Sequence space mutations

The choice of the particular upward pathway in the kinetic resolution of rac-19, that is, the specific order of choosing the sites in ISM, appeared arbitrary. Indeed, the pathway B C D F E, without utilizing A, was the first one that was chosen, and it led to a spectacular increase in enantioselectivity (Figure 2.15). The final mutant, characterized by nine mutations, displays a selectivity factor of E=115 in the model reaction [23]. This result is all the more remarkable in that only 20000 clones were screened, which means that no attempt was made to fully cover the defined protein sequence space. Indeed, relatively small libraries were screened. The results indicate the efficiency of iterative CASTing and its superiority over other strategies such as repeating cycles of epPCR. 

Even if we restrict our design to a small number of sites in the protein, the combinatorial possibilities quickly approach astronomical dimensions. If we consider mutations at 10 sites and a subset of 10 amino acids, we have 1010 possible variants. Although experimental approaches are under development that can actually search large subsets of protein sequence space, it is not at all a small feat to identify those variants that give rise to a stable structure and at the same time come close to the desired features. Therefore, computational approaches that, with some reliability, are able to pick those variants having a stable structure are desirable instruments in the protein engineer s toolbox. 

Figure 10. The molecular quasispecies and its support in sequence space. Due to unavoidable non-zero mutation rates, replicating populations form distributions of genotypes or polynucleotide sequences. As shown in the sketch these distributions are centered around a most frequent genotype called the master sequence. A population thus occupies a connected region in sequence space which, according to usual mathematical terminology, is called the support of the population.

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

If genotypes are ordered in sequence space, the support forms an area which consists of one, two or more connected compounds. T vo genotypes are connected when they are separated by a single point mutation, i.e. when they have Hamming distance one. 

Subsequent directed evolution work on Pseudomonas aeruginosa demonstrates that protein sequence space with respect to enantioselectivity is best explored by a three-step procedure (Reetz, 2001) (i) generation of mutants by error-prone PCR at a high-mutation rate (ii) identification of hot regions and spots in the enzyme by error-prone PCR and substantiation by simplified combinatorial multiple-cassette mutagenesis (iii) extension of the process of combinatorial multiple-cassette mutagenesis to cover a defined region of protein sequence space. 

The fitness function is simply the mapping between points in sequence space and their fitnesses. The fitness landscape is the combination of the fitness function and the neighbor relationship. With neighbors defined by point mutation, for an N-site molecule, the landscape is the N-dimensional surface that results from plotting the fitness function over an N-dimensional Cartesian coordinate sequence space (Fig. 13). 

For techniques not using mutation, the minimum affinity needed in the initial library is that which would be an adequate result from the search. There are no estimates for the repertoire sizes that cover target sequence space with higher minimal affinities however, arguments based on catalytic task space suggest that 10s molecules are sufficient for saturation [4], If enzymes are considered to bind to transitional states between reactants and products, then 10s ligands may be adequate for techniques without mutation. However, there is no estimate for the minimal affinity towards a transitional state needed to catalyze a reaction, so it is unclear what exactly is achieved by a library of this size. 

The accessibility of many novel structures one point mutation off most neutral nets is not simply a consequence of a net s percolating through sequence space. Different nets could shadow one another to a great extent, limiting the number of different structures easily accessed. Instead, networks meander so that the networks of the vast majority of all common structures are within only one point mutation of at least one point on the networks for every other common structure [64,75],... 

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