Simulations of SELEX against complex receptors with a condensed statistical model

The Effects of SELEX Conditions on the Resultant Aptamer Pools in the Selection of Aptamers Binding to Bacterial Cells

Aptamers of high affinity and specificity have a wide range of analytic and clinical applications. Selection of DNA or RNA aptamer molecules usually involves systematic evolution of ligands via exponential enrichment (SELEX), in which a random DNA or RNA library is incubated with a target molecule, and the oligonucleotides that bind the target are then separated from the nonbinders, PCR amplified, and used as refined libraries in the next round of selection. Conventional SELEX methodologies require the use of purified target molecules and their immobilization onto a solid support. However, purified targets from cells are not always available, and fixing the target to a support may alter its conformation. To overcome these problems, we have developed a SELEX technique using live bacterial cells in suspension as targets, for selecting DNA aptamers specific to cell-surface molecules. Through the selection of aptamers binding to Lactobacillus acidophilus and Streptococcus pyogenes, we report here optimization of this technique and show how varying selection conditions impact the characteristics of resultant aptamer pools, including the binding affinity, selectivity, and the secondary structures. We found that the use of larger starting library sequence diversity, gel purification of the subsequent pools, and the introduction of counter-selection resulted in a more efficient SELEX process and more selective aptamers. A SELEX protocol with lower starting sequence diversity, the use of heat denaturation, and the absence of counter-selection still resulted in high-affinity aptamer sequences specific to the target cell types; however, the SELEX process was inefficient, requiring 20 rounds, and the aptamers were not specific to the strain of the bacterial cells. Strikingly, two different SELEX methodologies yielded the same sequence that bound strongly to the target S. pyogenes cells, suggesting the robustness of the bacterial cell-SELEX technique.

The universal statistical distributions of the affinity, equilibrium constants, kinetics and specificity in biomolecular recognition

We uncovered the universal statistical laws for the biomolecular recognition/binding process. We quantified the statistical energy landscapes for binding, from which we can characterize the distributions of the binding free energy (affinity), the equilibrium constants, the kinetics and the specificity by exploring the different ligands binding with a particular receptor. The results of the analytical studies are confirmed by the microscopic flexible docking simulations. The distribution of binding affinity is Gaussian around the mean and becomes exponential near the tail. The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail. The intrinsic specificity for biomolecular recognition measures the degree of discrimination of native versus non-native binding and the optimization of which becomes the maximization of the ratio of the free energy gap between the native state and the average of non-native states versus the roughness measured by the variance of the free energy landscape around its mean. The intrinsic specificity obeys a Gaussian distribution near the mean and an exponential distribution near the tail. Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail. Our study provides new insights into the statistical nature of thermodynamics, kinetics and function from different ligands binding with a specific receptor or equivalently specific ligand binding with different receptors. The elucidation of distributions of the kinetics and free energy has guiding roles in studying biomolecular recognition and function through small-molecule evolution and chemical genetics.

Uncovering the principles and underlying mechanisms of biomolecular recognition and molecular binding process is crucial for understanding the function and evolution, yet challenging. We meet the challenge by quantifying the statistical natures of the relevant physical variables of biomolecular recognition using the analytical model combined with microscopic flexible docking simulation methods. We uncovered the universal statistical laws obeyed by the affinity, equilibrium constant, intrinsic specificity and kinetics for biomolecular recognition. The general statistical laws based on energy landscape theory can serve as a conceptual framework for molecular recognition in biological repertoires. They can be applied to molecular selection, in vitro evolution process, high throughput screening and virtual screening for drug discovery. The statistical laws in combinations with experiments provide quantitative signatures of a specific ligand binding to a specific receptor, these resultant laws as a guideline will contribute to drug design against a specific target. Our developed statistical methodology is general and applicable for all other biomolecular recognitions.

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. In conclusion, when N >> M (that realizes a deterministic process), it is desirable to adopt [R]=K(d1), and when N=M (that realizes a stochastic process), [R]=[Formula in text] only in the first round (where * represents the population average) and [R]=K(d1) in the subsequent rounds. Based on this strategy, the mole fraction of the fittest increases by (2q)(-r) times after the rth round. With realistic parameters, we calculated several quantities such as the optimal [R] values and number of rounds needed. These values were quite reasonable and consistent with observations, suggesting the validity of our theory.

A mathematical analysis of multiple-target SELEX

SELEX (Systematic Evolution of Ligands by Exponential Enrichment) is a procedure by which a mixture of nucleic acids can be fractionated with the goal of identifying those with specific biochemical activities. One combines the mixture with a specific target molecule and then separates the target-NA complex from the resulting reactions. The target-NA complex is separated from the unbound NA by mechanical means (such as by filtration), the NA is eluted from the complex, amplified by PCR (polymerase chain reaction), and the process repeated. After several rounds, one should be left with the nucleic acids that best bind to the target. The problem was first formulated mathematically in Irvine et al. (J. Mol. Biol. 222:739-761, 1991). In Levine and Nilsen-Hamilton (Comput. Biol. Chem. 31:11-25, 2007), a mathematical analysis of the process was given. In Vant-Hull et al. (J. Mol. Biol. 278:579-597, 1998), multiple target SELEX was considered. It was assumed that each target has a single nucleic acid binding site that permits occupation by no more than one nucleic acid. Here, we revisit Vant-Hull et al. (J. Mol. Biol. 278:579-597, 1998) using the same assumptions. The iteration scheme is shown to be convergent and a simplified algorithm is given. Our interest here is in the behavior of the multiple target SELEX process as a discrete "time" dynamical system. Our goal is to characterize the limiting states and their dependence on the initial distribution of nucleic acid and target fraction components. (In multiple target SELEX, we vary the target component fractions, but not their concentrations, as fixed and the initial pool of nucleic acids as a variable starting condition). Given N nucleic acids and a target consisting of M subtarget component species, there is an M × N matrix of affinities, the (i,j) entry corresponding to the affinity of the jth nucleic acid for the ith subtarget. We give a structure condition on this matrix that is equivalent to the following statement: For any initial pool of nucleic acids such that all N species are represented, the dynamical system defined by the multiple target SELEX process will converge to a unique subset of nucleic acids, each of whose concentrations depend only upon the total nucleic acid concentration, the initial fractional target distribution (both of which are assumed to be the same from round to round), and the overall limiting association constant. (The overall association constant is the equilibrium constant for the system of MN reactions when viewed as a composite single reaction). This condition is equivalent to the statement that every member of a certain family of chemical potentials at infinite target dilution can have at most one critical point. (The condition replaces the statement for single target SELEX that the dynamical system generated via the process always converges to a pool that contains only the nucleic acid that binds best to the target). This suggests that the effectiveness of multiple target SELEX as a separation procedure may not be as useful as single target SELEX unless the thermodynamic properties of these chemical potentials are well understood.

High affinity extremes in combinatorial libraries and repertoires

By generating a large diversity of molecules, the immune system selects antibodies that bind antigens. Sharing the same approach, combinatorial biotechnologies use a large library of compounds to screen for molecules of high affinity to a given target. Understanding the properties of the best binders in the pool aids the design of the library. In particular, how does the maximum affinity increase with the size of the library or repertoire? We consider two alternative models to examine the properties of extreme affinities. In the first model, affinities are distributed lognormally, while in the second, affinities are determined by the number of matches to a target sequence. The second model more explicitly models nucleic acids (DNA or RNA) and proteins such as antibodies. Using extreme value theory we show that the logarithm of the mean of the highest affinity in a combinatorial library grows linearly with the square root of the log of the library size. When there is an upper bound to affinity, this "absolute maximum" is also approached approximately linearly with root log library size, reaching the upper limit abruptly. The design of libraries may benefit from considering how this plateau is reached as the library size is increased.

Complex SELEX against target mixture: stochastic computer model, simulation, and analysis

Systematic evolution of ligands by exponential enrichment (SELEX) is an important technology in combinatorial chemistry and molecular biology of developing high affinity target-binding molecules (aptamers) from highly complex nucleic acid ligand libraries. Schematically, the SELEX is a series of iterative rounds of operations where in each operational round ligands are incubated with the target (e.g., a purified protein), and target-binding ligands are extracted and amplified. In the recent development of biological study and drug discovery, by incubating ligand libraries with complex target mixtures (e.g., cell fragments), the SELEX experiments have been explored to simultaneously develop aptamers for targets embedded in target mixtures: the complex SELEX. While holding the considerable advantages of saving experimental resources, practicing the complex SELEX has often accompanied with unstable experimental performances. It is therefore important to understand the behaviors of the new application. In this paper, we develop stochastic computer model, and customized computational algorithm to numerically mimic the complex SELEX. We model the ligand selection through the probability of ligand binding to complex targets at the binding equilibrium, and efficiency of separating target-binders for amplification. The customized computational algorithm allows us to simulate real experiments that operate on huge ligand libraries. We evaluate the ligand evolution, and aptamer enrichment of complex SELEX under various experimental conditions by stochastic simulations, and theorize the simulated results. We argue that the stochastic effects, which were not previously captured in the studies of complex SELEX, may significantly affect the results of experiments.