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

Mao B-Z transition deviceYurke and Turberfield molecular motor

Reif walking-rolling devices

Peng unidirectional walker Mao crawler

Sherman Biped walker

Shapiro Devices

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– Just like single strands but with different

parameters.

• Modeling complex structures

Parameters

• Single 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

Repeatm* = 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

• Initialize• While t ≤ T do

Physical Simulation

Collision Detection

Event Simulation• Hybridization• Dehybridization• Strand Displacement

t=t+Δt

mi MList do MCSimulation(mi)

mi,mj MList if collide(mi,mj) e=ColEvent(mi,mj) enqueue e in CQ

Algorithm

• Initialize• While t ≤ T do

Physical Simulation

Collision Detection

Event Simulation• Hybridization• Dehybridization• Strand Displacement

t=t+Δt

While CQ is nonempty e= dequeue(CQ) Hybridize(e) Update MList if potential_strand_displacement event enqueue SDQ

For no. of element in SDQ e = dequeue(SDQ) e* = StrandDisplacement(e) if e* is incomplete strand displacement enqueue e* in SDQUpdate MList

mi MList b bonds of mi

if potential_dehybridization(b)breakbond(b)

if any bond was brokenPerform a DFS on graph on mi

Every connected component is one new molecule formed

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– 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 strandsA partially complementry to B and C

• New strand addedPartially complementary to B

Acknowledgement

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