1. A Framework for Modeling DNA Based Nanorobotical Devices
Sudheer Sahu (Duke University)
Bei Wang (Duke University)
John H. Reif (Duke University)

2. DNA based Nanorobotical devices
Yurke and Turberfield molecular motor
Mao B-Z transition device
Reif walking-rolling devices
Sherman Biped walker
Shapiro Devices
Peng unidirectional walker
Mao crawler

3. Simulation Aid in design Work done in simulators:
Virtual Test Tubes [Garzon00]
VNA simulator [Hagiya]
Hybrisim [Ichinose]
Thermodynamics of unpseudo-knotted multiple interacting DNA strands in a dilute solution [Dirks06]

4. Simulator for Nanorobotics
Gillespi method mostly used in simulating chemical systems [Gillespi77,Gillespi01,Kierzek02]
Topology of nanostructures important
Physical simulations to model molecule conformations
Molecular level simulation
Two components/layers
Physical Simulation [of molecular conformations]
Kinetic Simulation [of hybridization, dehybridization and strand displacements based on kinetics, dynamics and topology]
Sample and simulate molecules in a smaller volume

5. Modeling DNA Strands Single strand Gaussian chain model
[Fixman73,Kovac82]
Freely Jointed Chain
[Flory69]
Worm-Like Chain
[Marko94,Marko95,Bustamante00,Klenin98,Tinoco02]

6. More modeling… Modeling double strands Modeling complex structures
Just like single strands but with different parameters.
Modeling complex structures

7. Parameters Single Strands: Double Strands:
l0=1.5nm, Y=120KBT /nm2 [Zhang01]
P= 0.7 nm [Smith96]
lbp = 0.7nm [Yan04]
D = 1.52 ×10-6 cm2s [Stellwagen02]
Double Strands:
l0 = 100 nm [Klenin98, Cocco02]
P = 50 nm, Y = 3KBT/2P [Storm03]
lbp= 0.34 nm [Yan04]
D = 1.07 × 10-6 cm2s [Stellwagen02]

8. Random Conformation Generated by random walk in three dimensions
Change in xi in time Δt, Δxi = Ri
Ri : Gaussian random variable distributed
W(Ri) = (4Aπ)-3/2 exp(-Ri/4A)
where A = DΔt

9. Energy Stretching Energy [Zhang01] (0.5Y)Σi (ui-l0)2
Bending Energy [Doyle05, Vologdskii04]
(KBTP/l0 )Σi cos(θi)
Twisting Energy [Klenin98]
Electrostatic Energy [Langowski06,Zhang01]

10. MCSimulation
Repeat
m* = RandomConformation(m)
ΔE = E(m*) – E(m)
x [0,1]
until ((ΔE<0) or (ΔE > 0 & x<exp(-ΔE/KBT))
m = m*
Bad!!!
Good!!!

11. Data Structure and Underlying Graph

12. Hybridization Nearest neighbor model
Thermodynamics of DNA structures that involves mismatches and neighboring base pairs beyond the WC pairing.
ΔG° = ΔH° – TΔS°
ΔH° = ΔH°ends+ΔH°init+Σk€{stacks}ΔHk°
ΔS° = ΔS°ends+ΔS°init+Σk€{stacks}ΔSk°
On detecting a collision between two strands
Probabilities for all feasible alignments is calculated.
An alignment is chosen probabilistically

13. Dehybridization Reverse rate constant kr=kf exp(ΔG°/RT)
Concentration of A = [A]
Reverse rate Rr=kr [A]
Change in concentration of A in time Δt
Δ[A] = Rr Δt
Probability of dehybridization of a molecule of A in an interval of Δt
= Δ[A] /[A] = krΔt

14. Strand Displacement Random walk
direction of movement of branching point chosen probabilistically
independent of previous movements
Biased random walk (in case of mismatches)
Migration probability towards the direction with mismatches is substantially decreased

15. Strand Displacement

16. Calculating probabilities of biased random walk
G°ABC , G°rABC , G°lABC
ΔG°r = G°rABC - G°ABC
ΔG°l = G°lABC - G°ABC
Pr = exp(-ΔG°r /RT)
Pl = exp(-ΔG°l /RT)

17. Algorithm mi MList do MCSimulation(mi) Initialize While t ≤ T do
Physical Simulation
Collision Detection
Event Simulation
Hybridization
Dehybridization
Strand Displacement
t=t+Δt
mi,mj MList
if collide(mi,mj)
e=ColEvent(mi,mj)
enqueue e in CQ

18. Algorithm Initialize While t ≤ T do Physical Simulation mi MList
While CQ is nonempty
e= dequeue(CQ)
Hybridize(e)
Update MList
if potential_strand_displacement event
enqueue SDQ
Initialize
While t ≤ T do
Physical Simulation
Collision Detection
Event Simulation
Hybridization
Dehybridization
Strand Displacement
t=t+Δt
mi MList
b bonds of mi
if potential_dehybridization(b)
breakbond(b)
if any bond was broken
Perform a DFS on graph on mi
Every connected component is one new molecule formed
Update MList
For no. of element in SDQ
e = dequeue(SDQ)
e* = StrandDisplacement(e)
if e* is incomplete strand displacement
enqueue e* in SDQ
Update MList

19. Algorithm Analysis In each simulation step:
A system of m molecules each consisting of n segments.
MCsimulation loop runs f(n) times before finding a good configuration.
In every run of the loop the time taken is O(n).
Time for each step of physical simulation is O(mnf(n)).
Collision detection takes O(m2n2)
For each collision, all the alignments between two reacting strands are tested. O(cn), if number of collisions detected are c.
Each bond is tested for dehybridization. O(bm), if no. of bonds per molecule is b. For every broken bond, DFS is required and connected components are evaluated. O(b2m)
Time taken in each step is O(m2n2+mn f(n) )

20. [Unsolved Problem???] Physical Simulation of Hybridization
What happens in the time-interval between collision and bond formation?
What is the conformation and location of the hybridized molecule?

21. Further Work Enzymes Hairpins, pseudoknots More accurate modeling
Ligase, Endonuclease
Hairpins, pseudoknots
More accurate modeling
Electrostatic forces
Loop energies
Twisting energies

22. Some snapshots….
3 strands
A is partially complementary to B and C

24. Some more snapshots…. 3 strands New strand added
A partially complementry to B and C
New strand added
Partially complementary to B

26. Acknowledgement
This work is supported by NSF EMT Grants CCF and CCF