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Some Mathematical Challenges from Life SciencesPart IIPeter Schuster, Universität WienPeter F.Stadler, Universität LeipzigGünter Wagner, Yale University, New Haven, CTAngela Stevens, Max-Planck-Institut für Mathematik in den Naturwissenschaften, LeipzigandIvo L. Hofacker, Universität WienOberwolfach, GE, 16.-21.11.2003

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Web-Page for further information:

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

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Mathematics and the life sciences in the 21st century

  • Selection dynamics
  • RNA evolution in silico and optimization of structure and properties
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A  B

B  A

Reactions in the continuously stirred tank reactor (CSTR)

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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

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Autocatalytic second order reactions are the basis of selection processes.

The autocatalytic step is formally equivalent to replication or reproduction.

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Replication in the flow reactor

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

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Complementary replication as the simplest copying mechanism of RNA

Complementarity is determined by Watson-Crick base pairs:

GCandA=U

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

Mean fitness or dilution flux,(t), is a non-decreasingfunction of time,

Solutions are obtained by integrating factor transformation

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

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

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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.

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City-block distance in sequence space

2D Sketch of sequence space

Single point mutations as moves in sequence space

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

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

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