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
U I (t ) 
 IN 
i
( )
n
i
 V (t) dt  (
I
0
t
t
 
0 0
c (t )   j | 
1
ji
t
i
t n 1
0

t
0
i
)
2
t
t
  V (t)V (t) dtdt 
0 0
I
I
VI (t )VI (t ) VI (t n ) dt n

dt 
Need to introduce V_
VI (t ) dt  | i 
Remove the lambdas
i
c (t )   j | (  )
n
ji
n
t
t 1
0
0
 
n
VI (t ) VI (t n ) dt n
dt  | i 
We don’t show the interm
we for consistency w belo
9. Finalize these
Quantum Interferences and Quantum Steering
T
N T
t2
N
N T
t3
t2
0
l 1 0
0
j 1 k 1 0
0
0
U ba (T )   vba (t1 )dt1    vbl (t2 )  vla (t1 )dt1dt2    vbj (t3 )  v jk (t2 ) vka (t1 )dt1dt2dt3 
6
5
4
3
2
1
| c (T )  c (T ) |
1
ba
2
ba
2
*
1
2
 c (T )  c (T )   cba
(T )  cba
(T ) 
1
1
2
 | cba
(T ) |2 2 Re[cba
(T )cba
(T )*] | cb2a (T ) |2
1
ba
2
ba
Interference
6
5
4
3
2
1
• 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

1
 E1
diag
exp


N
 Ej 
 kT

exp  


j 1
 kT 
Mixed state density matrix:
 (0)
Pure state:
 (0) |  (0) (0) | diag (1, 0,
Expectation value of observable:
F (U (T ))  Tr (U (T )  (0)U † (T )O)
Cost functional:
J ( (·))  F (U  (·) (T ))


,

 E
, exp   N
 kT



, 0)
R. Chakrabarti, R. Wu and H. Rabitz, Quantum Pareto Optimal Control. Phys. Rev. A, 2008.
Quantum System Learning Control: Critical Topology
J
i
  Tr O(T ),  (0)   (t )   0,
 (t )
O(T ),  (0)  0
2 N 2
 2J
H (t , t ') 
  l (t ) l (t ')
 (t ) (t ) l 1
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
D  S1  S2
k1m , k2m
DNA Melting
DNA Melting
Again
S1  P1  S1P1
k11 ,k21
k1 ,k2
S2  P2 
 S2 P2
2
2
Primer
Annealing
ke ,k  e
SP  E 
 E.SP
k n , k n
E.SP  N 
[ E.SP.N ] kcat
 E.D1
k n , k n
E.D1  N 
[ E.D1.N ] kcat
 E.D2
.
E.DN 
 E  DNA
kcat '
k11t ,k21 t
S1  S2  DNA
3/19/2016
School of Chemical Engineering, Purdue
University
27
The DNA Amplification Control Problem and Cancer Diagnostics
Wild Type
DNA
Mutated
DNA
 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
Sequence-dependent Model Development
f r
S1  S2 
D
k ,k
Reaction
Equilibrium Information
 G 
k f / kr  K  exp  

 RT 
Relaxation Time

ΔG – From Nearest
Neighbor Model
Similar to the Time constant in Process Control

1
kr  k f CS 1eq  CS 2eq

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
 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
S. Moorthy, K. Marimuthu and R. Chakrabarti, in preparation
Variation of rate constants
Flow representation of standard PCR cycling
 (t1 ,
, t30 )( p)  
fu3
ct30

f u2
bt30
cycle 30
u1

fu1
at30

fu3
ct1

fu2
bt1
 ( p)
fu1
at1
cycle 1
 [0.00, 3340, 0.00, 3340, 0.00, 0.00, 0.00]T
u2
 [30.0, 5.95, 30.0, 5.95, 0.04, 0.62, 53.5]T
u3
 [0.00, 19.0, 0.00, 19.0, 0.01, 0.90, 275]T
u  U  {u1 , u2 , u3}
Choose times t1 ,
, t30 : Lie brackets, analogy to parallel parking
From standard to generalized PCR cycling
Accessibility
May mention reachable set here rather than above
move / send to backup
R ( p)   tukk 
 tu11 ( p),
k  1, u1 ,
6. Decide what to sho
, uk U , t1 ,
, tk  0
u1
L  span{[ f u1 ,
,[ f uk 1 , f uk ]]}
L  span{[hik ,
,[hi2 , hi1 ]]}, hi { fu1 ,
L  span{[him ,
L  span{[him ,
,[hi2 , hi1 ]]}, hi { fu1 ,
Specify controls in finite set
u2
u3
u
, f um }
, f uk }
,[hi2 , hi1 ]]}, hi { f ( y ), g ( y )}
May show affine extension state equations in u,f,g format
T
dU T ( s )
   [  (t )] T [ 
0
ds
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

Min CDNA  t f   C
T (t )
st
max
DNA

2
dx
 f  x, u 
dt
x  CS1 , CS2 ,.....CE .D1 .....CDNA 
Tr
For N nucleotide template –
2N + 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
90
85
Temperature in Deg C
80
75
70
65
60
55
50
45
-10
10
30
50
70
Time in Seconds
90
110
130
150
Optimal control of PCR
95
90
85
Temperature in Deg C
80
Minimal time control?
Apply Lagrange cost
75
70
T
L
65
 dt
0
60
55
50
45
0
20
40
Annealing Time = 10 s
60
80
100
Time in Seconds
Annealing time = 12 s
120
140
Annealing time = 15 s
Optimal control of PCR
1
90
0.9
85
0.8
80
0.7
75
0.6
70
0.5
65
0.4
60
0.3
55
0.2
50
0.1
45
0
0
20
40
Annealing Time = 10 s
60
80
100
Time in Seconds
Annealing time = 12 s
120
140
Annealing time = 15 s
Efficiency
Temperature in Deg C
95
Optimal control of PCR
1
90
0.9
85
0.8
80
0.7
75
0.6
70
Competitive problems?
0.5
65
Check rank of Gramian
0.4
60
0.3
55
0.2
50
0.1
45
0
0
20
40
Annealing Time = 10 s
60
80
100
Time in Seconds
Annealing time = 12 s
120
140
Annealing time = 15 s
Efficiency
Temperature in Deg C
95
Optimal control of PCR
95
90
85
Cycle 1
Temperature in Deg C
80
Cycle 2
75
70
Geometric growth:
after 15 cycles,
DNA concentrations
are
65
60
red – 4×10-10 M
blue – 8×10-9 M
green – 2×10-8 M
55
50
45
0
20
40
Annealing Time = 10 s
60
80
100
Time in Seconds
Annealing time = 12 s
120
140
Annealing time = 15 s
Technology Development for Control of Molecular Amplification
 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
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
1  4
6
5
– (i) Amplitude of 2nd order
pathway via state 2:
4
3
2
T t2
2( 2)
U 41
   v42 (t2 )v21 (t1 )dt1dt2
1
(i)
0 0
6
– (ii) Transition amplitude for
3rd order pathway 1  2  5
5
4
 4
3
T t3 t 2
3( 2 , 5 )
U 41
    v45 (t3 )v52 (t2 )v21 (t1 )dt1dt2 dt3
0 0 0
2
(ii)
1
Interference Identification
VI ( s)  H I exp(i s)
H(t,s)
U (T )   ı  b | 
n
T
0

t2
0
VI (tn ) VI (t1 )dt1
U ba (T , s)  U ban exp(in s)
n 1

Uba (T ,  )   Uba (T , s) exp(i s) ds

Uba(T)
Decode
dU I (t , s )
 iVI ( s)U I (t , s)
dt
n
ba
Normal
Dynamics
Encode
H(t)
Encoded
Dynamics
dtn | a
Uba(T,s)
Linear Programming Formulation: Observable Max
 Quantum observable maximization:
  R†  R  diag{ 1 ,
,1 ;
; r ,
p1
  S †S  diag{1 ,
J (U )   |U ij |2  i  j
i, j
pr
, 1 ;
q1
,  r },
; s ,
, s }
qs
J (U )  Tr[( R†US ) ( R†US )]  Tr (U U †)
N
|U
j 1
ij
|2  1; i  1,
,N
ij
|2  1;
,N
N
| U
i 1
j  1,
 Translation to linear programming:
J (U )  J (x)  cT x
x(i 1) N  j  |Uij |2
Me
c(i 1) N  j   i  j
Ax  b
bi  1; i  1,
, 2N 1
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
N
N
max  cij xij ,
i 1 j 1
N
x
i 1
ij
 1,
N
x
j 1
ij
 1, xij  0, cij 
Foundation for Quantum System Learning Control. II:
Geometry of Search Space
5. Maximum weighted bip
• Maximum weighted bipartite matching
of
prob):Would
need to menti
indicate the two examples
\gamma_i,\lambda_j
slide, then show projected
one matrix G_thick, indicat
 ( s, t )
compatibility cond’n, and in
 iTr  T† ( s )F ( T ( s ))  (t ) 
start from points within pol
t
vertex (do not need to draw
T
d T ( s )
   [  (t )] T [  (t )] dt F ( T ( s))
0
ds
Replace w
dxT ( s )
1

M
F ( xT ( s ))
• Birkhoff polytope:
ds
• flows start from points within polytopes and proceed toM: 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
dx
 Ax
dt
Negative reciprocal of the maximum
Eigenvalue is the Relaxation time.
Kinetic rate constant control: general formulation
Kinetic rate constant control
m
dx
  ui gi ( x )
•general formulation dt
of i 1
rate constant control x  [ x1, , xn ]T ; xi  0,i  1, , n
u
 [u1 , , um ]T ; ui  0, i  1, , m
•temperature control
formulation
u2
um
1
 k 2u1
 k mu1m 1
T (t )  
Ea ,1
u (t )
R ln
k1
3. Use beamer for now? Finalize
Ea ,i 1
i 
Ea1
Decide wheth
not essential