Molecular Control Engineering Nonlinear Control at the Nanoscale Raj Chakrabarti PSE Seminar Feb 8, 2013
Dec 27, 2015
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