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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 1.410-7 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
Subdivision
Focal unit
Observed data
Population
Subpopulations
Gene
Allele frequency
Metacommunity
Local communities
Species
Species abundance
Neutral definition
Alleles are selectively neutral,
selection coefficient  0, or < 1/2Ne
Individuals have equal vital rates
Driving forces
Genetic drift (1/2Ne)
Mutation
Gene flow among subpopulations
Ecological drift (1/JM or 1/JL)
Speciation
Dispersal among local communities
Spatial structure
Population genetic structure
Species distribution and abundance
variation among local communities
Island biogeography
Model
Measurement
Island model
Stepping stone model
Isolation by distance
Mainland-island
Cline pattern (speciation phases)
Fis
Fst
Spatial autocorrelation
Dispersal limitation
Metacommunity-local community
 diversity
 diversity
Species-area power law
 = 4Ne
 = 4JM or  = 4JL
Average number of migrants (Nem)
Effective population size (Ne)
Distribution of allele frequency
Fixed (extinct) probability of an allele
Average number of migrants (JLm)
Effective community size (JM or JL)
Distribution of species abundance
Fixed (extinct) probability of a species
Assembly rules
Genetic drift/mutation
Genetic drift/migration
Genetic drift/migration/mutation
Ecological drift/speciation
Ecological drift/dispersal
Ecological drift/dispersal/ speciation
Mathematical tools
Statistical methods
Diffusion model
Coalescent theory
Statistical methods
Diffusion model
Phylogeny
Hu et al. 2006. Oikos 113:548-556.
Parameters
Infinite-allele Model
2

1 
1
Ft  (1  )   1   Ft 1 
N  N 

Ft: the probability that two randomly sampled alleles at generation t are identical.
: mutation rate.
1
1
F

2 N  1   1
2 N

D

1  2 N 1  
where
  2N
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
( x)   (1  x) 1 x 1
where
0  x 1
  2N
( x)dx  number of alleles within ( x, x  dx)
n 1
1
E (Sn )   
i 0   i
 n
E ( S n )   ln 1  
 
Ewens, W.J. 1972. The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3:87-112.
Age of Species
Time
p
t ( p)  2 N
ln( p)
1 p
Overestimate the age
of tree species in
orders!
p
Kimura, M. 1983. The neutral allele theory of molecular evolution. Cambridge Univ. Press.
Community Assembly Rules
Atmospheric inputs
Birth
Death
Growth
Competition
Predation
.....
Genetic
structure
Size
structure
Age
structure
Community
Patterns
Species
composition
Spatial
structure
Edaphic conditions
Geographical processes
Temporal
structure
Disturb:
Winds
Fires
Logging
Pollution
.....
Hubbell’s Neutral Theory
• Random walk
• Dispersal limitation
• Speciation
Metacommunity Neutral Model
Expected diversity
(Simpson index):
D
2 N


2 N  1   1
Expected # of species:
f ( n) 
x n
n
Species-abundance model:
 n
E ( S n )   ln 1  
 
Hubbell, S.P. 2001. The unified neutral theory of biodiversity and biogeography. Princeton Univ. Press.
Random Walk
0
j-1
Initial state
Extinction
0
1
2
3
…
0
1
0
0
0
... 0
1
q1
r1
p1
0
... 0
2
…
0 q2
... ...
r2
...
p2 ... 0
... ... ...
N
0
0
0
0
j
N
... 1
j+1
N
Fixation
N j j

p

p

j , j 1
 j
N N 1



j N j
q

p

 j
j , j 1
N N 1



r j  p j , j  1  p j , j 1  p j , j 1

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.
0
Extinction
j-1
j
Initial state
j+1
N
Fixation
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.
0
Extinction
j-1
j
Initial state
j+1
N
Fixation
Extinction time
Extinction probability
N = 20
tj
pj
N = 20
j
j
Differential Birth Rates
Zhang and Lin (1997) consider the case of differential birth rates
N j
uj

p

 j , j 1
N N  uj  j  1



j
N j

 p j , j 1 
N N  u ( j  1)  j



 p j , j  1  p j , j 1  p j , j 1

t
N = 100
j = 50
u
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
N j
uj

p

 j , j 1 N  vj  j N  uj  j  1



vj
N j

p

 j , j 1
N  vj  j N  u ( j  1)  j



 p j , j  1  p j , j 1  p j , j 1

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:
Q( N  j )
Pj  Q( N  j )
Nj
P
This leads to
, where v 
Q
N  vj  j
Effects of Birth and Death Rates on Coexistence Prob.
N = 50
j = 25
d
1.8
2.0
c
1.6
1.4
2.0 v
1.8
1.4 v
1.2
1.4
u
1.6
1.2
1.8
2.01.0
1.2
1.6
fixation
1.0
1.0
0.8
0.6
ej 0.4
0.2
0.0
1.0
extinction
1.0
1.2
1.4
1.6
u
1.8
2.0
N = 50
j = 45
f
2.0
e
1.8
1.4
2.0
1.4
1.2
1.4
u
1.2
1.6
1.8
2.01.0
v
1.2
1.6
fixation
1.0
1.0
0.8
ej 0.6
0.4
0.2
0.0
1.0
1.6
1.8
extinction
1.0
1.2
1.4
1.6
1.8
2.0
Effects of Birth and Death Rates on Coexistence Time
1.6
1.4
1.2
1.4
u
1.2
1.6
1.8
2.01.0
v
1.4
1.8
1.2
1500
tj 1000
500
1.0
1.0
2.0
1.6
1.8
2.0
N = 50
j = 25
1.0
1.2
1.4
1.6
1.8
2.0
N = 50
j = 25
f
1.6
1.4
1.2
1.4
u
1.2
1.6
1.8
2.01.0
v
1.4
1.8
1.2
tj
1200
1000
800
600
400
200
1.0
v
1.0
2.0
1.6
1.8
2.0
e
1.0
1.2
1.4
1.6
1.8
2.0
v=1
1500
1500
N = 50
j = 25
1500
t-u profiles for different death rates v’s.
500
500
tj
500
v = 1.2
v = 1.1
1.0 1.2 1.4 1.6 1.8 2.0
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 = 1.8
v=2
500
500
tj
500
u
v = 1.5
1500
v
1500
1500
t
1.0 1.2 1.4 1.6 1.8 2.0
1.0 1.2 1.4 1.6 1.8 2.0
1.0 1.2 1.4 1.6 1.8 2.0
u
u
u
Local Community Model


N j 
uj
 (1  m)
 m 
 p j , j 1 
N  vj  j 
N  uj  j  1






vj
N j



p

(
1

m
)

m
(
1


)
 j , j 1

N  vj  j 
N  u ( j  1)  j




 p j , j  1  p j , j 1  p j , j 1


Effects of Birth and Death Rates on Extinction
N = 50
j = 25
N = 50
j=5
1.0
0.8
ej 0.6
0.4
0.2
0.01.0
1.0
0.8
0.6
0.4
1.2
1.4
u
0.2
1.6
1.8
2.00.0
1.0
0.8
0.6
0.4
0.2
0.01.0
1.0
0.8
0.6
0.4
1.2
1.4
u
0.2
1.6
1.8
2.00.0
m
Effects of Birth, Death & Immigration Rates on Coexistence
v=1
tj
1e+15
8e+14
6e+14
4e+14
2e+14
0e+00
1.0
1.0
0.8
0.6
0.4
1.2
1.4
1.6
8e+05
6e+05
4e+05
2e+05
1.0
0.2
1.8
0.20
0.15
0.10
1.2
1.4
m
0.05
1.6
2.00.0
1.8
0.00
2.0
v=1.2
3e+13
1.0
0.8
0.6
2e+13
tj
1e+13
0e+00
1.0
0.4
1.2
1.4
u
1.6
0.2
1.8
2.00.0
1e+06
8e+05
6e+05
4e+05
2e+05
1.0
0.20
0.15
0.10 m
1.2
1.4
u
0.05
1.6
1.8
0.00
2.0
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
Local community model
N j j

p

 j , j 1
N N 1



j N j

 p j , j 1 
N N 1



 p j , j  1  p j , j 1  p j , j 1



N j
j

p

(
1

m
)
 m 

 j , j 1
N 
N 1




j 
N j


 m(1   ) 
 (1  m)
 p j , j 1 
N
N 1




 p j , j  1  p j , j 1  p j , j 1


N j
uj

p

 j , j 1 N  vj  j N  uj  j  1



vj
N j

p

 j , j 1
N  vj  j N  u ( j  1)  j



 p j , j  1  p j , j 1  p j , j 1




N j 
uj
 (1  m)
 m 
 p j , j 1 
N  vj  j 
N  uj  j  1






vj
N j



p

(
1

m
)

m
(
1


)
 j , j 1

N

vj

j
N

u
(
j

1
)

j





 p j , j  1  p j , j 1  p j , j 1


t
v
u
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.
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
t
v
u
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.