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

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

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]

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

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

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

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-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]

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

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]

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!!!

Data Structure and Underlying Graph

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

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

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

Strand Displacement

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)

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

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

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) )

[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?

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

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

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

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