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## DNA Based Self-assembly and Nanorobotics: Theory and Experiments

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**DNA Based Self-assembly and Nanorobotics: Theory and**Experiments Sudheer Sahu Department of Computer Science, Duke University Advisor: Prof. John Reif Committee: Prof. John Board, Prof. Alex Hartemink, Prof. Thom LaBean, Prof. Kamesh Munagala, Prof. Xiaobai Sun**Tile**[Yan03] [Park06] [Rothemund05] [Seeman91] 4x4 tile Self-assembly • Fundamental process in nature • Recent uses in nanoscale constructions • Algorithmic Self-assembly: Universal Computation [Winfree96]**Tile Assembly Models**• Winfree’s Tile Assembly Model GlueΣ Temperature • Strength function : g: Σ x Σ → R • g(σ,σ’) > 0 if σ = σ’ • = 0 otherwise Tile-system (T,S,g,) • Adleman’s Reversible Model • Winfree’s Kinetic Model • Generalized Models**Capabilities and Limitations of**Redundancy Based Compact Error Resilient Methods Sudheer Sahu, John H. Reif DNA12, LNCS 4287, 223-228, 2006 Submitted to Algorithmica**Algorithmic Self-Assembly**0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 [Rothemund04] Output 1 1 1 1 1 Output 2 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 Pad 1 1 0 1 0 0 0 0 0 0 0 Input 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 Input 1 0 0 0 0 0 0 1 0 Output 1 = Input 1 XOR Input 2 Output 2 = Input 1 XOR Input 2**Original tiles:**X Y Z XY YZ Error resilient tiles: Error checking pads Work in Error correction Original tiles: • Winfree’s seminal work • Goel’s snake tiling • Reif’s compact error resiliency • Nucleation Errors[Schulman05] Error resilient tiles: (Excerpted from Winfree 03) • Error rate 2 • Assembly size increased by 4**Error Model**• The assembly takes place in a kinetic manner, but the error analysis is done at equilibrium • When the system is in equilibrium, the probability that a mismatch occurs is ε • Independent errors • Emphasis on the correctness of complete assembly • Redundancy based compact error resilience scheme**V(i+1,j+1)**V(i,j+1) U(i,j+1) T(i,j) U(i,j+1) U(i+1,j+1) U(i,j) U(i+1,j) V(i,j) V(i+1,j) V(i+1,j) V(i,j) U(i,j) Two Dimensional Assembly V(I,j+1) Input: U(i,j), V(i,j) Output: U(i+1,j)=U(i,j)OP1V(i,j) V(i,j+1)=U(i,j)OP2V(i,j) T(i,j) U(i+1,j) U(I,j) V(I,j) Theorem: There exists a redundancy based compact error resilient scheme to reduce error from ε to ε2 for arbitrary boolean functions OP1 and OP2.**A Self-assembly Model with Time Dependent Glue Strength**Sudheer Sahu, Peng Yin, John Reif DNA11, LNCS 3892, 290-304, 2006 Submitted to Algorithmica**Time Dependent Glue Model**• Glue strength increases monotonically before becoming constant • Glue strength function • Time for maximum strength • Minimum interaction time**Implementation using Strand Displacement**A s3 B s1 s2**Self-Replication**• A-B acts as a catalyst for formation of C-D which in turn acts as a catalyst for the formation of A-B • Two states: • Dormant state • Replicating state • Low probability of going from dormant to replicating state**Tile Complexity**• Minimum number of distinct tile types required to construct a shape uniquely • nxn square [Standard model,Adleman01] • k x mk rectangle O(k+m) [Standard model, Aggarwal04] • Thin rectangle (kxn for k < ) [Standard Model, Aggarwal04] [Time dependent Glue Model]**Construction of thin rectangles**• Construct a j x n rectangle using O(j+n1/j ) type of tiles, where j > k. • The glue of bottom k rows become strong after mit, and the glue of top j-k rows (volatile rows) do not.**Shapes with holes**nxn square with a hole of k x k in center • Upper Bound in our model: • Grow four rectangles**[Yan et al 02]**[Shin et al 04] DNA based Nanorobotical devices • Advantages of DNA-based synthetic molecular devices: • simple to design and engineer • well-established biochemistry used to manipulate DNA nanostructures DNA-fuelled Molecular machine [Yurke et al 00] B-Z transition device [Mao, Seeman 99] DNA Biped walker [Sherman et al 04]**DNA motor powered by Nicking enzyme**[Bath et al 05] DNA based Nanorobotical devices • Major challenges: • Autonomous (without externally mediated changes per work-cycle) • Programmable (their behavior can be modified without complete redesign of the device) [Tian et al 05] Unidirectional DNA Walker [Yin et al 04]**A Framework for Modeling DNA Based Molecular Systems**Sudheer Sahu, Bei Wang, John Reif DNA12, LNCS 4287, 250-265, 2006 Submitted to Journal of Computational and Theoretical Nanoscience**Simulation**• Gillespi method mostly used in simulating chemical systems [Gillespi77, Gillespi01, Kierzek02] • Work done in nanorobotic simulators: • Virtual Test Tubes [Garzon00] • VNA simulator [Hagiya] • Hybrisim [Ichinose] • Thermodynamics of unpseudo-knotted multiple interacting DNA strands in a dilute solution [Dirks06] • Two components/layers • Physical Simulation [of molecular conformations] • Kinetic Simulation [of hybridization, dehybridization and strand displacements based on kinetics, dynamics and topology]**Modeling DNA Strands**• Single strand • Gaussian chain model [Fixman73,Kovac82] • Freely Jointed Chain [Flory69] • Worm-Like Chain [Marko94,Marko95, Bustamante00,Klenin98,Tinoco02] • Double strands • Similar but parameters differ • Complex structures**MCSimulation**• Generated by random walk in three dimensions • Change in xiin time Δt, Δxi = Ri • Ri : Gaussian random variable • W(Ri) = (4Aπ)-3/2 exp(-Ri/4A) where A = DΔt • Stretching Energy = (0.5Y)Σi (ui-l0)2 [Zhang01] • Bending Energy = (KBTP/l0 )Σi cos(θi) [Doyle05, Vologdskii04] • Twisting Energy [Klenin98] • Electrostatic Energy [Langowski06,Zhang01] Repeat m* = RandomConformation(m) ΔE = E(m*) – E(m) x ε[0,1] until ((ΔE<0) or (ΔE > 0 & x<exp(-ΔE/KBT)) m = m* Bad!!! Good!!!**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] = RrΔt • 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 • 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)**miMList do**MCSimulation(mi) mi,mj MList if collide(mi,mj) e=ColEvent(mi,mj) enqueue e in CQ 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 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) )**Extensions….**• 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? • Enzymes • Ligase, Endonuclease • Hairpins, pseudoknots • More accurate modeling • Electrostatic forces • Loop energies • Twisting energies**Autonomous Programmable Nanorobotic Devices Using DNAzymes**John Reif, Sudheer Sahu DNA13, LNCS 4848, 66-78, 2008 Submitted to Theoretical Computer Science**Polycatalytic Assemblies [Pei et al 06]**DNAzyme tweezer [Chen et al 04] DNAzyme crawler [Tian et al 05] DNAzyme based nanomechanical devices • Autonomous • Programmable • Require no protein enzymes**Our DNAzyme based designs**• DNAzyme FSA: a finite state automata device, that executes finite state transitions using DNAzymes • extensions to probabilistic automata and non-deterministic automata, • DNAzyme Router: for programmable routing of nanostructures on a 2D DNA addressable lattice • DNAzyme Doctor : a medical-related application to provide transduction of nucleic acid expression. • can be programmed to respond to the under-expression or over-expression of various strands of RNA, with a response by release of an RNA • operates without use of any protein enzymes.**FSA**0101110100 01 0101 0 0 1 1 0 0 010 010111 1 01011101 2 0 0101110 01011 1 010111010**x2**a2 a1 x1 0 b2 b1 x1 x2 1 x2 b2 b1 x1 a2 a1 x1 x2 a2 x2 a1 x1 x2 b2 b1 x1 a2 a1 x1 x2 a2 x2 a1 x1 t1 t2 t2 t1 t2 t1 t2 t1 1 0 0 0 1 0 DNAzyme FSA (inputs) Active Input: Input that is being read by FSA currently**x2**a2 a1 x1 0 b2 b1 x1 x2 1 DNAzyme FSA(State Transitions)**Complete Finite State Machine**• Non-deterministic finite automata • Probabilistic automata • Reusable system • No. of DNAzymes proportional to the no. of transitions**[Park et al 06 ]**[Rothemund 05] DNAzyme Router…. 0 Go right 1 Go down Input: 110110 Input: 0110100 • Input string acts as program for the robot • Non-destructive • Multiple robots walking together**DNAzyme Doctor (state diagram)**• Shapiro Device [uses protein enzymes]**Design Principle**• We need AND operation • We need a way to test for the under-expression and over-expression conditions**Detecting RNA Expression**y1,y2,y3,y4 characteristic sequence of RNAs R1, R2, R3, R4 A threshold concentration of y1, y2, y3, y4 is thrown in the solution, therefore lack of y3, y4 causes excess of y3 and y4, respectively.**A DNA Nanotransportation Device Powered by Polymerase Φ29**Sudheer Sahu, Thom LaBean, John Reif To be submitted**A C T G**P P T T W P Q T Polymerase Driven Nanomotor • Replication of DNA RNA template • Sustain life processes • Protector Strand Q • Stopper Sequence • Phi29 • Exceptional strand displacement abilities • Wheel slips on the track and not rolls**Results**Circular Track and Wheel Intertwined Track Circularization**Results**Nanomotor Nick Crossing Brakes