Neutral Theory:

From molecular evolution to community ecology

Fangliang He

Department of Renewable Resources

University of Alberta

“Nothing in biology makes sense except

in the light of evolution.”

Dobzhansky (1973)

A Brief History of (Molecular) Evolution

• In 1859, Charles Darwin proposed that:

(1) all organisms on earth evolved from a single proto-organism by

descend with modification,

(2) the primary force of evolution is natural selection.

• Natural selection is not the only force of evolution, may not even be a

dominant force. The alternative mechanisms: transmutation theory,

Lamarckism, geographical isolation, and nonadaptive evolution.

• Mutationism: represented by the post-Mandelian geneticist Morgan in 1920’s.

A strong critic of natural selection, argued for the importance of

advantageous mutations. Natural selection merely serves as a sieve to filter

deleterious mutations. (Also proposed that some part of morphological

evolution is caused by neutral mutation.)

Nei, M. 2005. Selectionism and neutralism in molecular evolution. Mol. Biol. Evol. 22:2318-2342.

• Neo-Darwinism: Represented by Fisher, Wright, Haldane, Dobzhansky in the

30-50’s. Natural selection is claimed to play much more role than mutation.

Two main reasons:

(1) the amount of genetic variation contained in natural populations

are so large that any genetic change can occur by natural selection

with no need of new mutations,

(2) math showed that the gene frequency change by selection the

change by mutation.

• Neutralism in 60’s: The foundation of Neo-Darwinism started to shake as

molecular data on evolution accumulated in the 60’s. Evidence:

(1) Amino acid sequences show that most amino acid substitutions in

a protein do not change the protein function (hemoglobins 血红蛋白 ,

cytochrome c 细胞色素 , fibrinopetides 纤维蛋白钛 ),

(2) Genetic variation within populations is much higher than

previously thought.

Neutral Theory of Molecular Evolution

Neutral theory: Most molecular polymorphism and substitutions are due to neutral

mutations and genetic drift. Genetic drift is the main force changing allele

frequencies.

Hemoglobins are evolving at a steady rate of 1.410-7 amino acid substitutions/yr ~

one nucleotide pair/2 yr (too high based on the cost of natural selection).

Kimura (1968) argued that many of the substituted alleles must be neutral.

King and Jukes independently proposed that most amino acid substitutions are

neutral. The inverse relationship between the importance of a protein or site

within a protein and its rate of evolution (Principle of Molecular Evolution).

Kimura, M. 1968. Evolutionary rate at the molecular level. Nature 217:624-626.King, J. L. and Jukes, T. H. 1969. Non-Darwinian evolution. Science 164:788-798.

Items Population genetics Macroecology

Operational units Population MetacommunitySubdivision Subpopulations Local communitiesFocal unit Gene SpeciesObserved data Allele frequency Species abundance

Neutral definition Alleles are selectively neutral, Individuals have equal vital ratesselection coefficient 0, or < 1/2Ne

Driving forces Genetic drift (1/2Ne) Ecological drift (1/JM or 1/JL)

Mutation SpeciationGene flow among subpopulations Dispersal among local communities

Spatial structure Population genetic structure Species distribution and abundancevariation among local communities

Model Island model Island biogeographyStepping stone modelIsolation by distance Dispersal limitationMainland-island Metacommunity-local communityCline pattern (speciation phases)

Measurement Fis diversity

Fst diversity

Spatial autocorrelation Species-area power law

Parameters = 4Ne = 4JM or = 4JL

Average number of migrants (Nem) Average number of migrants (JLm)

Effective population size (Ne) Effective community size (JM or JL)

Distribution of allele frequency Distribution of species abundance Fixed (extinct) probability of an allele Fixed (extinct) probability of a species

Assembly rules Genetic drift/mutation Ecological drift/speciationGenetic drift/migration Ecological drift/dispersalGenetic drift/migration/mutation Ecological drift/dispersal/ speciation

Mathematical tools Statistical methods Statistical methodsDiffusion model Diffusion modelCoalescent theory Phylogeny

Hu et al. 2006. Oikos 113:548-556.

Infinite-allele Model

1

2 11

1)1( tt F

NNF

Ft: the probability that two randomly sampled alleles at generation t are identical.

: mutation rate.

1

1

12

1

NF N2where

121

2

N

ND Heterozygosity expected from

mutation vs random drift.

Kimura, M. & Crow, J.F. 1964. The number of alleles that can be maintained in a finite population. Genetics 49:725-738.

Frequency of Spectrum

Ewens, W.J. 1972. The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3:87-112.

11)1()( xxx 10 x

N2where

)( within alleles ofnumber )( dxx, xdxx

1

0

1)(

n

in i

SE

n

SE n 1ln)(

Age of Species

)ln(1

2)( pp

pNpt

p

Tim

e

Kimura, M. 1983. The neutral allele theory of molecular evolution. Cambridge Univ. Press.

Overestimate the age of tree species in orders!

CommunityPatterns

Speciescomposition

Spatialstructure

Temporalstructure

Agestructure

Sizestructure

Atmospheric inputs

Edaphic conditionsGeographical processes

Disturb:

WindsFires

LoggingPollution

.....

Geneticstructure

Community Assembly Rules

BirthDeath

GrowthCompetition

Predation.....

Hubbell’s Neutral Theory

• Random walk

• Dispersal limitation

• Speciation

Metacommunity Neutral Model

112

2

N

ND

Expected diversity (Simpson index):

Hubbell, S.P. 2001. The unified neutral theory of biodiversity and biogeography. Princeton Univ. Press.

Expected # of species:

Species-abundance model:

n

xnf

n)(

n

SE n 1ln)(

Random Walk

j-1 j+1j0 N

Extinction FixationInitial state

1...0000

..................

0...0

0...0

0...0001

222

111

prq

prq

0

2

1

N

…

0 1 2 3 … N

1,1,,

1,

1,

1

1

1

jjjjjjj

jjj

jjj

pppr

N

jN

N

jpq

N

j

N

jNpp

Coexistence of Neutral Species

Probability of Extinction

Competitive exclusion in the two species system

occurs if the red species starting from j is finally

absorbed into either state N (the red species wins)

or 0 (the blue species wins).

The probability to extinction is defined as chance

for the focal species traveling from j to 0.

j-1 j+1j0 N

Extinction FixationInitial state

Time to Extinction

The time to extinction is defined as the average #

of steps for the focal species traveling from j to N

or 0.

j-1 j+1j0 N

Extinction FixationInitial state

j

tj

N = 20

j

pj

N = 20

Extinction probability Extinction time

Differential Birth Rates

Zhang and Lin (1997) consider the case of differential birth rates

1,1,,

1,

1,

1

)1(

1

jjjjjj

jj

jj

ppp

jjuN

jN

N

jp

jujN

uj

N

jNp

t

u

N = 100j = 50

u 1, higher birth rate in focal species

Zhang, D. Y. & Lin, K. 1997. The effects of competitive asymmetry on the rate of competitive displacement: how robust is Hubbell’s community drift model? J Theor Biol 188:361-367.

Differential Death Rates

Yu et al (1998) consider

the case of differential

death rates by simulation.

Yu, D. W., Terborgh, J. W. & Potts, M. D. 1998. Can high tree species richness be explained by Hubbell’s null model? Ecol Lett 1:193-199.

Differential Birth & Death Rates

Metacommunity model

1,1,,

1,

1,

1

)1(

1

jjjjjj

jj

jj

ppp

jjuN

jN

jvjN

vjp

jujN

uj

jvjN

jNp

)(

)(

jNQPj

jNQ

jvjN

jN

P is the prob of sampling an individual from the

focal species (abundance = j) and Q is the prob

of sampling an individual from the other species

(abundance = N-j). P+Q =1. If we randomly

sample one individual from the community (to

kill), the prob that the sampled individual belongs

to the other species (not the focal species) is:

This leads to , where Q

Pv

Effects of Birth and Death Rates on Coexistence Prob.

1.01.2

1.41.6

1.82.01.0

1.2

1.4

1.6

1.8

2.0

0.00.20.40.60.81.0

v

u

ej

u

v

1.0 1.2 1.4 1.6 1.8 2.01

.01

.21

.41

.61

.82

.0

c

extinction

fixation

d

N = 50j = 25

1.01.2

1.41.6

1.82.01.0

1.2

1.4

1.6

1.8

2.0

0.00.20.40.60.81.0

u

v

ej

1.0 1.2 1.4 1.6 1.8 2.01

.01

.21

.41

.61

.82

.0

ef

extinction

fixation

N = 50j = 45

Effects of Birth and Death Rates on Coexistence Time

N = 50j = 25

1.01.2

1.41.6

1.82.01.0

1.2

1.4

1.6

1.8

2.0

5001000

1500

v

u

tj

1.0 1.2 1.4 1.6 1.8 2.01

.01

.21

.41

.61

.82

.0

N = 50j = 25

1.0 1.2 1.4 1.6 1.8 2.0

1.0

1.2

1.4

1.6

1.8

2.0

v

1.01.2

1.41.6

1.82.01.0

1.2

1.4

1.6

1.8

2.0

200400600800

10001200

v

u

tj

e f

t-u profiles for different death rates v’s.

N = 50j = 25

vu

t1.0 1.2 1.4 1.6 1.8 2.0

50

01

50

0

1.0 1.2 1.4 1.6 1.8 2.0

50

01

50

01.0 1.2 1.4 1.6 1.8 2.0

50

01

50

0

1.0 1.2 1.4 1.6 1.8 2.05

00

15

00

1.0 1.2 1.4 1.6 1.8 2.0

50

01

50

0

v = 1 v = 1.1tj

v = 1.5 v = 1.8 v = 2

u u u

tj

1.0 1.2 1.4 1.6 1.8 2.0

50

01

50

0

v = 1.2

Local Community Model

1,1,,

1,

1,

1

)1()1(

)1(

1)1(

jjjjjj

jj

jj

ppp

mjjuN

jNm

jvjN

vjp

mjujN

ujm

jvjN

jNp

Effects of Birth and Death Rates on Extinction

1.01.2

1.41.6

1.82.00.0

0.2

0.4

0.6

0.8

1.0

0.00.20.40.60.81.0

u

ej

m1.0

1.21.4

1.61.8

2.00.0

0.2

0.4

0.6

0.8

1.0

0.00.20.40.60.81.0

u

N = 50j = 5

N = 50j = 25

1.01.2

1.41.6

1.82.00.00

0.05

0.10

0.15

0.20

2e+054e+056e+058e+05

1.01.2

1.41.6

1.82.00.0

0.2

0.4

0.6

0.81.0

0e+002e+144e+146e+148e+141e+15

v=1

1.01.2

1.41.6

1.82.00.0

0.2

0.4

0.6

0.81.0

0e+00

1e+13

2e+13

3e+13

1.01.2

1.41.6

1.82.00.00

0.05

0.10

0.15

0.20

2e+054e+056e+058e+051e+06

v=1.2

tj

tj

u

m

u

m

Effects of Birth, Death & Immigration Rates on Coexistence

Summary

1. The nearly neutral model which generalizes Hubbell’s neutral theory.

2. Birth and death have compensatory effects on coexistence but their

effects are not symmetric. Birth rates must be slightly higher than death

rates to maintain maximum coexistence.

3. The nearly neutral models provide a potential theory for reconciling

neutral and niche paradigms (?)

4. Immigration cannot prevent eventual extinction of a species but will

always increase the time of coexistence.

5. Nearly neutral systems have substantially shorter time of coexistence

than that of neutral systems. This reduced time provides a promising

solution to the “problem of time”.

Metacommunity model

1,1,,

1,

1,

1

)1()1(

)1(

1)1(

jjjjjj

jj

jj

ppp

mjjuN

jNm

jvjN

vjp

mjujN

ujm

jvjN

jNp

1,1,,

1,

1,

1

)1(

1

jjjjjj

jj

jj

ppp

jjuN

jN

jvjN

vjp

jujN

uj

jvjN

jNp

1,1,,

1,

1,

1

1

1

jjjjjj

jj

jj

ppp

N

jN

N

jp

N

j

N

jNp

1,1,,

1,

1,

1

)1(1

)1(

1)1(

jjjjjj

jj

jj

ppp

mN

jNm

N

jp

mN

jm

N

jNp

Local community model

The variation in birth and death rates

(subject to niche differentiation) determines

the fitness landscape. The focal species j

has a higher fitness (higher u, lower v) than

the other species. In this case, there is a

strong competitive exclusion, thus a short

coexistence time.

v

u

t

The effects of birth rate (u) and death

rate (v) on species coexistence are not

symmetric. A slightly higher birth rate

than the death rate is needed to

maintain a maximum coexistence.

x

The Problem of Time

v

u

t

Future Work

1. Derive macroecological patterns (species-abundance

distribution, species-area curves etc) as functions of u,

v and m.

2. Investigate trade-off in u and v.

3. Develop methods for testing the nearly neutral theory.