Some Mathematical Challenges from Life Sciences

Part II

Peter Schuster, Universität WienPeter F.Stadler, Universität Leipzig

Günter Wagner, Yale University, New Haven, CTAngela Stevens, Max-Planck-Institut für Mathematik in den

Naturwissenschaften, Leipzigand

Ivo L. Hofacker, Universität Wien

Oberwolfach, GE, 16.-21.11.2003

Web-Page for further information:

http://www.tbi.univie.ac.at/~pks

1. Mathematics and the life sciences in the 21st century

2. Selection dynamics

3. RNA evolution in silico and optimization of structure and properties

S tock S olution [a] = a0 R eaction M ixture [a],[b]

A

A

A

A

AAA

AA

A

A

A

A

A

A

A

A

A

A

B

BB

B

B

B

B

B

B

B

B

B

F low ra te r = 1R-

Reactions in the continuously stirred tank reactor (CSTR)

A B

B A

Reversible first order reaction in the flow reactor

2 .0 4 .0 6 .0 8 .0 1 0 .0

F lo w ra te r [ t ] -1

1 .0

0 .8

0 .6

1 .2C

once

ntra

tion

a

[a

] 0

A B

0 .2 5 0 .5 0 1 .0 00 .7 5 1 .2 5

F lo w ra te r [ t ] -1

1 .0

0 .8

0 .6

1 .2C

once

ntra

tion

a [

a] 0

A

A + B

B

2 B

= 0 = 0 .0 0 1 = 0 .1

=

Autocatalytic second order and uncatalyzed reaction in the flow reactor

0 .1 00 .0 80 .0 60 .0 40 .0 2 0 .1 2 0 .1 4 0 .1 6 0 .1 8

F lo w ra te r [ t ] -1

1 .0

0 .8

0 .6

1 .2C

once

ntra

tion

a

[a

] 0

A + 2 B

A

3 B

B

= 0 = 0 .0 0 1 = 0 .0 0 2 5 = 0 .0 0 7

=

Autocatalytic third order and uncatalyzed reaction in the flow reactor

Pattern formation in autocatalytic third order reactions

G.Nicolis, I.Prigogine. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations. John Wiley, New York 1977

A u to ca ta ly tic th ird o rd er re a ctio n s

A + 2 X 3 XD ire c t, , o r h id d e nin th e re a c t io n m e c h a n is m(B e lo u s o w -Z h a b o tin s k i i r e a c tio n ) .

M u lt ip le s te a d y s ta te s

O s c il la t io n s in h o m o g e n e o u s s o lu tio n

D e te rm in is tic c h a o s

T u r in g p a tte rn s

S p a t io te m p o ra l p a tte rn s (sp ira ls )

D e te rm in is tic c h a o s in s p a c e a n d t im e

Autocatalytic second order reactions are the basis of selection processes.

The autocatalytic step is formally equivalent to replication or reproduction.

A u tocata ly tic secon d ord er rea ction s

A + I 2 ID irec t, , o r h id d e n inth e re ac tio n m ec h a n ism

C h em ica l se lf-en h an cem en t

S electio n o f la ser m od es

S electio n o f m o lecu lar sp ecie s com p etin g for co m m on sou rces

C o m b u stio n an d ch em istry o f f lam es

Stock Solution [A ] = a0 R eaction M ixture: A ; I , k=1,2,...k

A + I 2 I1 1

A + I 2 I2 2

A + I 2 I3 3

A + I 2 I4 4

A + I 2 I5 5

k 1

k 2

k 3

k 4

k 5

d 1

d 2

d 3

d 4

d 5

Replication in the flow reactor

P.Schuster & K.Sigmund, Dynamics of evolutionary optimization, Ber.Bunsenges.Phys.Chem. 89: 668-682 (1985)

F lo w ra te r = R-1

Con

cent

rati

on o

f st

ock

solu

tion

a 0

A + I1A

+ I

+ I

1 2

A

A + I 2 I2 2

A + I 2 I3 3

A + I 2 I4 4

A + I 2 I5 5

A + I 2 I1 1

k > k > k > k > k1 2 3 4 5

Selection in the flow reactor: Reversible replication reactions

F lo w ra te r = R-1

Con

cent

rati

on o

f st

ock

solu

tion

a 0

A + I1

A

A + I 2 I2 2

A + I 2 I3 3

A + I 2 I4 4

A + I 2 I5 5

A + I 2 I1 1

k > k > k > k > k1 2 3 4 5

Selection in the flow reactor: Irreversible replication reactions

G

G

G

G

C

C

C G

C

C

G

C

C

G

C

C

G

C

C

G

C

C

C

C

G

G

G

G

G

C

G

C

P lu s S tran d

P lu s S tran d

M in u s S tran d

P lu s S tran d

P lu s S tran d

M in u s S tran d

3 '

3 '

3 '

3 '

3 '

5 '

5 '

5 '

3 '

3 '

5 '

5 '

5 '+

C o m p le x D isso c ia tio n

S y n th es is

S y n th es is

Complementary replication as the simplest copying mechanism of RNAComplementarity is determined by Watson-Crick base pairs:

GC and A=U

Reproduction of organisms or replication of molecules as the basis of selection

d x / d t = x - x

x

i i i

j j

i , j

f

f

i

j

f i

x - i

j jx = 1 ,2 ,. .. ,n

[ I ] = x 0 ; i i i = 1 ,2 ,. ..,n ;

I i

I1

I2

I1

I2

I1

I2

I i

I n

I i

I nI n

+

+

+

+

+

+

(A ) +

(A ) +

(A ) +

(A ) +

(A ) +

(A ) +

fn

f i

f1

f2

I mI m I m++(A ) +(A ) +f m

fm fj = m a x { ; j= 1 ,2 ,. ..,n }

x m (t) 1 fo r t

[A ] = a = c o n s ta n t

Selection equation: [Ii] = xi 0 , fi > 0

 

   

 Mean fitness or dilution flux, (t), is a non-decreasing function of time,

  

 

Solutions are obtained by integrating factor transformation 

fxfxnifxdt

dx n

j jj

n

i iiii 11

;1;,,2,1,

0var22

1

fffdt

dxf

dt

d in

ii

ni

tfx

tfxtx

j

n

j j

iii ,,2,1;

exp0

exp0

1

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

0 .2

0

0

0 .4

0 .6

0 .8

1

T im e [G e n e ra tio n s]

Fra

ctio

n of

adv

anta

geou

s va

rian

t

s = 0 .1

s = 0 .01

s = 0 .02

Selection of advantageous mutants in populations of N = 10 000 individuals

s = ( f2-f1) / f1; f2 > f1 ; x1(0) = 1 - 1/N ; x2(0) = 1/N

Changes in RNA sequences originate from replication errors called mutations.

Mutations occur uncorrelated to their consequences in the selection process and are, therefore, commonly characterized as random elements of evolution.

G

G

G

C

C

C

G

C

C

G

C

C

C

G

C

C

C

G

C

G

G

G

G

C

P lus S trand

P lus S trand

M inus S trand

P lus S trand

3 '

3 '

3 '

3 '

5 '

3 '

5 '

5 '

5 '

P o in t M u ta tio n

In ser tio n

D ele tio n

G AA AA UC C C G

G AAUC C A C G A

G AA AAUC C C G UC C C G

G AAUC C A

The origins of changes in RNA sequences are replication errors called mutations.

Chemical kinetics of replication and mutation as parallel reactions

I j

In

I2

I i

I1 I j

I j

I j

I j

I j

I j +

+

+

+

+

(A ) +

f j Q j1

f j Q j2

f j Q j i

f j Q j j

f j Q jn

Q (1 - ) i j-d (i,j) d (i ,j) = lp p

p .. ... ... .. E r ro r ra te p e r d ig it

d (i,j ) . ... H a m m in g d is ta n c e b e tw e e n I i a n d I j

. ... ... ... . C h a in le n g th o f th e p o ly n u c le o tid el

d x / d t = x - x

x

i j j i

j j

f

f x

j

j j i

Q j i

Q i j i = 1

[A ] = a = co n s ta n t

[I i] = x i 0 ; i = 1 ,2 ,... ,n ;

Single point mutations as moves in sequence space

.... GC UC ....C A

.... GC UC ....GU

.... GC UC ....GA .... GC UC ....C U

d = 1H

d = 1H

d = 2H

2D Sketch of sequence spaceCity-block distance in sequence space

Mutation-selection equation: [Ii] = xi 0, fi > 0, Qij 0

 

  

Solutions are obtained after integrating factor transformation by means of an eigenvalue problem

 

fxfxnixxQfdt

dx n

j jj

n

i iij

n

j jiji 111

;1;,,2,1,

)0()0(;,,2,1;exp0

exp01

1

1

0

1

0

n

i ikikn

j kk

n

k jk

kk

n

k iki xhcni

tc

tctx

njihHLnjiLnjiQfW ijijiji ,,2,1,;;,,2,1,;;,,2,1,; 1

1,,1,0;1 nkLWL k

spaceS equen ce

Con

cent

ratio

n

M a s te r s e q u e n c e

M u ta n t c lo u d

The molecular quasispecies in sequence space

e 1

e1

e3

e 3

e 2e 2

l 0

l 1

l 2

x 3

x 1

x 2

The quasispecies on the concentration simplex S3= 1;3,2,1,03

1 i ii xix

Quasispecies as a function of the replication accuracy q

E rro r ra te p = 1 -q0 .0 0 0 .0 5 0 .1 0

Q u asispecies U n ifo rm d is tribu tio n