altruistic" genes, that is _genes

which determine _a behaviour

disadvantageous

to the

carrier,

but beneficial for other individuals

(benefits

and

disadvantages

_are measured in term of individual

fitness).

Nevertheless,

_in nature _{we can} find _some

examples

of altruistic behaviour: the social _or-

ganization

in

himenoptera

and in

hisoptera iii, parental

care [2]

(including

"self

sacrificing"

behaviour such _as

injury-feigning),

homeostatic

regulation

of

population density

_[Si

Ii.e.

terri- torial behaviour

capable

of

adjusting population density

to the available food

supply), warliing

calls in birds

[3j,

^and so on. A recent

report

_concems territorial defense

by partes

^{of lions.}

Female lions

living

in

Serengeti

National Park and

Ngorongoro

Crater

(Tanzania)

live in groups that hunt

together

and defend _{a common}

territory against

other groups. Heinsohn and Packer

report

_{in [fil} ^that some honesses _{act more}

aggressively

and

swiftly

in

defense,

while others tend to

lag

behind. Territorial contests between lions _are

violent, frequently leading

to serious

injury

or death of the combatants. On the other

hand,

_a

pride

^of lions without

territory

has small

probability

to feed itself and

reproduce.

No consistent relations _were observed between

physi-

cal size and defense-readiness.

Furthermore,

when engaging in aid of _a

pride-mate mounting

a

defense,

_no consistent relation _was found between the readiness of the

helper,

and the kin relation between the

helper

and the first defender. Territorial defense in groups of female lions therefore _seems to be _{a case} of altruistic behaviour where kin relation ares net

play

_a role.

©

Les

Éditions

_de

Physique

1996

446 JOURNAL DE

PHYSIQUE

I N°4

SO

far,

there is _no evidence of _a

genetical

basis for altruistic behaviour.

However,

it is

an

interesting problem

in itself to understand how interactions between individuals may be reflected _on the action of individual natural selection.

The

biological

world may be divided in _a

hierarchy

of

increasing complexity:

_{gene, genome,}

cell, organism,

_group,

species, type,

_up to the whole

biosphere.

Selective _pressure acts _on each of these

levels;

in the

following

^{I will} call _a level _on which natural selection

operates

a seiection

unit. A

phenomenon

_{or a} behaviour which increases the chances of survival for _a selection unit may be

disadvantageous

at another level. This is what

happens

_in altruism: the _presence of

altruists _increases the chances of the group which ii

belongs

to. The

general problem

is net

only

to understand what is the level of selection

action,

but aise how the diiferent levels may interact under the selective pressure.

Classically,

two main mechanisms have been

proposed

to

explain

the

origin

and evolution of altruism: kin selection _11,

2, ii,

and _group selection

[4, Si.

In bath mechanisms it is necessary ^to suppose that individual fitness

depends

net

only

on individual

genotype,

^{but also} on the rest of

population.

In kin selection the individual

probability

of survival is

proportional

to the relatedness that the individual shares with the altruist. The selection mechanism thus is limited

only

to related

individuals,

and the selection unit is the individual. In group

selection,

the interaction _is between individuals

belonging

to the

same group, and groups are defined

by

external constraints

(for example spatial boundaries), disregarding

_any relatedness between individuals. Here there _are two selection levels and two

selection units, the individual and the group.

In the mortel discussed below _we

disregard

relatedness between individuals. The selection unit is the

individual,

but the individual fitness

depends

_on the

composition

of the group.

Although

this mortel _is

extremely simple,

_we are able to observe

numerically

the maintenance of altruistic _genes.

2. The Model

We consider _a

population Q,

made up of _a fixed numbers of

individuals,

M. This

population

is divided into groups

(demes)

of variable size. We _assume that

generations

_are

non-overlapping;

that

heredity

acts

according

to the usual Mendehan mechanism and that there is _a behavioural locus with two alleles: A

(recessive),

E

(dominant).

An individual of

genotype

AA is _an

altruist,

while AE and EE _are not.

To each individual _{a in a}

generation

we associate its

reproduction probability fia),

which in the mortel is the _{same as} the survival

probability

of this genome in the

population

_{over one}

step. The fitness

fia) depends

_on two factors:

1. _on its

genotype:

if the individual is altruist

(genotype AA),

its fitness _is reduced

by

_a

factor

il r)

with respect ^{to the other members of its} group;

2. if the individual

(whatever

its

genotype) belongs

to _a group with _a

large enough

fraction

z of AA individuals

Ii.e.

x >

x*),

its fitness is enhanced

by

_a factor

il

+

c)

with respect to groups which do net

satisfy

this condition.

The factor _{r is} called intrademic seiection rate and the factor _{c is} called interdemic seiection rate.

The transition between

generations

is _a discrete _process,

involving

^the

following

three

steps:

1. Selection and

reproduction. Reproduction

_may be asexual _or sexual. In the

following,

we will consider

only

the asexual _case. If _a is _an individual of the

population fit,

the

probability

that _an individual a' _E

Qt+i

is one of its

oifspring

is

given by:

Fia'

_e

_Qi+i ja

e

ni)

=

/1°~

,

Lp=i fifl) ii)

where

f

is the fitness defined in Table I and the _sum is _over the whole

population.

The

oifspring

of the individuals of _one deme _in one

generation

^form one deme in the next

generation.

2. Mutation. Each

genotype

_may mutate into another _one; if _u is the mutation _rate per

allele,

then the mutation

probabilities p(_~,

are

given

in Table II.

3. Deme

splitting.

There is _a maximal finite size of _a

deme,

call it M*. If _a deme _grows to be

larger

than M* it is

split

in two, and the members of the

original

deme _are

randomly

distributed into the two new demes. This rule _is motivated

by

the

analogy

with social

animais,

which

typically

live in groups that _are net toc

large.

It is also _{a necessary}

ingredient

for

purely modelling

_reasons: if there _{is no} maximal _size of

demes,

_{sooner or} later _one of the demes will take _over trie whole

population,

and trie interdemic selection

ceases to operate.

Then,

_{we compute} trie fraction of AA individuals for each deme and for trie whole

population,

and iterate trie process.

Table I. Table

of

^the

jitness.

ROMS: indi~iduai

genotypes;

^coiumns:

geneticai

_{group compo-}

sition.

Î X<X* X>X*

AA _r

il r) il

₊

c)

EA,EE

1 1+c

Table II. Mutation

probabiiity

_pg-g,. ROMS:

genotypes g';

coiumns:

genotypes

_g.

Î AA AE EE

AA

il u) vil u)

_u

AE

2u(1 u) il u)

_{+ u}

2u(1 u)

EE _u

vil u) il u)

3. Numerical Results

As _a consequence of trie

ergodic

theorem of stochastic _{processes [9]}

(which

in _our model holds for mutation rates diiferent from

zero),

in _a few

generations

trie

system

reaches _a

stationary prob- ability distribution, independent

_on trie initial conditions. This

steady

state is characterized

by

_a

dynamical equihbrium

between trie intrademic and the interdemic

dynamics;

fluctuations in the concentration of altruists _in diiferent demes _{are very}

large.

We therefore present results

both about distributions and average values.

The altruistic _gene is

kept

in the

population by

the relative

advantage

it confers _{to its group} in the interdemic

competition.

Its distribution _{is not}

trivial,

_at least in the

phases

in which its

concentration _is diiferent from _zero. This _is shown in

Figures 1, 2, 3,

4.

448 JOURNAL DE

PHYSIQUE

I N°4

o.5

04

0.3

0.2

oi

O.O

0 0.2 0.4 06 0.8

Fig.

l.

Histogram

of altruist concentration _{xtot m} the

population.

Interdemic selection rate _{c =} o-o- M

=

500,

M* ₌ 50, _r

= 0.2, _u

= 0.03.

o.5

04

0.3

02

o-i

O.O

0 0.2 0.4 0.6 0.8

Fig.

2.

Histogram

of aItrui5t concentration _{xtot m} the

population.

Interdemic selection _rate

c = o-à-

M = 500, M* ₌ 50, r = 0.2, _u

= 0.03.

We found _a transition in the

equilibrium

_average value of the fraction _xtot of AA individuals

m the whole

population,

_as function of the mterdemic selection rate c,

keeping

fixed the other

parameters (the

intrademic selection rate r, the mutation rate u, the threshold concentration of AA individuals

~*).

This _is shown in

Figure

5.

0.125

o.ioo

o 075

o.osa

0.025

0.000

0 02 0.4 0.6 08

Fig.

3.

Histogram

of altruist concentration _{xtot m} the

population.

Interdemic selection rate _{c =} 2.0.

M = 500, M*

= 50, _r

=

0.2,u

= 0.03.

o.15

o.io

o 05

0.00

0 02 0A 0.6 0.8

Fig.

4.

Histogram

of altruist concentration _xtot

m the

population.

Interdemic selection rate _{c =} 10.0. M ₌ 500, M*

= 50, _r

= 0.2, _u

= 0.03.

If the mutation rate increases, the transition becomes _more and _more broad: both intrademic and interdemic selection have _{no more} eifect. This result reminds of the _error threshold in the

quasi-species

mortel of M.

Eigen

_[8]. See

Figure

6 below.

Another

interesting quantity

to

study

is the deme _size. In the mortel this

quantity

is a result of the

dynamics

of the

system,

it is not _a

parameter.

^{We found} that at the transition

Ii.e.

_in

450 JOURNAL DE

PHYSIQUE

I N°4

0.6

0.4

02

O.O

0 0 5 5 2 2.5

Fig.

5.

Average

value of altruist concentration _{xtot m} the

population

_~ersus interdemic selection

rate _c. M

= 500, M* ₌ 50, _{r =} 0.2, _{u =} 0.03.

05

0.4

0.3

0.2

o-1

00

0 2 3 4 5

Fig.

6.

Average

value of altruist concentration _{xtot m} the

population

_~ersus interdemic selection

rate c. M

= 500, ^Jf* ₌

50,

_{r =} 0.2. _u

=

0.05, 0.08, o-1,

0.3.

correspondence

of the interdemic selection value

c*)

the average deme size reaches _a minimum.

Moreover,

_{we can see a}

change

in the

shape

of the distribution above and below the transition.

See

Figures 7, 8,

9.

0 03

0.02

coi

0.00

0 20 40 60 80 100

Fig.

7.

Histogram

of deme _size below the transition: _c

= o-o- M

= 500, M*

= 50, _{u =} 0.03,

r = 0.2.

0 06

0 04

0 02

0.00

0 20 40 60 80 100

Fig.

8.

Histogram

of deme

size above the transition: _{c =} 2.0. M

=

500,

M*

= 50, _{u =} 0.03,

r = 0.2.

4. Discussion

We have studied _a mortel where _a

population

of M individuals _is divided in demes of smaller

size. The fitness of _an individual

depends

_on its

genetic make-up

and _on the _presence of

altruistic individuals _in the deme.

452 JOURNAL DE

PHYSIQUE

I N°4

20

18

16

14

0 0.5 1.5 2 2.5

Fig.

9.

Average

value of deme size _~ersus interdemic selection rate _c. M

= 500, ^M* = 50, _{u =} 0.03,

r = 0.2.

Numerical results demonstrate the maintenance of the altruistic _gene; in

particular,

there _is

a transition in the value of concentration of altruists _{as a} function of interdemic selection rate.

The

interesting

feature of this result is that in the simulated mortel the selection unit is the individual. The individual

probability

of

reproduction depends

also _on the

genetic composition

of the

deme,

and for this _reason the effects of individual selection

"propagate"

to the deme level. The individual

probability

of

reproduction

is enhanced if the concentration of

altruists,

in its

deme,

is above the threshold: this may be considered _as a kind of "interaction" between

individuals,

without

involving

_any kin

relationship

between them.

Since the genome of _our mortel consists of

just

one locus with

only

three

possible

states

(AA, AE, EE)

and _we consider asexual

reproduction,

the

only meaning

one coula

give

to kin

relationship

would be the

genetic overlap

at this locus between individual in _a deme. A kin selection mechanism would

imply

that altruism _{is more}

frequent

in

genetically homogeneous

demes. In _our mortel altruism _{is more} promirent ⁱⁿ

genetically inhomogeneous demes,

and the most

genetically homogeneous

are the _ones with the smallest fraction of altruists.

In _our mortel deme selection is _a consequence of individual selection: _a deme _is the collection of individuals who survived

(reproduced)

in the

previous generation.

This is the _reason which

allows _us to consider the deme size _{as a} variable and not as a

parameter.

We _saw that in the range of _c

corresponding

to the

transition,

the deme _size reaches _{a minimum;} it _means that

altruism _is most successful in small demes: this _{is a}

result,

not _a

starting pointl

The other important result is that the transition becomes broader with

increasing

mutation rate: this is

analogous

to the _error threshold in the quasi-species mortel

[8],

^{where there is} a

transition between _a locahzed

phase

and _a

disperse

_one. Below the threshold the

equilibrium

distribution of _{sequences is} locahzed _near the "master" sequence, and the transition

happens

when the individual selective _pressure, which tends to localize _sequences, cannot anymore

balance the

dispersive

effect of mutation. In _our

mortel,

the

changes

in the

genetical composition

of

population

_are due not

only

to mutation and to individual

selection,

but also to the effects

on selective _process of the "interaction" _in the

reproduction probability.

Hence _our numerical

results

suggest

^{that the} error-threshold remains also in presence of interaction between the selective units.

I have also studied in collaboration with Luca Peliti and Maurizio Serva _a

simplified

mortel where selection acts

directly

_on the deme

level,

with

intensity proportional

to the interactions

between the individuals in the deme

[loi. Analytical

results _on this mortel

fully

_support the numerical results _on the _more detailed mortel studied here.

Acknowledgments

thank Luca Peliti for

suggesting

the

problem,

and also Maurizio Serva and

Ugo

Bastolla for

enlightening discussions,

and the Nordic Institute of Theoretical Atomic

Physics (NORDITA)

in

Copenhagen

for invitation to the Summer School

"Physics

of

Biological Systems"

and fi- nancial

support.

References

iii

Hamilton

W-D-,

J. Theor. Biot. 7

(1964)1-16.

Hamilton

W-D-,

J. Theor. Biot. 7

(1964)

17-32.

[2]

Maynard

Smith

J.,

The

Theory

of Evolution

(Cambridge University Press, 1958); Maynard

Smith

J.,

Evolution and the

Theory

of Games

(Cambridge University Press, 1982).

[3]

Maynard

Smith

J.,

Amer. Nat. 99

(1965)

59-63.

[4] Eshel

I.,

Theor.

Pop.

Biot. 3

(1972)

258-277.

[Si

Wynne-Edwards V.C.,

Nature 200

(1963)

623-626.

[6] Heinsohn R. and Packer

C.,

Science 269

(1995)

1260-1262.

[7] Haldane

J-B-S-,

The Causes of Evolution

(New

York:

Harper

and

Row, 1932).

[8]

Eigen M., Naturwissenchaften

58

(1971)

465-523.

[9] Feller

W.,

An Introduction to

Probability Theory

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Applications,

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J.

Wiley, 1968).

[10]

^Donato

R.,

Serva M. and Peliti

L.,

An

Analytically

Solvable Model for the Maintenance of Altruistic Genes

(in preparation).