Molecular Control Engineering

Nonlinear Control at the Nanoscale

Molecular Control Engineering

Nonlinear Control at the Nanoscale

Raj Chakrabarti

PSE Seminar Feb 8, 2013

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 microscopicinputs 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

Molecular Design

Control of Biochemical Reaction Networks

femtoseconds,angstroms

femtoseconds,angstroms

milliseconds, micrometersmilliseconds, micrometers

picoseconds,nanometerspicoseconds,nanometers

Quantum Control of Chemical Reaction Dynamics

Parallel Parking and Nonlinear Control

Tight spots: Move perpendicular to curb through sequences composed of Left, Forward + Left, Reverse + Right, Forward + Right, Reverse

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

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

8. Finalize these

Vector Fields

Control with Linear Vector Fields

Lie Brackets and Directions of Motion

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?

From classical control to the coherent control of chemical processes

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

Coherent Control versus Catalysis

probability densityprobability density

timetime interatomic distanceinteratomic 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.

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

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

Bilinear and Affine Control Engineering

Few-Parameter Control of Quantum DynamicsFew-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

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 DynamicsOptimal 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

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

1

2

0 0 0

0 0 0

( )

( )  ( ) ( ) ( ) 

( ) ( ) ( ) ( ) n

I

t t t

N I I I

t t tn n nI I I

U t

i iI V t dt V t V t dt dt

iV t V t V t dt dt

1

0

1

0 0

( ) | ( )  |

( ) | ( ) ( ) ( )  |n

t

ji I

t tn n n nji I I

ic t j V t dt i

ic t j V t V t dt dt i

Remove the lambdas

Need to introduce V_I

We don’t show the intermediate states here; shouldwe for consistency w below?

9. Finalize these

Understanding Interferences

• Mechanism identification techniques have been devised to efficiently extract important constructive and destructive interferences

32 2

1 1 2 1 1 2 3 2 1 1 2 31 1 10 0 0 0 0 0

( ) ( ) ( ) ( ) ( ) ( ) ( )tt tT T TN N N

ba ba bl la bj jk kal j k

U T v t dt v t v t dt dt v t v t v t dt dt dt

1

2

3

4

5

6

1

2

3

4

5

6

InterferenceInterference

*1 2 2 1 2 1 2

1 2 1 2 22

| ( ) ( ) | ( ) ( ) ( ) ( )

( ) | 2Re[ ( ) ( )*] ( || )|

ba ba ba bab ba

ba ba ba b

a

a

c T c T c T c T c T c T

c T c T c T c T

Quantum Interferences and Quantum Steering

V. Bhutoria, A. Koswara and R. Chakrabarti, Quantum Gate Control Mechanism Identification, in preparation

1

1

1(0)   exp , ,exp

exp

N

Nj

j

EEdiag

E kT kT

kT

(0) | (0) (0) |  (1,0, ,0)diag

Mixed state density matrix:

Pure state:

†( ( )) ( ( ) (0) ( ) )F U T Tr U T U T OExpectation value of observable:

(·)( (·)) ( ( ))J F U T Cost functional:

Control of Molecular Dynamics

HCl

CO

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.

( ( 0,(

))

), (0)O TJ i

Trt

t

( ), ) 0(0O T

2 2 2

1

( , ') ( ) ( ')( ) ( )

N

l ll

JH t t t t

t t

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 index

• 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

From Quantum Control to Bionetwork Control

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

04/19/23 School of Chemical Engineering, Purdue University 27

DNA Melting

PrimerAnnealing

Single Strand – Primer Duplex

Extension

DNA MeltingAgain21

, 21 SSDmm kk

DNASS tt kk 12

11 ,

21

22,

22

22

21 PSPS kk

DNAEDE

DENDENDE

DENSPENSPE

SPEESP

kcatN

kcatkk

kcatkk

kk

nn

nn

ee

'

.

.

.]..[.

.]..[.

.

21,

1

1,

,

11,

11

12

11 PSPS kk

Wild Type DNA

Mutated DNA

The DNA Amplification Control Problem and Cancer DiagnosticsThe DNA Amplification Control Problem and Cancer Diagnostics

Can’t maximize concentration of target DNA sequence by maximizing any individual kinetic parameter

Analogy between a) exiting a tight parking spot

b) maximizing the concentration of one DNA sequence in the presence of single nucleotide polymorphisms

Can’t maximize concentration of target DNA sequence by maximizing any individual kinetic 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

/ expf r

Gk k K

RT

ΔG – From Nearest Neighbor Model

1 2

1

eq eqr f S Sk k C C

,

1 2f rk k

S S D

τ – Relaxation time(Theoretical/Experimental)

Solve above equations to obtain rate constants

Reaction

Equilibrium Information

Relaxation Time Similar to the Time constant in Process Control

Sequence-dependent Model Development

K. Marimuthu and R. Chakrabarti, Sequence-Dependent Modeling of DNA Hybridization Kinetics: Deterministic and Stochastic Theory, in preparation

σ – Nucleation constant for resistance to form the first base pair

The forward rate constant is a fixed parameter

Estimate σ, forward rate constant offline based on our experimental data

Compute and hence kf, kr for a given DNA sequence using

Sequence-dependent rate constant prediction

S. Moorthy, K. Marimuthu and R. Chakrabarti, in preparation

Variation of rate constants

3 32 1 2 1

30 30 30 1 1 11 30

cycle 1cycle 30

( , , )( ) ( )u uu u u uf ff f f f

ct bt at ct bt att t p p

1

2

3

1 2 3

[0.00, 3340, 0.00, 3340, 0.00,0.00,0.00]

[30.0, 5.95, 30.0, 5.95, 0.04, 0.62, 53.5]

[0.00, 19.0, 0.00, 19.0, 0.01, 0.90, 275]

{ , , }

T

T

T

u

u

u

u u u u

U

Flow representation of standard PCR cycling

1 30Choose times : Lie brackets, analogy to parallel par ing, , kt t

Accessibility

1 1span{[ , ,[ , ]]}

k ku u uL f f f

2 1span{[ , ,[ , ]]}, { ( ), ( )}

mi i i iL h h h h f y g y

May mention reachable set here rather than above

May show affine extension state equations in u,f,g format

PCR gradient, mentioning PMP and definition of \phi(t) (can then indicate below that gradient components in 2nd cycle will be ~ null)Project flow w Gramian in terms of \phi(t) – for comments on model-free learning control of competitive problems below)

Then transition to full OCT – for nonlinear problem, application of vector fields in arbitrary combinations

Specify controls in finite set

• Reachable set

May remove / send to backup6. Decide what to show, finalize

0

( )[ ( )] [ ( )]    ( ( ))

T TTT

dU st t dt F U s

ds

1

2

3

1 2 3

[, , , , ]

[, , , , ]

[, , , , ]

{ , , }

T

T

T

u

u

u

u u u u

U

2 1 1span{[ , ,[ , ]]}, { , , }

m ki i i i u uL h h h h f f

From standard to generalized PCR cycling

1

11 1( )    1,    , ,( ), ,    , , 0k

kt t k ku up k u u t tp U R

2 1 1span{[ , ,[ , ]]},   { , , }

k mi i i i u uh h h h f f L

1 2 1

2max

( )

.

,

, ,..... .....

DNA f DNAT t

Tr

S S E D DNA

Min C t C

dxst f x u

dt

x C C C C

For N nucleotide template – 2N + 13 state equations

Typically N ~ 103

Optimal Control of DNA Amplification

R. Chakrabarti et al. Optimal Control of Evolutionary Dynamics, Phys. Rev. Lett., 2008K. Marimuthu and R. Chakrabarti, Optimally Controlled DNA amplification, in preparation

Optimal control of PCR

Optimal control of PCR

0

T

L dt

Minimal time control?Apply Lagrange cost

Optimal control of PCR

Optimal control of PCR

Competitive problems?

Check rank of Gramian

Optimal control of PCR

Cycle 1 Cycle 2

Geometric growth:after 15 cycles,DNA concentrationsare

red – 4×10-10 Mblue – 8×10-9 Mgreen – 2×10-8 M

Next steps: application of nonlinear programming dynamic optimization strategies for longer sequences, competitive problems

Future work: robust control, real-time feedback control using parameter distributions we obtain from experiments

Technology Development for Control of Molecular Amplification

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

• Insert more slides here:• A) Affine control system (edit slide above to

precede bilinear w affine?)

• B) possibly Magnus expansion vis-à-vis controllability. Possibly geometric picture of Lie brackets, Ad formula vis-à-vis CBH

• 6 level system, Pif transition

– (i) Amplitude of 2nd order pathway via state 2:

– (ii) Transition amplitude for 3rd order pathway

41

4521

T t

dtdttvtvU0 0

21121242)2(2

41

2

)()(

T t t

dtdtdttvtvtvU0 0 0

321121252345)5,2(3

41

3 2

)()()(

1

2

3

4

5

6

1

2

3

4

5

6

(i)

(ii)

Pathway Examples

Interference Identification

( ) exp( )I IV s H i s

1

( , ) exp( )nba

nbaU T s U in s

2

1 10 0( ) | ( ) ( ) |

T tnnI n I nbaU T ı b V t V t dt dt a

( , ) ( , ) exp( ) ba baU T U T s i s ds

H(t)

H(t,s)

Uba(T)

Uba(T,s)

Enc

ode

NormalDynamics

EncodedDynamics

Dec

ode

{Un ba

}

( , )( ) ( , )I

I I

dU t siV s U t s

dt

Quantum observable maximization:

Translation to linear programming:

2

1

|  | 1; 1, ,N

ijj

U i N

2

1

|   | 1; 1, ,N

iji

U j N

2( 1)  |  |i N j ijU x

2

,

( ) |  |ij i ji j

J U U

( ) ( ) TJ U J x c x

1;   1, , 2 1ib i N

1

1

†1 1

†1 1

{ , , ; ; , , },

{ , , ; ; , , }

r

s

r r

p p

s s

q q

R R diag

S S diag

Ax b

( 1)i N j i j c

† † †) [( ) ( ) ] ( )(U Tr R US R US Tr U UJ

Mention riemannian geometry working paper

Linear Programming Formulation: Observable Max

K. Moore, R. Chakrabarti, G. Riviello and H. Rabitz, Search Complexity and Resource Scaling for Quantum Control of Unitary Transformations. Phys. Rev. A, 2010.

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

The analogy to the “assignment problem”

1 1 1 1

max , 1, 1, 0,N N N N

ij ij ij ij ij iji j i j

c x x x x c

•Maximum weighted bipartite matching of \gamma_i,\lambda_j

•Birkhoff polytope: •flows start from points within polytopes and proceed to optimal vertex

†

0

( , )( ) ( ( )) ( )

( )[ ( )] [ ( )]    ( ( ))

T T

T TTT

s tiTr s F s t

td s

t t dt F sds

Replace w polytope formulation

5. Maximum weighted bipartite matching (assignment prob):Would need to mention Birkhoff polytope and then indicate the two examples shown in notes in a separate slide, then show projected flow on polytope in terms of just one matrix G_thick, indicate it is inverse metric due to compatibility cond’n, and indicate in bullet point that flows start from points within polytopes and proceed to optimal vertex (do not need to draw the polytopes now)

1( )( ( ))T

T

dx sM F x s

ds

M: inverse Gramian, Riemannian metricon polytope

Foundation for Quantum System Learning Control. II: Geometry of Search Space

R. Chakrabarti and R. Wu, Riemannian Geometry of the Quantum Observable Control Problem

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

bionetwork and biomolecular amplification control; sequence dependence of rate constants

4. Consolidate wrt KM’s prelim slides

Axdt

dx

Negative reciprocal of the maximum Eigenvalue is the Relaxation time.

Kinetic rate constant control

•general formulation of rate constant control

•temperature control formulation

•general formulation of rate constant control

•temperature control formulation

1

1

2 2 1

1m

m m

u k u

u k u

1

1

1

( )

[ , , ] ;  0,  1, ,

[ , , ] ;  0,  1, ,

m

i ii

Tn i

Tm i

dxu g x

dt

x x x x i n

u u u u i m

1

, 1a ii

a

E

E

,1

1

( )( )

ln

aET t

u tR

k

Decide whether to explicitly show the form of the g_i(x)’s here;not essential

3. Use beamer for now? Finalize

Kinetic rate constant control: general formulation