Molecular Control Engineering Nonlinear Control at the Nanoscale

Molecular Control Engineering Nonlinear Control at the Nanoscale Raj Chakrabarti PSE Seminar Feb 8, 2013

What is Molecular Control Engineering? Control engineering: Manipulation of system dynamics through nonequilibrium modeling and optimization. Inputs and outputs are macroscopic variables. Molecular control engineering: Control of chemical phenomena through microscopic inputs and chemical physics modeling. Adapts to changes in the laws of Nature at these length and time scales. Aims § Reaching ultimate limits on product selectivity § Reaching ultimate limits on sustainability § Emulation of and improvement upon Nature’s strategies

Approaches to Molecular Design and Control Quantum Control of Chemical Reaction Dynamics Control of Biochemical Reaction Networks Molecular Design femtoseconds, angstroms picoseconds, nanometers milliseconds, micrometers

Parallel Parking and Nonlinear Control § Stepping on gas not enough: can’t move directly in direction of interest § Must change directions repeatedly § Left, Forward + Right, Reverse enough in most situations § Tight spots: Move perpendicular to curb through sequences composed of Left, Forward + Left, Reverse + Right, Forward + Right, Reverse

Vector Fields 8. Finalize these

Control with Linear Vector Fields

Lie Brackets and Directions of Motion

From classical control to the coherent control of chemical processes § FMO photosynthetic protein complex transports solar energy with ~100% efficiency § Phase coherent oscillations in excitonic transport: exploit wave interference § Biology exploits changes in the laws of nature in control strategy: can we?

Coherent Control versus Catalysis § Potential Energy Surface with two competing reaction channels § Saddle points separate products from reactants § Dynamically reshape the wavepacket traveling on the PES to maximize the probability of a transition into the desired product channel probability density time interatomic distance

C. Brif, R. Chakrabarti and H. Rabitz, New J. Physics, 2010. C. Brif, R. Chakrabarti and H. Rabitz, Control of Quantum Phenomena. Advances in Chemical Physics, 2011.

Femtosecond Quantum Control Laser Setup 2011: An NSF funded quantum control experiment collaboration between Purdue’s Andy Weiner (a founder of fs pulse shaping) and Chakrabarti Group

Prospects and Challenges for Quantum Control Engineering

Coherent Control of State Transitions in Atomic Rubidium

Bilinear and Affine Control Engineering R. Chakrabarti, R. Wu and H. Rabitz, Quantum Multiobservable Control. Phys. Rev. A, 2008.

Few-Parameter Control of Quantum Dynamics § Conventional strategies based on excitation with resonant frequencies fails to achieve maximal population transfer to desired channels § Selectivity is poor; more directions of motion are needed to avoid undesired states

Optimal Control of Quantum Dynamics § Shaped laser pulse generates all directions necessary for steering system toward target state § Exploits wave-particle duality to achieve maximal selectivity, like coherent control of photosynthesis

Understanding Interferences Need to introduce V_ Remove the lambdas We don’t show the interm we for consistency w belo 9. Finalize these

Quantum Interferences and Quantum Steering Interference • Mechanism identification techniques have been devised to efficiently extract important constructive and destructive interferences V. Bhutoria, A. Koswara and R. Chakrabarti, Quantum Gate Control Mechanism Identification, in preparation

Control of Molecular Dynamics HCl CO Mixed state density matrix: Pure state: Expectation value of observable: Cost functional: R. Chakrabarti, R. Wu and H. Rabitz, Quantum Pareto Optimal Control. Phys. Rev. A, 2008.

Quantum System Learning Control: Critical Topology R. Wu, R. Chakrabarti and H. Rabitz, Critical Topology for Optimization on the Symplectic Group. J Opt. Theory, 2009 R. Chakrabarti and H. Rabitz, Quantum Control Landscapes, Int. Rev. Phys. Chem. , 2007 K. W. Moore, R. Chakrabarti, G. Riviello and H. Rabitz, Search Complexity and Resource Scaling for the Quantum Optimal Control of Unitary Transformations. Phys. Rev. A, 2011.

Quantum Robust Control R. Chakrabarti and A. Ghosh. Optimal State Estimation of Controllable Quantum Dynamical Systems. Phys. Rev. A, 2011.

Improving quantum control robustness Check sign, fix in

From Quantum Control to Bionetwork Control • Nature has also devised remarkable catalysts through molecular design / evolution • Maximizing kcat/Km of a given enzyme does not always maximize the fitness of a network of enzymes and substrates • More generally, modulate enzyme activities in real time to achieve maximal fitness or selectivity of chemical products

The Polymerase Chain Reaction: An example of bionetwork control Nobel Prize in Chemistry 1994; one of the most cited papers in Science (12757 citations in Science alone) Produce millions of DNA molecules starting from one (geometric growth) Used every day in every Biochemistry and Molecular Biology lab ( Diagnosis, Genome Sequencing, Gene Expression, etc. ) Generality of biomolecular amplification: propagation of molecular information - a key feature of living, replicating systems

Single Strand – Primer Duplex Extension DNA Melting Again DNA Melting Primer Annealing 12/6/2020 School of Chemical Engineering, Purdue University 27

The DNA Amplification Control Problem and Cancer Diagnostics Wild Type Mutated DNA § Can’t maximize concentration of target DNA sequence by maximizing any individual kine parameter § Analogy between a) exiting a tight parking spot b) maximizing the concentration of one DNA sequence in the presence of single nucleotide polymorphisms

PCR Temperature Control Model Sequence-dependent annealing DNA targets Cycling protocol

Sequence-dependent Model Development Reaction Equilibrium Information ΔG – From Nearest Neighbor Model Relaxation Time Similar to the Time constant in Process Control τ – Relaxation time (Theoretical/Experimental) Solve above equations to obtain rate constants K. Marimuthu and R. Chakrabarti, Sequence-Dependent Modeling of DNA Hybridization Kinetics: Deterministic and Stochastic Theory, in preparation

Sequence-dependent rate constant prediction v σ – Nucleation constant for resistance to form the first base pair v The forward rate constant is a fixed parameter v Estimate σ, forward rate constant offline based on our experimental data v Compute t and hence kf, kr for a given DNA sequence using S. Moorthy, K. Marimuthu and R. Chakrabarti, in preparation

Variation of rate constants

Flow representation of standard PCR cycling

From standard to generalized PCR cycling Accessibility May mention reachable set here rather than above move / send to backup 6. Decide what to sho Specify controls in finite set May show affine extension state equations in u, f, g format Then transition to full OCT – for nonlinear problem, application of vector fields in arbitrary com PCR gradient, mentioning PMP and definition of phi(t) (can then indicate below that gradien Project flow w Gramian in terms of phi(t) – for comments on model-free learning control of

Optimal Control of DNA Amplification For N nucleotide template – 2 N + 13 state equations Typically N ~ 103 R. Chakrabarti et al. Optimal Control of Evolutionary Dynamics, Phys. Rev. Lett. , 2008 K. Marimuthu and R. Chakrabarti, Optimally Controlled DNA amplification, in preparation

Optimal control of PCR 95 Temperature in Deg C 85 75 65 55 45 -10 10 30 50 70 Time in Seconds 90 110 130 150

Optimal control of PCR 95 Temperature in Deg C 85 Minimal time control? Apply Lagrange cost 75 65 55 45 0 20 40 Annealing Time = 10 s 60 80 Time in Seconds Annealing time = 12 s 100 120 140 Annealing time = 15 s

Optimal control of PCR 95 1 0. 9 85 0. 8 75 0. 6 0. 5 65 0. 4 0. 3 55 0. 2 0. 1 45 0 0 20 40 Annealing Time = 10 s 60 80 100 120 140 Time in Seconds Annealing time = 12 s Annealing time = 15 s Efficiency Temperature in Deg C 0. 7

Optimal control of PCR 95 1 0. 9 85 0. 8 75 0. 6 65 Competitive problems? 0. 5 Check rank of Gramian 0. 4 0. 3 55 0. 2 0. 1 45 0 0 20 40 Annealing Time = 10 s 60 80 100 120 140 Time in Seconds Annealing time = 12 s Annealing time = 15 s Efficiency Temperature in Deg C 0. 7

Optimal control of PCR 95 85 Temperature in Deg C Cycle 1 Cycle 2 75 Geometric growth: after 15 cycles, DNA concentrations are 65 red – 4× 10 -10 M blue – 8× 10 -9 M green – 2× 10 -8 M 55 45 0 20 40 Annealing Time = 10 s 60 80 Time in Seconds Annealing time = 12 s 100 120 140 Annealing time = 15 s

Technology Development for Control of Molecular Amplification v Next steps: application of nonlinear programming dynamic optimization strategies for longer sequences, competitive problems v Future work: robust control, real-time feedback control using parameter distributions we obtain from experiments

Summary • Can reach ultimate limits in sustainable and selective chemical engineering through advanced dynamical control strategies at the nanoscale • Requires balance of systems strategies and chemical physics • New approaches to the integration of computational and experimental design are being developed

Reviews of our work Quantum control § R. Chakrabarti and H. Rabitz, “Quantum Control Landscapes”, Int. Rev. Phys. Chem. , 2007 § C. Brif, R. Chakrabarti and H. Rabitz, “Control of Quantum Phenomena” New Journal of Physics, 2010; Advances in Chemical Physics, 2011 § R. Chakrabarti and H. Rabitz, Quantum Control and Quantum Estimation Theory, Invited Book, Taylor and Francis, in preparation. Bionetwork Control and Biomolecular Design § “Progress in Computational Protein Design”, Curr. Opin. Biotech. , 2007 § “Do-it-yourself-enzymes”, Nature Chem. Biol. , 2008 § R. Chakrabarti in PCR Technology: Current Innovations, CRC Press, 2003. § Media Coverage of Evolutionary Control Theory: The Scientist, 2008. Princeton U Press Releases

Pathway Examples • 6 level system, Pif transition – (i) Amplitude of 2 nd order pathway via state 2: (i) – (ii) Transition amplitude for 3 rd order pathway (ii)

Interference Identification Normal Dynamics H(t, s) Uba(T) Decode {Unba} Encode H(t) Encoded Dynamics Uba(T, s)

Linear Programming Formulation: Observable Max § Quantum observable maximization: § Translation to linear programming: Me K. Moore, R. Chakrabarti, G. Riviello and H. Rabitz, Search Complexity and Resource Scaling for Quantum Control of Unitary Transformations. Phys. Rev. A, 2010

The analogy to the “assignment problem” § Maximum weighted bipartite matching (assignment prob): Given N agents and N tasks Any agent can be assigned to perform any task, incurring some cost depending on assignment Goal: perform all tasks by assigning exactly one agent to each task so as to maximize/minimize total cost

Foundation for Quantum System Learning Control. II: Geometry of Search Space • Maximum weighted bipartite gamma_i, lambda_j 5. Maximum weighted bip matching of prob): Would need to menti indicate the two examples slide, then show projected one matrix G_thick, indicat compatibility cond’n, and in start from points within pol vertex (do not need to draw Replace w • Birkhoff polytope: • flows start from points within polytopes and proceed to. M: in on p optimal vertex R. Chakrabarti and R. B. Wu, Riemannian Geometry of the Quantum Observable Control Problem, 2013, in preparation.

R. Chakrabarti, Notions of Local Controllability and Optimal Feedforward Control for Quantum Systems. J. Physics A: Mathematical and Theoretical, 2011.

Quantum Estimation

Sequence-dependent rate constant prediction Negative reciprocal of the maximum Eigenvalue is the Relaxation time.

Kinetic rate constant control: general formulation Kinetic rate constant control • general formulation of rate constant control • temperature control formulation 3. Use beamer for now? Finalize Decide wheth not essential