Mathematical consideration of massive estimation of dissociation rate constant for genotype-phenotype linking molecules bound to targets through washing/selection and next-generation sequencing.

As one of methods for in vitro selection, a flow reactor type washing/selection system seems to be effective, where a ligand library is composed of "genotype-phenotype linking molecules". In this system, high affinity ligands are selected by their respective "residual ratio" given by exp(-k×t), where k is the dissociation rate constant and t is the washing time. In this paper, we mathematically considered the following possibility.

Inference of fitness values and putative appearance time points for evolvable self-replicating molecules from time series of occurrence frequencies in an evolution reactor.

We have established a translation-coupled RNA replication system within a cell-like compartment, and conducted an experimental evolution of the RNA molecules in the system. Then, we obtained a time series of occurrence frequencies of 91 individual genotypes through random sampling and next-generation sequencing. The time series showed a complex clonal interference and a polymorphic population called the "quasispecies".

Periodic Pattern of Genetic and Fitness Diversity during Evolution of an Artificial Cell-Like System.

Genetic and phenotypic diversity are the basis of evolution. Despite their importance, however, little is known about how they change over the course of evolution. In this study, we analyzed the dynamics of the adaptive evolution of a simple evolvable artificial cell-like system using single-molecule real-time sequencing technology that reads an entire single artificial genome.

Evolutionary dynamics of a polymorphic self-replicator population with a finite population size and hyper mutation rate.

Self-replicating biomolecules, subject to experimental evolution, exhibit hyper mutation rates where the genotypes of most offspring have at least a one point mutation. Thus, we formulated the evolutionary dynamics of an asexual self-replicator population with a finite population size and hyper mutation rate, based on the probability density of fitnesses (fitness distribution) for the evolving population. As a case study, we used a Kauffman's "NK fitness landscape".

Detection of ultra-low levels of DNA changes by drinking water: epidemiologically important finding.

The safety of drinking water is essential to our health. In this context, the mutagenicity of water needs to be checked strictly. However, from the methodological limit, the lower concentration (less than parts per million) of mutagenicity could not be detected, though there have been of interest in the effect of less concentration mutagens.

Systematic genome sequence differences among leaf cells within individual trees.

Background: Even in the age of next-generation sequencing (NGS), it has been unclear whether or not cells within a single organism have systematically distinctive genomes. Resolving this question, one of the most basic biological problems associated with DNA mutation rates, can assist efforts to elucidate essential mechanisms of cancer.

Results: Using genome profiling (GP), we detected considerable systematic variation in genome sequences among cells in individual woody plants.

Estimation of statistical binding properties of ligand population during in vitro selection based on population dynamics theory.

During in vitro selection process, it is very valuable to monitor the binding properties of the ligand population in real time, particularly the population average of the association constant in the population. If this monitoring can be realized, the selection process can be controlled in a rational way. In this paper, we present a simple method to estimate the binding properties of the ligand population during in vitro selection.

Probabilistic model based error correction in a set of various mutant sequences analyzed by next-generation sequencing.

To analyze the evolutionary dynamics of a mutant population in an evolutionary experiment, it is necessary to sequence a vast number of mutants by high-throughput (next-generation) sequencing technologies, which enable rapid and parallel analysis of multikilobase sequences. However, the observed sequences include many errors of base call. Therefore, if next-generation sequencing is applied to analysis of a heterogeneous population of various mutant sequences, it is necessary to discriminate between true bases as point mutations and errors of base call in the observed sequences, and to subject the sequences to error-correction processes.

Theoretical consideration of selective enrichment in in vitro selection: optimal concentration of target molecules.

We considered an in vitro selection system composed of a peptide-ligand library and a single target protein receptor, and examined effective strategies to realize maximum efficiency in selection. In the system, a ligand molecule with sequence s binds to a target receptor with probability of [R]/(K(ds)+[R]) (specific binding) or binds to non-target materials with probability of q (non-specific binding), where [R] and K(ds) represent the free target-receptor concentration at equilibrium and dissociation constant K(d) of the ligand sequence s with the receptor, respectively. Focusing on the fittest sequence with the highest affinity (represented by K(d1) ≡ min{K(ds)|s=1,2,…,M}) in the ligand library with a library size N and diversity M, we examined how the target concentration [R] should be set in each round to realize the maximum enrichment of the fittest sequence.

A mapping of an ensemble of mitochondrial sequences for various organisms into 3D space based on the word composition.

To visualize a bird's-eye view of an ensemble of mitochondrial genome sequences for various species, we recently developed a novel method of mapping a biological sequence ensemble into Three-Dimensional (3D) vector space. First, we represented a biological sequence of a species s by a word-composition vector x(s), where its length [absolute value]x(s)[absolute value] represents the sequence length, and its unit vector x(s)/[absolute value]x(s)[absolute value] represents the relative composition of the K-tuple words through the sequence and the size of the dimension, N=4(K), is the number of all possible words with the length of K. Second, we mapped the vector x(s) to the 3D position vector y(s), based on the two following simple principles: (1) [absolute value]y(s)[absolute value]=[absolute value]x(s)[absolute value] and (2) the angle between y(s) and y(t) maximally correlates with the angle between x(s) and x(t).

Toward an evolutionary containment of evolving pathogen-receptors by using an ensemble of multiple mutant ligands: from the viewpoint of fitness landscape in sequence space.

It is known that even if a ligand peptide is designed to bind to a target receptor on the surface of a pathogen such as viruses, bacteria or cancer cells, it is likely that some receptors are subject to random mutation and thus the ligand has a reduced ability to bind to these receptors. This issue is known as drug-resistant or escape mutants. In this paper, we present an idea to inhibit the evolving receptors by using an ensemble of all possible single- or double-point mutant sequences of the ligand peptide.

Biophysical connection between evolutionary dynamics and thermodynamics in in vitro evolution.

We analyzed a mathematical model of in vitro evolution conducted by repetition of mutagenesis and selection processes. The selection process consists of the selective enrichment and subsequent sampling as follows: each mutant with fitness W is amplified by the Boltzmann factor exp(rW/k(B)T(the)), where the fitness W is defined as the negative Gibbs free energy (-ΔG) in a reaction of the phenotypic molecules and r is the round number of the selective enrichment; then, an arbitrary mutant is randomly chosen from the resulting mutant population and it becomes a new parent in the next generation. As a result, we found that the evolutionary dynamics is described in a mathematical framework similar to thermodynamics: the "evolution constant" k(E) and "evolutionary temperature" T(evo) play key roles similar to the Boltzmann constant k(B) and thermodynamic temperature T(the), respectively.

A visualization of 3D proteome universe: mapping of a proteome ensemble into 3D space based on the protein-structure composition.

To visualize a bird's-eye view of an ensemble of proteomes for various species, we recently developed a novel method of mapping a proteome ensemble into Three-Dimensional (3D) vector space. In this study, the "proteome" is defined as the entire set of all proteins encoded in a genome sequence, and these proteins were dealt with at the level of the SCOP Fold. First, we represented the proteome of a species s by a 1053-dimensional vector x(s), where its length ∣x(s)∣ represents the overall amount of all the SCOP Folds in the proteome, and its unit vector x(s)/∣x(s)∣ represents the relative composition of the SCOP Folds in the proteome and the size of the dimension, 1053, is the number of all possible Folds in the proteome ensemble given.

A mathematical consideration of the word-composition vector method in comparison of biological sequences.

To measure the similarity or dissimilarity between two given biological sequences, several papers proposed metrics based on the "word-composition vector". The essence of these metrics is as follows. First, we count the appearance frequencies of all the K-tuple words throughout each of two given sequences.

A trade-off relationship between energetic cost and entropic cost for in vitro evolution.

In this paper, we consider two complementary cost functions for the landscape exploring processes to obtain the global optimum sequence through in vitro evolution protocol: one is the entropic cost C(etp), which is based on the deviation from the uniformity of a mutant distribution in sequence space, and the other is the energetic cost C(eng), which is based on the total number of sequences to be generated and evaluated. Based on a prior knowledge about the structure of a given fitness landscapes, the conductor of the experiment can think up the efficient search algorithm (ESA), which requires the minimum number of points (=sequences) to be searched up to the global optimum. For five typical fitness landscapes, we considered their respective (putative) ESA, C(etp)(*) and C(eng)(*) based on the ESA.

Biomolecular information gained through in vitro evolution.

An in vitro evolution is a simplified Darwinian evolution in well-controlled surroundings. This evolution process can be modeled as a hill-climbing or adaptive walk on a fitness landscape in sequence space. The evolving molecular system gains at least two kinds of information originating from the converged sequences and the fitness increment of the evolving biopolymer as the adaptive walker.

Toward the fast blind docking of a peptide to a target protein by using a four-body statistical pseudo-potential.

In vitro molecular evolution creates a lot of peptide aptamers that bind to each target protein. In many cases, their binding sites on a protein surface are not known. Then, predicting the binding sites through computation within a reasonable time is desirable.

Quantifying epistatic interactions among the components constituting the protein translation system.

In principle, the accumulation of knowledge regarding the molecular basis of biological systems should allow the development of large-scale kinetic models of their functions. However, the development of such models requires vast numbers of parameters, which are difficult to obtain in practice. Here, we used an in vitro translation system, consisting of 69 defined components, to quantify the epistatic interactions among changes in component concentrations through Bahadur expansion, thereby obtaining a coarse-grained model of protein synthesis activity.

Hierarchical distribution of ascending slopes, nearly neutral networks, highlands, and local optima at the dth order in an NK fitness landscape.

We obtained several structural features of an NK fitness landscape by analytical approach. Particularly, we focused on spatial distributions of "ascending slopes", "highlands", "nearly neutral networks", and "local optima" along the fitness coordinate W, from the viewpoint of adaptive walks with step-width d , where d is the number of mutated sites (Hamming distance) after a generation. The parameter k governs the degree of the ruggedness on the NK landscape, and we handled cases where k is moderate against the sequence length.

Fitting protein-folding free energy landscape for a certain conformation to an NK fitness landscape.

The NK fitness landscape is a mathematical landscape model with a parameter k that governs the degree of ruggedness of the landscape. We presented a procedure to fit a given landscape to the NK fitness landscape by introducing the "apparent k-value"k(app). In this paper, we defined the protein free energy (DeltaG) landscape in amino acid sequence space, where DeltaG is the folding free energy from a random coil to a "certain conformation".

Extracting characteristic properties of fitness landscape from in vitro molecular evolution: a case study on infectivity of fd phage to E.coli.

We have developed a methodology for extracting characteristic properties of a fitness landscape of interest by analyzing fitness data on an in vitro molecular evolution. The in vitro evolution is required to be conducted as the following "adaptive walk": a single parent sequence generates N mutant sequences as its offsprings, and the fittest individual among the N offsprings will become a new parent in the next generation. N is the library size of mutants to be screened in a single generation.

Experimental rugged fitness landscape in protein sequence space.

The fitness landscape in sequence space determines the process of biomolecular evolution. To plot the fitness landscape of protein function, we carried out in vitro molecular evolution beginning with a defective fd phage carrying a random polypeptide of 139 amino acids in place of the g3p minor coat protein D2 domain, which is essential for phage infection. After 20 cycles of random substitution at sites 12-130 of the initial random polypeptide and selection for infectivity, the selected phage showed a 1.

Directed evolution by accumulating tailored mutations: thermostabilization of lactate oxidase with less trade-off with catalytic activity.

We assumed that adverse effects posed by introducing multiple mutations could be decomposed into those of each of the component mutations and that the risk could be reduced by the accumulation of mutations that were finely tuned for directed improvement of a specific property. We propose here a directed evolution strategy for improving a specific property with less effect on other ones. This strategy is composed of fine-tuning of mutations and their accumulation by our original mutation-assembling method.

Modified substrate specificity of pyrroloquinoline quinone glucose dehydrogenase by biased mutation assembling with optimized amino acid substitution.

A biased mutation-assembling method-that is, a directed evolution strategy to facilitate an optimal accumulation of multiple mutations on the basis of additivity principles, was applied to the directed evolution of water-soluble PQQ glucose dehydrogenase (PQQGDH-B) to reduce its maltose oxidation activity, which can lead to errors in blood glucose determination. Mutations appropriate for the reduction without fatal deterioration of its glucose oxidation activity were developed by an error-prone PCR method coupled with a saturation mutagenesis method. Moreover, two types of incorporation frequency based on their contribution were assigned to the mutations: high (80%) and evens (50%), in constructing a multiple mutant library.

Biased mutation-assembling: an efficient method for rapid directed evolution through simultaneous mutation accumulation.

We have developed an efficient optimization technique, 'biased mutation-assembling', for improving protein properties such as thermostability. In this strategy, a mutant library is constructed using the overlap extension polymerase chain reaction technique with DNA fragments from wild-type and phenotypically advantageous mutant genes, in which the number of mutations assembled in the wild-type gene is stochastically controlled by the mixing ratio of the mutant DNA fragments to wild-type fragments. A high mixing ratio results in a mutant composition biased to favor multiple-point mutants.