Abstract
Selection–mutation dynamics is studied as adaptation and neutral drift on abstract fitness landscapes. Various models of fitness landscapes are introduced and analyzed with respect to the stationary mutant distributions adopted by populations upon them. The concept of quasispecies is introduced, and the error threshold phenomenon is analyzed. Complex fitness landscapes with large scatter of fitness values are shown to sustain error thresholds. The phenomenological theory of the quasispecies introduced in 1971 by Eigen is compared to approximation-free numerical computations. The concept of strong quasispecies understood as mutant distributions, which are especially stable against changes in mutations rates, is presented. The role of fitness neutral genotypes in quasispecies is discussed.
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Notes
- 1.
The expression hypersurface points at the fact that fitness landscapes are surfaces in high-dimensional space. Since we shall be dealing here almost exclusively with such high-dimensional objects, we drop the prefix ‘hyper.’
- 2.
The genotype space in Wright’s seminal paper (Wright 1932) is a space of genes, whereas we use virus genomes as elements of genotypes space. Accordingly, genotype space is identical with the space of DNA or RNA sequences of the chain length of virus genomes.
- 3.
Here, ‘aa’ stands for ‘amino acid residue.’
- 4.
Considering single nucleotides as sites in structural RNA elements requires complementarity of the nucleobase at another locus for the formation of a base pair, and accordingly, the two sites are strongly coupled epistasis (see Sect. 2).
- 5.
- 6.
A matrix W is primitive if (i) all the elements of matrix W are nonnegative and (ii) some finite power W m is a positive matrix, which means that all entries of W m are strictly positive.
- 7.
- 8.
By the notion ‘exact,’ we mean here ‘without approximations.’ In order to make clear that numerical computations can never be exact in the strict sense, we put exact between apostrophes.
- 9.
The use of binary sequences (\( \kappa = 2 \)) facilitates several operations and implies no loss of generality. Natural four-letter sequences \( (\kappa = 4 ) \) can be encoded by binary sequences of twice the chain length.
- 10.
For sufficiently long sequences, the particular choice ϑ = 0.01, 0.001 or 0.0001 is unimportant because the results for small values are very similar and converge to a limit (see Fig. 9), but for short chains, the concentration values of the uniform distribution \( \mathcal{U} \) set a lower limit for \( \bar{x}_{m} (p) \). For example, in case of \( l = 10 \), the value \( \bar{x}_{m} (\frac{1}{2}) = 1/2^{l} = 1/1024 \) is compatible only with the choice \( \vartheta = 1/100 \) because \( \vartheta = 1/1000 \) is too close to \( \bar{x}_{m} (\frac{1}{2}) \).
- 11.
The stationary concentrations of the phenomenological approach are denoted by the ‘hat’ symbol: \( \hat{x}_{m}^{(0)} ,\;\hat{x}_{j}^{(0)} ,\;\hat{y}_{k}^{(0)} ,\;\hat{c}^{(0)} \), etc.
- 12.
This agreement is not accidental as a simple consideration shows: The lowest mutation rate for merging two classes is \( (p_{{{\text{m}}g}}^{(\theta )} )_{0} \), the p-value where \( \varDelta_{0} = |\bar{y}_{0} - \bar{y}_{l} | = |\bar{x}_{m} - \bar{x}_{ - m} | = \theta \). Since the concentration of the complementary sequence of the master sequence with \( {\text{d}}_{{{\mathsf{X}}_{m} {\mathsf{X}}_{ - m} }}^{\text{H}} = l \) is commonly very small, \( \bar{x}_{ - m} \ll \bar{x}_{m} \), we find for \( \vartheta = \theta \): \( \varDelta_{0} \approx \bar{x}_{m} \) and \( p_{\text{tr}}^{(\vartheta )} \approx \hbox{min} (p_{{{\text{m}}g}}^{(\theta )} )_{k} = (p_{{{\text{m}}g}}^{(\theta )} )_{0} \).
- 13.
The single-peak linear landscape with \( h = 1 \) is identical with the single peak fitness landscape. The error threshold for h = 5 extends almost to \( p = \frac{1}{2} \), and landscapes with \( h > 5 \) do not support error thresholds at all.
- 14.
Numerical computations of eigenvalues and eigenvectors become highly demanding with respect to CPU time and memory above \( l = 10 \). For \( l = 20 \), the diagonalization of the W-matrix with about the size \( 10^{6} \times 10^{6} \) requires certain tricks (Niederbrucker and Gansterer 2011), and for \( l = 50 \), the dimension of W is more than \( 10^{15} \times 10^{15} \) and diagonalization is far beyond current technical capacities.
- 15.
Thirteen years after this publication, the phenomenon has been observed in quasispecies of digital organisms (Wilke et al. 2001) and was called survival of the flattest.
- 16.
Naïvely, we would expect a band of one-error sequences at higher concentration than the two-error sequence.
- 17.
For class k = 1, we omit the master sequence \( {\mathsf X}_{m} \), which trivially is the fittest sequence in the one-error neighborhood, and search only in class \( k = 2 \).
- 18.
The adjacency matrix of a graph, A, is a symmetric square matrix that has an entry \( a_{jk} = a_{kj} = 1 \) whenever the graph has an edge between the nodes for \( {\mathsf X}_{j} \) and \( {\mathsf X}_{k} \) and zero entries everywhere else.
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Acknowledgements
The calculations reported here were done with the package Mathematica 9.0. For the simulations based on Gillespie’s algorithm, we made use of the open-source SSA-program within X-Cellerator package. We are grateful to Bruce Shapiro for making this software public.
The author wishes to acknowledge support by the University of Vienna, Austria, and the Santa Fe Institute, Santa Fe, USA. A number of colleagues have helped in discussions. I am particulary grateful to Reinhard Bürger, Esteban Domingo, Christoph Flamm, Leticia Gonzalez-Herrero, Ivo Hofacker, Markus Oppel, David Saakian, and Peter Stadler.
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Schuster, P. (2015). Quasispecies on Fitness Landscapes. In: Domingo, E., Schuster, P. (eds) Quasispecies: From Theory to Experimental Systems. Current Topics in Microbiology and Immunology, vol 392. Springer, Cham. https://doi.org/10.1007/82_2015_469
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