Models of Populations by John Maynard Smith Presented by Elena Fanea - - PowerPoint PPT Presentation

Models of Populations

by John Maynard Smith Presented by Elena Fanea

CPSC 605 December 8, 2002

The Message

Simple models of population growth and evolutionary change can explain the answer to some important questions

Questions

1.

?

• 2. How accurate must the hereditary

process be ?

• 3. How frequencies fluctuate in a finite

population ?

Outline

Models of population growth Selection in an asexual population The accuracy of replication Genetic drift in finite populations

Population growing asexually by binary fission Ex: bacteria contains x individuals at time t r·dt = Probability of division in dt r = const - intrinsic rate of increase

Hypothesis

• 1. Models of Population Growth

rt x x e x x rx dt dx t rx x

rt t

+ = fi = fi = æ æ Æ æ =

Æ

ln ln

d

d d

Logarithmic growth

r = const ?

Synchrony Inherited differences between cells

• Lotka’s theorem

Natural Selection

Malthus fi Darwin & Wallace – natural selection K=carrying capacity

˜ ¯ ˆ Á Ë Ê - ⋅ = K x rx dt dx 1

Logistic Equation

• 2. Selection in an Asexual Population

x K1 y K2 Populations limited by different resources They will coexist indefinitely => no selection

˜ ˜ ¯ ˆ Á Á Ë Ê - ⋅ =

1 1

1 K x x r dt dx

{

˜ ˜ ¯ ˆ Á Á Ë Ê - ⋅ =

2 2

1 K y y r dt dy

Selection in an Asexual Population (Cont.)

Populations have the same resources

˜ ˜ ¯ ˆ Á Á Ë Ê +

• ⋅

=

1 1

1 K y x x r dt dx

{

˜ ˜ ¯ ˆ Á Á Ë Ê +

• ⋅

=

2 2

1 K y x y r dt dy

K1>K2 => x until x+ y = K1 y 0

} =>

x selectively eliminates y

Selection in an Asexual Population Conclusion

Natural selection replacement IIF The two types are limited by the same factors (are competing for resources)

• 3. The Accuracy of Replication

Replication process Exact Too inaccurate No evolution

Critical accuracy Q depends on:

• Success of non-optimal copies
• Average number of copies produced by optimal

particles In practice, evolution is unlikely if Q<1/2

The Accuracy of Replication – Cont.

Practical implication -> genome's size limited

A genome of n nucleotides, u =error of replication/nucleotide Q=(1-u)n @ e –nu => nu<1

Error rates

u non-enzymic replication 10-1-10-2 RNA replication (without “proof-reading” ) 10-3-10-4 DNA replication (with “proof-reading”) 10-9-10-10

The Accuracy of Replication – Cont.

Population of replicating RNA molecules S- optimal sequence, unique, produces copies at rate R >r Q-the probability to produce an exact copy of itself

RQx dt dx =

+

( )

1 1

1 x Q R rx dt dx ⋅

• +

=

( ) ( )

1 1 1

x x D rx Rx dt x x d +

• +

= +

The Accuracy of Replication – Cont.

If deaths = births, can optimal molecules survive?

( )

p r Rp x x rx Rx D

• +

= + + = 1

1 1

( )

[ ]

1 x p r Rp RQ dt dx ⋅

• +
• =

( )

p R r p Q

• +

= fi 1

at equilibrium

R r Q @

p=proportion of optimal molecules

• 4. Genetic Drift in Finite Populations

=fluctuations of proportions of different kinds

• f populations

fl type a - no selection - type A ‡ p - frequencies - q Np - # of individuals - Nq

Genetic Drift in Finite Populations (Cont.)

Next generation p’, q’ – frequencies E(p’)=p – expected value - unchanged E(q’)=q V=E(p’-p’)2 => V=E(p’-p)2 – variance s(p’)=V1/2 –standard deviation

Genetic Drift in Finite Populations (Cont.)

Family size has a Poisson distribution Each offspring was assigned randomly to one parent, independently Has a probability p of being a The probabilities of 0, 1, 2…N offspring of type a : pN, NpN-1q, N(N-1)/2pN-2q2, …, qN

Genetic Drift in Finite Populations (Cont.)

Standard deviation Variance Mean Frequencies of a individuals Number of a individuals The binomial theorem: Npq

N pq

Npq Np p N pq

Genetic Drift in Finite Populations (Cont.)

Each new individual is equally likely to be produced by any one of the N parents After 2N generations, all will descend from the same individual Standard error >= N

Genetic Drift in Finite Populations Conclusion

No natural selection=>fluctuant frequencies Population => fluctuations One type will ultimately become fixed, by chance

Simulation

Summary

Mathematical models can describe Population growth Selection in an asexual population The accuracy of replication Genetic drift in finite populations And many other aspects of evolution

References

John Maynard Smith – Evolutionary Genetics,

second edition, Oxford University Press, 2000 chapter 2