Kinetic Tile Assembly Model (kTAM)

The kinetic tile assembly model(KTAM) is a more realistic model that considers the rate of association and dissociation of basic molecular elements (so-called monomers, or Tiles) within the original framework provided by the socalled abstract Tile Assembly Model ATAM.

In practice, DNA self-assembly entails a more complicated process than the simple model described previously. For instance, there are an infinite number of different types of monomer (tile) and the bonding strength for each edge is not a discrete integer, but a continuous number. Therefore, a different model is required for a realistic simulation of this process.

KTAM describes the dynamics of assembly according to an inclusive set of reversible chemical reactions: A tile can attach to an assembly anywhere that it makes even a weak bond, and any tile can dissociate from the assembly at a rate dependent on the total strength with which it adheres to the assembly

A Kinetic Model of DNA Self-Assembly

The self-assembly of two-dimensional lattices from a heterogeneous mix of DX molecules is a far more complicated system than the hybridization of two oligonucleotides. Rather than having just three species to consider (ssDNA, ssDNAA, and dsDNA), we now have an infinite number of species (all possible aggregates). For each aggregate of tiles with available sites, there are association reactions and dissociation reactions. Note that at every available site, there is an association reaction for every possible monomer, regardless of whether the monomer is the “correct” one or not; to understand when correct behavior can be expected, we must look closely at the kinetics of all the reactions. The model we develop here can be seen as an extention of Erickson (1980), which considers the self-assembly of an isotropic two-dimensional lattice consisting of a single unit type. To model the kinetics of self-assembly, we make several simplifying assumptions:

1. Monomer concentrations will be held constant. Further, all monomer types will be held at the same concentration. Primarily we make this assumption because the analysis is easier. Later we show how the results found with the assumption can be used to understand the more general case when the assumption is not true.

2.Aggregates do not interact with each other; thus the only reactions to model are the addition of a monomer to an aggregate, and the dissociation of a monomer from an aggregate. Potential drawbacks of this assumption will be discussed at the very end.

3. As in the hybridization of oligonucleotides, we assume that the forward rate constants for all monomers are identical. In particular, the forward rate constants for correct and incorrect additions are identical.

4. As in the hybridization of oligonucleotides, we assume that the reverse rate depends exponentially on the number of base-pair bonds which must be broken, and that mismatched sticky ends make no base-pair bonds. This amounts to assuming that binding on multiple edges is cooperative and that mismatched sticky ends do not affect the dissociation rate in any way.

References

Erik Winfree, Algorithmic Self-Assembly of DNA, PdD thesis, Caltech, 1998.