A Framework for Modeling DNA Based Nanorobotical Devices

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A Framework for Modeling DNA Based Nanorobotical
Devices

• Sudheer Sahu (Duke University)
• Bei Wang (Duke University)
• John H. Reif (Duke University)

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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
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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

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Simulator for Nanorobotics

• Gillespi method mostly used in simulating
  chemical systems. Gillespi77,Gillespi01,Ki
  erzek02
• 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

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Modeling DNA Strands

• Single strand
• Gaussian chain model
• Fixman73,Kovac82
• Freely Jointed Chain
• Flory69
• Worm-Like Chain
• Marko94,Marko95,Bustamante00,Klenin98,Tinoco02
  

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More modeling

• Modeling double strands
• Just like single strands but with different
  parameters.
• Modeling complex structures

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Parameters

• Single Strands
• l01.5nm, Y120KBT /nm2 Zhang01
• P 0.7 nm Smith96
• lbp 0.7nm Yan04
• D 1.52 10-6 cm2s-1
  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-1
  Stellwagen02

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Random Conformation

• Generated by random walk in three dimensions
• Change in xi in time ?t, ?xi Ri
• Ri Gaussian random variable distributed
• W(Ri) (4Ap)-3/2 exp(-Ri/4A)
• where A D?t

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Energy

• Stretching Energy
  Zhang01
• (0.5Y)Si (ui-l0)2
• Bending Energy Doyle05, Vologdskii04
• (KBTP/l0 )Si cos(?i)
• Twisting Energy
  Klenin98
• Electrostatic Energy Langowski06,Zhang01

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MCSimulation

• Repeat
• m RandomConformation(m)
• ?E E(m) E(m)
• x 0,1
• until ((?Elt0) or (?E gt 0 xltexp(-?E/KBT))
• m m

Bad!!!
Good!!!
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Data Structure and Underlying Graph
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Hybridization

• Nearest neighbor model
• Thermodynamics of DNA structures that involves
  mismatches and neighboring base pairs beyond the
  WC pairing.
• ?G ?H T?S
• ?H ?Hends?HinitSkstacks?Hk
• ?S ?Sends?SinitSkstacks?Sk
• On detecting a collision between two strands
• Probabilities for all feasible alignments is
  calculated.
• An alignment is chosen probabilistically

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Dehybridization

• Reverse rate constant krkf exp(?G/RT)
• Concentration of A A
• Reverse rate Rrkr A
• Change in concentration of A in time ?t
• ?A Rr ?t
• Probability of dehybridization of a molecule of A
  in an interval of ?t
• ?A /A kr?t

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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

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Strand Displacement
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Calculating probabilities of biased random walk

• GABC , GrABC , GlABC
• ?Gr GrABC - GABC
• ?Gl GlABC - GABC
• Pr exp(-?Gr /RT)
• Pl exp(-?Gl /RT)

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Algorithm

• Initialize
• While t T do
• Physical Simulation
• Collision Detection
• Event Simulation
• Hybridization
• Dehybridization
• Strand Displacement
• tt?t

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Algorithm
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
• tt?t

• 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
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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(m2n2mn f(n) )

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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?

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Further Work

• Enzymes
• Ligase, Endonuclease
• Hairpins, pseudoknots
• More accurate modeling
• Electrostatic forces
• Loop energies
• Twisting energies

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Some snapshots.

• 3 strands
• A is partially complementary to B and C

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Some more snapshots.

• 3 strands
• A partially complementry to B and C
• New strand added
• Partially complementary to B

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Acknowledgement

• This work is supported by NSF EMT Grants
  CCF-0523555 and CCF-0432038