Secondary Structure Design ofMulti-state DNA Machines Based on

Sequential Structure Transitions

Hiroki Uejima

Masami Hagiya

OverviewBackground and related work Thermo-dynamical model of DNA secondary structures DNA nanomachines

Hairpin-based multi-state machine Basic idea Conformational addressing of molecular memory

Sequence design Selectivity Ordinality (sequentiality)

Prediction of transition paths Analysis and improvement of Morgan-higgs heuristics

Background and related work

Thermo-dynamical model of DNA secondary structures (1)

Secondary structures are made of base pairs. They are stable with respect to free energy.

Nearest neighbor model (Zimm et al., 1964) Summing up stacking energies of adjacent base pairs an

d mismatched pairs

Folding problem (Zuker et al., 1981)

5’ 3’TTC…GCA

3’5’Base sequence

(linear structure)

Secondary structure

folding

inverse folding

Thermo-dynamical model of DNA secondary structures (2)

Inverse folding problem (Hofacker et al., 1994) Optimization with the fold function for evaluation

Search for sub-optimal structures (Wuchty et al., 1999) Enumeration of (sub-optimal) structures whose energy is unde

r mfe+Computation of the partition function (McCaskill, 1990)

Computation of the frequency of a structure

Estimation of the energy barier between structures (Flamm et al., 2000)

DNA nanomachines

Various DNA nanomachine DNA motor by B-Z transiton

(Seeman et al., 1999) molecular tweezer

(Yurke et al., 2000) three-state machine

(Simmel et al., 2002) PX-JX2

(Yan et al., 2002) Hybridization inhibition by bulge loop

(Tuberfield et al., 2003)

Designing DNA sequence with bistable structures (Flamm et al., 2001)

Yurke’s DNA Tweezers

Simmel’s 3-state machine

PX-JX2 by Yan

HybridizationInhibition byBulge loops(Tuberfield et al.,2003)

Hairpin-based multi-state machine

Molecular machine that allowssuccessive state transitions

input1

input2

input3

2

1

2

3

1

3

３ 3

2

2

……

……

……

Input order sensitive

Prototype of multi-state molecular machine and its application

……

……

Conformational addressing ofHierarchical molecular memory

Kameda et al.

Using bulge loops to inhibit hybridization (comparison)

More robust than

our machine

Our machine is simpler Consisting of only

a single strand If it works robustly,

then it can be used as

yet another kind of

building block for DNA machines.

More complexexample ofhairpin-basedmachine(not discussedin the paper)

Sequence design

Design criteria

A hairpin is opened only by a corresponding input oligomer --- selectivity if its predecessor has been opened --- ordinality (sequentiality)

Input oligomer

hairpin

Stickyend

Reduction of the design criteria

Two criteria Selectivity Ordinality (sequentiality)

These criteria should be satisfied by any number of hairpin sequences concatenated in any order.But Only combinations of a sticky end and a hairpin are nec

essary to be verified to guarantee the selectivity. Only combinations of two hairpin sequences are neces

sary to be verified to guarantee the ordinality.

Selectivity

For any combinations of A sticky end and a hairpin, and An input oligomer (also a sticky end and a hairpin)

It is checked whether the mfe (minimum free energy) structure is similar to the target structure.

1-a, 2-a and 2-b in the next slide are exhaustively checked. 1-a: The hairpin is opened. 2-a, 2-b: The hairpin is not opened.

1-b is checked with respect to the ordinality

1-a

2-a

1-b

2-b

1

2

a b

Taking the sub-optimal structures into account

In addition to the condition that the target structure is similar to the mfe structure,

The total frequency of the structures similar to the target should be maximized.

The sub-optimal structures that should be taken into account

mfeδ

Secondary structure

Free energy

Set of sub-optimal secondary structures

Target structure

The frequency of this region is maximized.

Computing the total frequency

Let T be the target structure for sequence xCompute the sub-optimal structures,S1, S2, …, Sn, of x Use the subopt_pf function, which is derived from

the subopt function of the Vienna package

Compute the frequency F(S) of the structure S User pf_fold and energy_of_struct of Vienna

The distance d(S, T) between structures, S and T, is the size of the symmetric difference of S and T, i.e., d(S, T) = |(ST)∖(ST)|Compute the sum of P(Si) such that d(Si, T) ≤ D

Ordinality (Sequentiality)

The hairpin should be opened

neither by (A) nor (B).

(A) (B)

Problem on ordinality

() ()

In general, () is more stable than ().

We should measure the tendency of the transition.

Mountain height and valley depth

()()

The mountain should be as high as possible and the valley should be as shallow as possible

G1 = 3.06 [kcal/mol] ， G2 = 4.05[kcal/mol]

G1

G2

When the hairpin is opened

(') (')

The mountain should be as low as possible, and the valley should be as deep as possible

G1 = 1.20 [kcal/mol] ， G2 = 15.60 [kcal/mol]

G1

G2

Condition for oligomer invasion:mountain height and valley depth

(α)

(β)

G1

G2

Mountain height G1>B=4.0 and valley depth G2<V=3.0

How to obtain G1: The heuristics for predicting the transition path with the minimum G1 (Morgan and Higgs 1998) is run multiple times, and the minimum value of G1 is adopted. Explained later.

Overview of sequence design

1. Generate the required number of sequences for hairpin structures, using the inverse_fold function of the Vienna package (sequence generation)

2. Validate ordinarity and selectivity for all combinations of sequences (sequence validation)

a) If a condition is not satisfied, related sequences are replaced with newly generated ones

b) If the conditions are satisfied, compute the total frequency of the structures similar to the target, and output the sequence set

Flowchart of sequence designStart Genereate hairpin sequences

When oligomer invadesbarrier height>B and valley depth<V

Validate selectivity

Sequences hold selectivity

Output the successful frequences of selectivity and ordinality

Goal

No

No

Yes

Yes

Example of a sequence set

ordinality min barrier height: 4.369999

ordinality max valley depth: 2.650002

((((((((((((((((((((.......))))))))))))))))))))

CGAATACTCCTAACGTTGCGCGTTTGACGCAACGTTAGGAGTATTCG -23.440001

CTTTGAACACGTGAGTGGCATTACGAGTGCCACTCACGTGTTCAAAG -23.879999

CTTGTCCACCACGTTATGTTTCCGGGTAACATAACGTGGTGGACAAG -22.090000

selectivity (open): 0.767506 0.868699

selectivity (closed): 0.486210 0.733504

ordinality: 0.000384 0.022198

Prediction of secondary structure transition paths

Structure transition path

G1

G2

Secondary structure

Manimize G1

fast transition⇒

Need for prediction of transition paths

Free energy

Various kinds of transition paths

One step in a transition path addition or deletion of a base pair

Direct path --- shortest path

Locally optimized direct path In each step, the transition with the minimum

energy increment (maximum decrement) is taken.

Globally optimized direct path

Globally optimized path

Computation of transition paths

In order to be used for evaluating sequences, structure transition paths should be computed accurately and efficiently Simple search is not tractable

Locally optimized direct path (LDP) Easily computed by the definition

Globally optimized direct path (GDP) Direct paths are generated by M-H heuristics (Morgan

et al., 1998) and the path with the minimum G1 is taken as an approximate solution

Globally optimized path (GP) Currently not available

Morgan and Higgs’s Heuristics

Base pairs in B

Incompatible base pairs in A

６ １，２７ ２８ ３９ ３，４１０ ４，５

Among base pairs in B, choose the pair with the minimum number of incompatible base pairs in A

Among base pairs in B, choose the pair with the minimum number of incompatible base pairs in ADelete the incompatible base pairs in A and add the base pair in BIf more base pairs in B can be added, do soLet the resulting structure be B'Unless B' is identical to A, repeat the procedure on B'

Morgan and Higgs’s Heuristics

６ １，２

７ ２８ ３９ ３，

４１０

４，５

６ １

８ ３９ ３，

４１０

４，５

６ １

９ ４１０

４，５

６ １

１０

５ １０

５

６ １７８ ３９ ３，

４１０

４，５

６ １

８９ ４１０

４，５

６ １

９１０

５

６

１０

５ １０

０ ０ ０

０ ０ ０

１ １ １

１ １

Representation of M-H heuristicsas a graph algorithm

６

７

８

９

１０

１

２

３

４

５

B　　　　　　　　　 A

1. Delete a vertex in A

2. Delete edges incident to the delete vertex

3. If a vertex in B has no incident edge, then delete the vertex

4. Repeat the above procedure until all the vertices are deleted

Base pair as vertex, incompatibility as edge bipartite graph

Decomposition intoconnected components

６

７

８

９

１０

１

２

３

４

５

B　　　　　　　　　 A 1. Delete a vertex in A

2. Delete edges incident to the delete vertex

3. If a vertex in B has no incident edge, then delete the vertex

4. Repeat the above procedure until all the vertices are deleted

Base pair as vertex, incompatibility as edge bipartite graph

Each connected component can be handled separately

The order to processconnected components

Connected component can be classified as follows

hh

f

hf

Do first， then ， and finally Among --- small h first, and (among equal h) large f first Among --- arbitrarily Among --- large h first, and (among equal h) small f first

A counter example for whichM-H fails

Structures corresponding to the counter example

1234

5 6 7

89

101112

13

14

(A) (B)

Prediction of globally optimized direct paths

Direct paths are randomly generated by M-H, and the path with the minimum G1 is taken as an approximate solution M-H only cares the number of base pairs

In the case of hairpin-based machines, the energy mountaintop height of LDP is close to and often a bit smaller than GDP approximated by M-HJudging from this result, the energy landscape around such simple structural transformation seems relatively smooth

Future work

Generalize the criteria for sequence design make the programs applicable to various design

problems

Verification of the validity of the criteria Comparison with experimental results Comparison with computer simulations Comparison with thermo-dynamical analyses

More detailed analysis of structure transition paths Kinetic analysis --- analysis of nucleation

Incorporation of other aspects of structures Various physical properties of double strands can

inhibit hybridization