Post on 15-Mar-2020
1
Dynamics of Large Fluctuations:
from Chaotic Attractors to Ion Channels
Igor Khovanov
● The Concept of Optimal Path
● Types of Chaos and the Structural Stability (roughness) of models
● Large Fluctuations, Optimal Force and Control
● Ion Transport
● Conclusions
University of Warwick
2
Dynamics of Large Fluctuations:
from Chaotic Attractors to Ion Channels
● The Concept of Optimal Path
● Types of Chaos and the Structural Stability (roughness) of models
● Large Fluctuations, Optimal Force and Control
● Ion Transport
● Conclusions
3
Optimal PathBrownian Motion
Free Diffusion
2-Dimensional
Diffusion Under Constraints
(interactions)
The microfluidic cell with two counter-
propagating flows that create a shear-flow
with zero mean velocity in the central region.
2
4
1( , )4
6
r
Dt
B
er t Z
k T
t
D
D
R
1
( )
( , ) B
U r
k Tr t Z e
4
The concept of optimal paths
L Boltzmann (1904),
L Onsager and S Machlup (1953)
The concept can be applied to non-equilibrium multi-dimensional systems
x
x0x1
U(x,t) – potential
x(t) – fluctuations
x(t)
xopt(t)
Different manifestations of fluctuations:
diffusion and large fluctuations
activrelax tt
5
Optimal Path
Fluctuational paths in the state (phase) space
Transition probability
Path
Path
Path
optxPath
1x
jx
2xxf
xi1x
nx
1nx
The states
xi and xf are
attractors
),|,( iiff tt xx ])([])([ opt
j
j tt xx
The most probable (optimal) path via the principle of the least action
6
Optimal Path is a deterministic trajectoryAssume Langevin Description with White Gaussian Noise
SCdttCt
f
i
t
t
jj2
1exp)(
2
1exp])([
2ξξ
])([ jtxThe probability of fluctuational path is related to the probability ])([ jtξ
jt)(ξ
For Gaussian noise:
Since the exponential form, the most probable path has a minimal S=Smin
Changing to dynamical variables:
)()()(,0),(),( stDsttt xxx QξxKx
D
tSConsttD
opt
optif
)(exp)();(,0limit In the
xxxx
)( tSS ξ Action)(),( tt ξxKx
),()( tt xKxξ )( txS
2
min ),(min)( tdttxSS opt xKx
of random force to have a realization
Deterministic minimization problem
7
Optimal Path
Wentzell-Freidlin (1970) small noise picture 0D
txSpHS xt ,, Least-action paths are defined by the Hamiltonian
,,
);,(2
1
xp
px
xpKpQp
HH
tH
S
px
with the boundary conditions, that the action is constant,
that is the momenta are zero on sets (invariants) of a
dynamical system:
.0,:
,0,:
fff
iii
ttstateFinal
ttstateInitial
pxx
pxx
Hamiltonian gives an infinite
number of extreme trajectories,
the optimal path (if it exists)
has a minimal action
Double optimization
The pattern of
extreme paths
8
Extreme Paths
Wentzell-Freidlin (1970) small noise picture 0D
Fluctuational behaviour
measured and calculated
for an electronic model of
the non-equilibrium system
9
Optimal path
Wentzell-Freidlin (1970) small noise picture 0D
Noise-induced escape in Duffing oscillator
x
x
Basin of attraction
of the state (xs2,ys2)
Basin of attraction
of the state (xs1,ys1)
( )U xThe potential
2 4
(4
)2
xU x
x
The optimal path connects the stable state and the
saddle (boundary) state
2 2( , )s sx y
0 0( , )s sx y
1 1( , )s sx y
Activation part
Relaxation part
3 ( )x bx x Dx t x
( ) 0p t
( ) 0p t
10
Dynamics of Large Fluctuations:
from Chaotic Attractors to Ion Channels
● The Concept of Optimal Path
● Types of Chaos and the Structural Stability (roughness) of models
● Large Fluctuations, Optimal Force and Control
● Ion Transport
● Conclusions
11
Structural Stability (roughness) of models. An example
Mathematical Model is an idealization and a simplification
An electrical circuit with a tunnel diode.(see http://www.scholarpedia.org/article/Van_der_Pol_oscillator)
3 4 5 .( .) .i vv vv v A lumped model of the circuit is
0 0
1( )
( )
L
L
V v iC
i
i v E I
V
L
231 1
0VV V VC LC
Introduce 3; ;
tx t
LC
L
V C
2( 0)1 xx x x Van der Pol equation
Roughness: Small change (uncertainties) in parameters and in nonlinearities cause small
change in the state space
Li
12
Structural Stability (roughness) of models. An example
Van der Pol oscillator. State (bifurcational) diagram 2( 0)1 xx x x
0
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
x
y x
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
x
y
x
The limit cycleThe equilibrium state (fixed point)
0 Corresponds to the Andronov-Hopf bifurcation
The roughness is observed outside of a vicinity of the bifurcation
2-Dimensioal models are rough (structurally stable)
13
Structural Stability (roughness)
“Mathematical” conclusion (so far, see for example Shilnikov L.P. et al Methods of Qualitative Theory in
Nonlinear Dynamics. Part II. World Sci. 2001)
Multi-dimensional rough systems are
Morse-Smale systems which have the
limiting sets in the form of
equilibrium states and
(quasi)periodic orbits (cycles, tori);
such models may only have a finite
number of them.
Morse-Smale systems do NOT admit
homo (hetero)-clinic trajectories
(tangencies of manifolds) of saddle sets
Homoclinic loop of a saddle-focus
Smale (1963) : Rough systems with
dimension of the state space greater than
two are not dense in the space of
dynamical systems.
14
Sets of a Morse-Smale System
Saddle-node
Stable focus Cycle
Torus
Saddle Cycle
Homoclinic Loop
15
Chaotic systems
http://www.scholarpedia.org/: Chaos. There is currently no text in this page.
http://mathworld.wolfram.com/: "Chaos" is a tricky thing to define…
The complicated aperiodic trajectory of low-dimensional (3-dimensional and higher)
dynamical systems
http://en.wikipedia.org: Dynamical systems that are highly sensitive to initial conditions
Edward Lorenz: When the present determines the future, but the approximate present
does not approximately determine the future.
Positive Lyapunov exponents
16
The Chaotic Lorenz attractor
( )x y x y rx y x z z xy bz
Butterfly effect
The Lorenz attractor
demonstrates a sensitive
dependence on initial
conditions
The largest Lyapunov
exponent is positive
The attractor has no any
stable set (points, cycles)
The attractor is fractal
6
01 02(0) (0) 10 x x
1
2 2
0.897
0 14.56
10 8/3 28b r
2.05FD
Often assumed:
1( ) exp( )t t x
17
The Chaotic Lorenz attractor
( )x y x
y rx y x z
z xy bz
Lorenz system: 100 initial points in small sphere
10
28
8 / 3
r
b
1
2
3
0.8971
0.0155
14.5446
0 500 1000 1500 2000 2500 300010
-6
10-4
10-2
100
102
time
dis
tan
ce
exp(0.00345 t)
1( ) exp( )t t x
( )tx
18
Chaos and Noise
How investigate systems with noise?
One possible programme of investigation was suggested byL. Pontryagin, A. Andronov, and A. Vitt, Zh. Exp. Teor. Fiz. 3, 165 (1933)
The (slightly modified) programme
1. Classification all sets of dynamical systems and the bifurcations of
the sets (without fluctuations)
2. Checking set’s stability (robustness) in respect of fluctuations.
Consider noise-induced deviations
The results (so far) of the first step of the programme
The observation of new (in comparison with 2D case) sets: e.g. Smale Horseshoe and new
regime: Deterministic chaos; Statistical description of low-dimensional dynamics etc
Still open problems: Complete Description of sets and their bifurcations.
Shilnikov’s group results:
“…A complete description of dynamics and bifurcations of systems with homoclinic
tangencies is impossible in principle.”
S. Gonchenko, D. Turaev and L. Shilnikov, Nonlinearity (2007) 20, 241-275
19
Types of Chaos
Chaotic Attractors
☺ Hyperbolic The distinct feature are
phase space is locally spanned by the same fixed number of stable and unstable
directions in each points of the set;
the existence of SRB probability measure;
the structural stability of the set and shadowing of trajectories.
K Quasi-hyperbolic There is a localized non-hyperbolicity (“bad set”), away from this
set the system is hyperbolic and trajectory spends most of the time on a hyperbolic part
L Non-hyperbolic Tangencies of manifolds and dimension variability of manifolds
are dominated in phase space; the co-existence of a large number (often infinite) of
different sets; quasi-attractor
20
Types of Chaos
Types of chaotic systems in “real life”
L Hyperbolic
There are not typical, in fact there is no (strictly proven) any example of
hyperbolic chaotic system described by ODE
K Quasi-hyperbolic
There are rare, but real; the famous example is the Lorenz System
J Non-hyperbolic
The majority of systems are non-hyperbolic
21
Dynamics of Large Fluctuations:
from Chaotic Attractors to Ion Channels
● The Concept of Optimal Path
● Types of Chaos and the Structural Stability (roughness) of models
● Large Fluctuations, Optimal Force and Control
● Ion Transport
● Conclusions
22
Chaos and Noise
Our Aim: Dynamics of Large Fluctuations versus types of
chaotic attractors
Approach: starting with noise-free dynamics to noisy dynamics,
is problematic, since we have no complete description of attractors.
Another approach is based on simplification of initial task:
We consider noise-induced significant changes in dynamics only;
we analyse Large Fluctuations (deviation) from chaotic attractor;
The noise-induced deviations must exceed diffusion along trajectories
The optimal path concept (formalism) allows to solve an optimal control problem, since it
defines both the optimal path and the optimal fluctuational force.
23
Large Fluctuations and Optimal Control
)()()(,0
),(),(
stDst
tt
xxx Q
ξxKx
2
min ),(min)( tdttxSS opt xKxix
fx
Formally the deterministic minimization problem can be formulated in the Hamiltonian form:
,,
);,(2
1
qp
pq
qpKpQp
HH
tH
.,0,:
;,0,:
ffff
iiii
tttstateFinal
tttstateInitial
pxq
pxq
The variable q(t) corresponds to the optimal paths x(t)opt,
The variable p(t) defines the optimal fluctuational force x(t)opt and the
energy-minimal function f(x,t).New way to solve the deterministic control problem via analysis of large fluctuations and vice versa.
This form coincides with Hamiltonian formulation of deterministic optimal (energy-minimal) control problem:
24
Quasi-hyperbolic attractor
Lorenz system
1 2 1
2 1 2 1 3
3 1 2 3 2 ( )
q q q
q rq q q q
q q q bq D t
x
10, 8/3, 24.08b r
Stable points
Saddle cyclesChaotic Attractor
Consider noise-induced
escape from the chaotic
attractor to the stable
point in the limit
The task is to determine the most probable (optimal) escape path
0D
25
Quasi-hyperbolic attractor
25
Lorenz Attractor
Saddle point Separatrices
The saddle point and its separatrices belong to chaotic attractor and form “bad set” or non-hyperbolic part of the attractor
Homoclinic loop -> Horseshoe
13.92r
Loops between separatrices G1 and G2 and stable manifolds of cycles L1 and L2 generate The Lorenz attractor – quasi-hyperbolic attractor
G1
G2
L1L2
24.06r
26
Large Fluctuations in Chaotic Systems
Hamilton formalism of Large Fluctuations: The problem of initial conditions
P1P2
)()()(,0
),(),(
stDst
tt
xxx Q
ξxKx
,,
);,(2
1
qp
pq
qpKpQp
HH
tH
:
, 0, ;
:
, 0, .
i i i i
f f f f
Initial state
t t t
Final state
t t t
q x p
q x p
?
The points P1 and P2
The solution of the problem is the use of prehistory approach, i.e. analysis of real escape trajectories
27
Prehistory approach
12 July 2007
)()()(,0
),(),(
stDst
tt
xxx Q
ξxKx
1. Select the regimei.e. rare large fluctuations
xf
0D
activrelax tt
2. Record all trajectories xj(t) arrived to the final state and build the prehistory probability density ph(x,t)
The maximum of the density corresponds to the most probable (optimal) path3. Simultaneously noise realizations x(t) are collected and give us the optimal fluctuational force
There is no any explicit initial states in the approach
28
Escape from quasi-hyperbolic attractor
12 July 2007
Escape trajectories
WS is the stable manifold and
G1 and G2 are separatrices of the
saddle point O
L1 and L2 are saddle cycles
T1 and T2 are trajectories which are
tangent to WS
The escape process is connected with the non-hyperbolic structure of attractor: stable and unstable manifolds of the saddle point
Noise-induced tail
q3
q2q1
The distribution of escape trajectories (exit distribution)
29
Non-autonomous nonlinear oscillator
22 3 40
0
( , )2 ( )
( , ) sin2 3 4
0.05 0.597 1 0.13 0.95
U q tq q D t
q
U q t q q q q h t
h
x
G
G
The motion is underdamped
The potential U(q)
is monostable
Non-hyperbolic attractor
Stable cycle
Saddle cycle
Chaotic attractor
q
q
Poincare section
Initial conditions are
on the chaotic attractor
Model of particle in
the cooling beam
Noise significantly changes (deforms)
the probability density of the attractorThe escape corresponds to the noise-induced
jumps between saddle cycles of chaotic sets
q
time
Saddle cycle
Saddle cycles
( , )2 ( )
U q tq q D t
qx
G
Trajectories
corresponding to
the noise-induced
escape
Prehistory
probability of
trajectoriesq
q
q
q
Deterministic pattern of noise-induced escape from a chaotic attractor
0D
65 10D
Fractal structure
Smooth structure
-1.5 -1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Via Large Fluctuation to Energy-minimal Deterministic Control:
Migration between states ( , )( )
U q tq q f t
q
G
f (t) is a control function corresponding to the
optimal fluctuational force xopt(t)
Optimal force
App. by
App. by pulses
Optimal force + LF perturbation
Feedback control
2
1 2 3 4sin exp( ( ) )a a t a t a
q
q
The cost (energy)
of control
21inf ( )
2 f FJ f t dt
1 2 3 4 5
J
Type of the control function
1
5
32
Suppression of the noise-induced escape
Control force as an inverse optimal fluctuational force in a particular time moment
Mean time between escapes as a function of noise intensity
Suppression of
escapes
Saddle cycle of period 5
Saddle cycle of period 3
Saddle cycle of period 1
Stablecycle
Optimal force
Optimal path
The optimal force and paths are used for suppressing the escape
( , )
2 ( ) ( )opt c
U q tq q D t t t t
qx x
G
( )optq t
( ) ( )opt t f tx
o without control
x with control
33
Large Fluctuations and Types of Chaos. Control problem
Summary
For a quasi-hyperbolic attractor, its non-hyperbolic part plays an essential
role in the escape process. The saddle point and its manifolds form the non-
hyperbolic part. The fluctuations lead escape trajectory along stable and
unstable manifolds.
For a non-hyperbolic attractor, saddle cycles embedded in the attractor and
basin of attraction are important. Escape from a non-hyperbolic attractor
occurs in a sequence of jumps between saddle cycles.
The analysis of large fluctuation provides an alternative way to solve a non-
linear control problem. The optimal path and optimal fluctuational force,
determined by using large fluctuations approach, corresponds to the
solution of deterministic energy-minimal control problem.
34
Dynamics of Large Fluctuations:
from Chaotic Attractors to Ion Channels
Igor Khovanov
● The Concept of Optimal Path
● Types of Chaos and the Structural Stability (roughness) of models
● Large Fluctuations, Optimal Force and Control
● Ion Transport
● Conclusions
University of Warwick
35
Potassium channel KcsA
MD simulations of KcsA
36
Main steps in line with the tutorial: http://www.ks.uiuc.edu/Research/smd_imd/kcsa/
1. Building the full protein using the information available from the x-ray structure from
MacKinnon group, 2.0 A resolution, Y. Zhou, J. H. Morais-Cabral, A. Kaufman, and R.
MacKinnon, “Chemistry of ion coordination and hydration revealed by a K+ channel-Fab complex at
2.0 A resolution,” Nature, vol. 414, pp. 43–48, Nov. 2001.
2. Building a phospholipid bilayer;
3. Inserting the protein in the membrane;
4. Solvating of the entire system.
5. Relaxing the membrane to envelop the protein and to
let it assume a natural conformation.
KcsA is a tetramer composed of
four identical subunits
37
MD simulations of KcsA
Empirical CHARMM Force Field, VMD, NMAD
Is it a Morse-
Smale System?
38
All-Atom Molecular Dynamics (MD)
Hamiltonian Equation:
1 2
1
6 7
( ,,2
,
)
10 10
Nk k
N
k k
i i i
i i i
m
d dH H
dt m dt
H U
U
N
ir
p prr r
r p p
p r
One-Atom Brownian Dynamics (BD). Generalized Langevin Equation:
0
( )( ) ) ( )
1( ) (
( ) (
0) ( )
t
ii i i
i
B
t d t
t tk T
Vm t
rv M v R
r
M R R( )tM
)( iV x
From MD to Experiments via BD
39
Overdamped Equilibrium Dynamics
One-Atom Brownian Dynamics (BD).
( )tξ
)( iV x
Typical assumption (See B. Roux, M. Karplus, J. of Chem. Phys., Vol 95, 4856, 1991)
Overdapmed Markovian diffusion:
( ) 21( )
( ) (
( ) damping coefficient
( ) white Gaussian no
)
s
()
i e
i Bi
i i i i i
i
V Tkt
m mt
t
rr ξ
r r r
r
ξ
40
Potential of mean force (PMF)
Biased simulations (Umbrella Sampling)
0( , ) ( , ) ( ) ( )bH H U V r p r p z z
Biased
HamiltonianInitial
Hamiltonian
Biasing
PotentialResulting PMF
Will the resulting PMF be
a potential we are looking for?
41
Potential of mean force (PMF)
Biased simulations (Meta-Dynamics)
0( , ) ( , ) ( ) ( )bH H U V r p r p z z
Biased
HamiltonianInitial
Hamiltonian
Biasing
PotentialResulting PMF
Will the resulting PMF be
a potential we are looking for?
Ion permeation and PMF
42
Abstract. “… the heights of the kinetic barriers for potassium ions to move through the
selectivity filter are, in nearly all cases, too high to predict conductances in line with
experiment. This implies it is not currently feasible to predict the conductance of
potassium ion channels, but other simpler channels may be more tractable.”
From
P.W. Fowler et al
J Chem Theory Comput
2013, 9, 5176-5189.
43
Ions activation dynamics
Ions trajectories Power spectrum
1/ f
44
Ions activation dynamics
Transition from site to site
-0.01
-0.005
0
0.005
0.01
0.015
7 8 9 10 11 12 13
Selectivity filter
z
z
45
Ions activation dynamics
-0.01
-0.005
0
0.005
0.01
0.015
7 8 9 10 11 12 13
z
z
Noise-induced escape in Duffing oscillator
x
x
Basin of attraction
of the state (xs2,ys2)
Basin of attraction
of the state (xs1,ys1)
Activation part
Relaxation part
3 ( )x bx x Dx t x
z
Conclusion:
Ion dynamics is underdamped
and non-Markovian
0
( )( ) ) ( )
1( ) (
( ) (
0) ( )
t
ii i i
i
B
t d t
t tk T
Vm t
rv M v R
r
M R R
46
Ion permeation
Network of residues control the permeation
47
Ion permeation
There is a conductive conformation.
Lessons of KcsA channel
Advantage of MD technique
provide a bridge from the structure to functions
Pitfalls of MD technique
high degree of uncertainty in each step of the technique
“standard” approaches are not (always) reliable
require “experience” in the use of MD
48