# (665i) An Optimization-Based Approach for the Design of Self-Assembled DNA Tiles

- Conference: AIChE Annual Meeting
- Year: 2016
- Proceeding: 2016 AIChE Annual Meeting
- Group: Engineering Sciences and Fundamentals
- Session:
- Time:
Thursday, November 17, 2016 - 10:45am-11:00am

In structural DNA nanotechnology, DNA strands exceed their

traditional role in gene expression and are being used as programmable building

blocks for nanoscale structures. This self-assembly is driven by hybridization

following the Watson-Crick rule.[1-3] Depending on the design of the DNA

strands, structures with a rich set of nanoscale features can be

self-assembled. A special class of DNA structures are the so-called DNA tiles,[4]

which are self-assembled from a relatively small number of DNA strands.

Experimental methods to self-assemble DNA tiles for a given design of DNA

strands are well developed. However, a challenge is to find the optimal design

of the individual DNA strands to self-assemble a desired DNA tile.

In general, the design of DNA strands for self-assembly of tiles

involves two steps, 1) structural design and, 2) sequence generation. For

structural design, the aim is to select optimal locations for so-called crossovers,

which involves bridging of two DNA double helices (see Figure 1). Each

crossover connects two double helices by a pair of shared single strands

DNA.[1,3] For sequence generation, an optimal sequence of nucleotides for each

DNA strand needs to be selected to enable the formation of the selected

crossovers. Generally speaking, efficient algorithms for sequence generation

exist. However, for the structural design, existing methods typically only

provide a modeling framework for prediction and visualization of a stable

structure for a given structural design (i.e., for solving the 'forward'

problem).[6-9] In contrast, finding a structural design that allows for

self-assembly of a desired final structure with high stability (i.e., solving

the 'inverse' problem) remains challenging.

The objective of this work is to investigate an optimization-based

approach for the optimal structural design of self-assembled DNA tiles. A

semi-empirical model from literature [8] is used to predict the potential

energy of a self-assembled DNA tile.

First, minimization of the potential energy for a given design

(i.e., locations of crossovers) is studied for three different types of

tiles.[5,10,11] The optimization problem was implemented in Matlab (2014a,

Mathworks, Inc.) using the solver *fmincon *and

in General Algebraic Modeling System (GAMS) Release 24.4.6. (GAMS Development

Corporation, Washington, DC, USA, 2015) using different non-linear programming

(NLP) solvers. Figure 2 illustrates the structural configurations of various

self-assembled DNA tiles that were found after minimization of the potential

energy. Different local minima were identified depending on the randomly chosen

initial guesses. In all cases, the structure with the lowest potential energy

corresponded to the intended structure reported in literature. Furthermore, in

all cases, a significant part of the initial guesses converged to local minima.

The structural configurations corresponding to those local minima could be

significantly different from the intended structure (Figure 1), which

illustrates the practical importance of avoiding local. Both a multi-start

optimization method and deterministic global optimization using GAMS/BARON were

investigated to identify the global minimum.

Second, self-assembly of the DNA tensegrity triangle [5,12](Figure

1) is chosen as the case to investigate model-based optimization of the

structural design. This tile has been synthesized experimentally[5] and has

practical relevance as template for protein crystallization. The optimization

problem aims to minimize the potential energy of the tile, to maximize

stability, using the orientation and position of the helices and the locations

of the crossovers as degrees of freedom. Since the length of the DNA strands is

typically short, decisions regarding the locations of crossovers are discrete

decisions (represented by integer variables). Therefore, the optimization

problem is formulated as a mixed-integer nonlinear programming (MINLP) problem.

The optimization problem was implemented in GAMS using the solver SBB, which is

based on branch-and-bound. The solution obtained with the MINLP formulation was

compared to the solution obtained by exhaustive enumeration of the integer

variables and solving an NLP problem at each node. The lowest potential energy

could be obtained by using 27 nucleotides between each crossover for an

equilateral triangle design, which matches experimental findings[5] and was

also found using branch-and-bound with a 10-fold smaller CPU time compared to

exhaustive enumeration. Finally, the design problem was also solved with the

number of nucleotides between crossovers on each side of the triangle as a

decision variable. The proposed MINLP optimization method was again able to

identify the same mixed-integer solution compared to exhaustive enumeration at

a much smaller CPU time (200-fold), which demonstrates the strength of a

branch-and-bound algorithm for structural design of self-assembled DNA tiles.

Figure 1: An example of a self-assembled DNA tile (tensegrity triangle)

and detailed view of the structure of crossovers that interlink DNA helices.

Figure 2: Identified local minima when minimizing the

potential-energy of three different types of DNA tiles with fixed locations of

crossovers. The percentage of randomly chosen initial guesses that led to each

structure is indicated below the potential energy of the structure

**Reference**

[1] Seeman, N. C. An overview of structural DNA nanotechnology. *Mol. Biotechnol.* **37,**246-257

(2007).

[2] Seeman, N. C. DNA in a material world. *Nature* **421,**427-431

(2003).

[3] Seeman, N. C. DNA nanotechnology: novel DNA constructions. *Annu. Rev. Biophys. Biomol.
Struct.*

**27,**225-248

(1998).

[4] Park, S. H. *et
al.* Programmable

DNA self-assemblies for nanoscale organization of ligands and proteins.

*Nano Lett.*

**5,**729-733

(2005).

[5] Zheng, J. *et
al.* From

molecular to macroscopic via the rational design of a self-assembled 3D DNA

crystal.

*Nature*

**461,**74-77

(2009).

[6] Birac, J. J., Sherman, W. B., Kopatsch, J., Constantinou, P.

E. & Seeman, N. C. Architecture with GIDEON, a program for design in

structural DNA nanotechnology. *J.
Mol. Graph. Model.*

**25,**470-480

(2006).

[7] Castro, C. E. *et
al.* A

primer to scaffolded DNA origami.

*Nat.*

Methods

Methods

**8,**221-229

(2011).

[8] Zhu, J., Wei, B., Yuan, Y. & Mi, Y. UNIQUIMER 3D, a

software system for structural DNA nanotechnology design, analysis and

evaluation. *Nucleic
Acids Res.*

**37,**2164-2175

(2009).

[9] Pan, K. *et
al.* Lattice-free

prediction of three-dimensional structure of programmed DNA assemblies.

*Nat. Commun.*

**5,**(2014).

[10] Wei, B. & Mi, Y. A new triple crossover triangle (TXT)

motif for DNA self-assembly. *Biomacromolecules* **6,**2528-2532

(2005).

[11] He, Y. *et
al.* Self-Assembly

of Hexagonal DNA Two-Dimensional ( 2D ) Arrays.

*J. Am. Chem. Soc.*12202-12203

(2005).

[12] Liu, D., Wang, M., Deng, Z., Walulu, R. & Mao, C.

Tensegrity: construction of rigid DNA triangles with flexible four-arm DNA

junctions. *J.
Am. Chem. Soc.*

**126,**2324-2325 (2004).