Statistics of protein library construction

Andrew E. Firth 1 Wayne M. Patrick 0 0 Center for Fundamental and Applied Molecular Evolution, Emory University , Atlanta, GA 30322 , USA 1 Department of Biochemistry, University of Otago , PO Box 56, Dunedin , New Zealand Summary: We have investigated the statistics associated with constructing and sampling large protein-encoding libraries. Using fairly simple statistics we have written algorithms for estimating the diversity in libraries generated by the most commonly used protocols, including error-prone PCR, DNA shuffling, StEP PCR, oligonucleotide-directed randomization, MAX randomization, synthetic shuffling, DHR, ADO and SISDC. Availability: Web interface and C++ source code available at http://guinevere.otago.ac.nz/stats.html Contact: Supplementary information: Complete mathematical notes, model assumptions and justification, users' guide and worked examples at above website. - INTRODUCTION Directed evolution is a powerful strategy for generating new proteins with desirable properties. Central to the technique is the generation of large sequence libraries. There are a number of methods for generating molecular diversity in these libraries (reviewed by Lutz and Patrick, 2004). However, to maximize the chances of finding a desired and rare improved variant, it is important to understand the statistics of library construction. Previously, we introduced a suite of algorithms for calculating library statistics for a variety of protocols. Since then, the equations and programs have been used a number of times (e.g. Hughes et al., 2005). However, the programs were a little unwieldy and required compiling by the user. In this short paper we present an improved and easy-to-use web interface, which will return a variety of library statistics and graphics for user-defined library sizes, mutation rates, sequence lengths, etc. These statistics may be used to direct experimental design (e.g. to determine what library size is required to sample a given amount of diversity, or to optimize the mutation rate to maximize diversity) and to interpret results (e.g. by estimating how many distinct sequences are represented in a given library). We note that more detailed models of some of the processes involved in library construction have been published (reviewed by Moore and Maranas, 2004). However, these models are not generally accessible to most laboratory researchers, can be CPU-intensive, and are less widely applicable than the generic tools that we present here. The web interface is available at http://guinevere.otago.ac.nz/stats. html. Users are referred to our original paper (Patrick et al., 2003) for experimental details, usage examples and a few caveats. Users interested in the mathematics behind the programs are invited to read the mathematical notes on our website. In the remainder of this short paper, we introduce the three main programs, GLUE, PEDEL and DRIVeR, and list situations in which they may be useful. EQUALLY PROBABLE VARIANTS The simplest program, GLUE, is broadly applicable to any protocol where all possible variants are equally likely to occur in the library. Examples include oligonucleotide-directed randomization, MAX randomization, synthetic shuffling, DHR, ADO and SISDC. Given the total number of possible variants, GLUE may be used to calculate (1) the expected number of distinct variants represented in a given library, (2) the library size required to sample a given fraction of the variants or (3) the library size required to have a given probability of sampling all possible variants. For example, if there are 1 million possible variants (e.g. an oligonucleotide-directed randomization involving four NNK codons allows 324 = 1 048 576 variants), GLUE shows that a library of 3 million transformants will be 95% complete, while a library of 17 million transformants has a 95% probability of being 100% complete. ERROR-PRONE PCR (epPCR) In this protocol, random base substitutions are introduced into a parent sequence. Although most recent examples of directed evolution use epPCR in conjunction with recombination-based strategies such as DNA shuffling, it is still commonly encountered as a means of generating random diversity at any position in a gene. The program PEDEL can be used to calculate the expected number of distinct variants present in a library, given the library size, mean substitution rate and parent sequence length. On the web page, the user may produce plots of the expected number of distinct daughter sequences as a function of library size and substitution rate. The user can also produce statistics and plots for the total number of variants with exactly x mutations, the expected size of the sub-library comprising those sequences with exactly x mutations, the completeness of each sub-library, and the redundancy of each sub-library. For example, given a library of 107 clones, a parent sequence length of 600 nt, and a mean substitution rate of 2 bases per daughter sequence, PEDEL calculates that the library is expected to contain 4.5 106 distinct sequences. These comprise 1.3, 1.8, 0.9, 0.4 and 0.1 million distinct sequences with, respectively, exactly 2, 3, 4, 5 and 6 mutations, together with the parent sequence, the 1800 distinct sequences with exactly 1 mutation, and 4.5 104 sequences with >6 mutations. The rest of the 107 clones break down into 1.4, 2.7 and 1.4 million redundant sequences with, respectively, exactly 0, 1 and 2 mutations. PEDEL uses a generic Poisson model of sequence mutations. All base substitutions are assumed equally likely. In reality, polymerases favour some substitutions over others. This will reduce the number of distinct sequences compared with the PEDEL predictions. The effect is limited by the fact that, for low substitution rates, the library tends to saturate all possible variants while, for high substitution rates, there are so many possible variants that, even with substitution bias, nearly every library member is distinct (Patrick et al., 2003). Note that, by using sequential PCR amplifications with two different polymerases with opposite substitution biases, it is possible to produce unbiased libraries. Another possible source of bias results from the uneven representation of mutations introduced early and late in the epPCR process. However, in practice one might use 109 identical template sequences, amplify them to perhaps 1015 product molecules in the epPCR, but usually only end up with a library of <107 variants after ligation and transformation. Under such conditions, amplification bias would have a typical frequency of 1 in 109, and would be undetectable in the final library of 107. In addition, different parent molecules may be copied a different number of times, but empirically the end result is a library with a Poisson distribution of mutations (Cadwell and Joyce, 1992). DNA SHUFFLING AND StEP PCR The program DRIVeR is applicable to libraries generated by recombining two highly homologou (...truncated)