Selection Theory II

Chris Kempes

• What are the basic ways in which we think about evolution?

•

Fitness Landscape00

0...0

000.

..110

0...0 ......

sequence

fitness

Quasispecies equationdxi

dt=

nX

j=1

xjfjqj,i � �xi

xi Frequency of a given sequence i

Fitness of a given sequence i (e.g. relative growth rate)fi

Probability of mutating from sequence i to sequence j

qi,j

� =nX

i=1

xifi

Steady State Solutions

f0f1

Steady State Solutions

f0

f0 > 1

f1 = 1

f1

Steady State Solutions

f0

f0 > 1 q0,0 = (1� u)L

f1 = 1

f1

Steady State Solutions

f0

f0 > 1 q0,0 = (1� u)L

q0,1 = 1� q0,0f1 = 1

f1

Steady State Solutions

f0

f0 > 1 q0,0 = (1� u)L

q1,1 = 1

q0,1 = 1� q0,0

q1,0 = 0f1 = 1

f1

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutions

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutionsf0 > 1

f1 = 1

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutionsf0 > 1

q1,1 = 1

q1,0 = 0

f1 = 1

q0,0 = (1� u)L ⌘ q

q0,1 = 1� q

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutionsf0 > 1

q1,1 = 1

q1,0 = 0

f1 = 1

q0,0 = (1� u)L ⌘ q

q0,1 = 1� q

dx0

dt= x0f0q � x0�

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutionsf0 > 1

q1,1 = 1

q1,0 = 0

f1 = 1

q0,0 = (1� u)L ⌘ q

q0,1 = 1� q

dx0

dt= x0f0q � x0�

dx1

dt= x0f0 (1� q) + x1 � x1�

dxi

dt=

nX

j=1

xjfjqj,i � �xi

� =nX

i=1

xifi

Steady State Solutionsf0 > 1

q1,1 = 1

q1,0 = 0

f1 = 1

q0,0 = (1� u)L ⌘ q

q0,1 = 1� q

dx0

dt= x0f0q � x0�

dx1

dt= x0f0 (1� q) + x1 � x1�

� = f0x0 + x1

Steady State Solutionsdx0

dt= x0f0q � x0�

dx1

dt= x0f0 (1� q) + x1 � x1�

� = f0x0 + x1

Steady State Solutionsdx0

dt= x0f0q � x0�

dx1

dt= x0f0 (1� q) + x1 � x1�

� = f0x0 + x1

x0 + x1 = 1

Steady State Solutionsdx0

dt= x0f0q � x0�

dx1

dt= x0f0 (1� q) + x1 � x1�

� = f0x0 + x1

x0 + x1 = 1

dx0

dt= x0 [f0q � 1� x0 (f0 � 1)]

Steady State Solutionsdx0

dt= x0 [f0q � 1� x0 (f0 � 1)]

Steady State Solutionsdx0

dt= x0 [f0q � 1� x0 (f0 � 1)]

x⇤0 =

f0q � 1

f0 � 1

Steady State Solutionsdx0

dt= x0 [f0q � 1� x0 (f0 � 1)]

x⇤0 =

f0q � 1

f0 � 1

f0q > 1x⇤0 > 0

Error Threshold

f0f1

f0q > 1 log (f0) > �L log (1� u)

u <log (f0)

L