dynamics of large fluctuations: from chaotic attractors to ...1 dynamics of large fluctuations: from...

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






● 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


Fluctuational behaviour

measured and calculated

for an electronic model of

the non-equilibrium system

9

Optimal path


Noise-induced escape in Duffing oscillator

x

x

Basin of attraction

of the state (xs2,ys2)

Basin of attraction


( )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






● 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

http://www.scholarpedia.org/

http://mathworld.wolfram.com/

http://en.wikipedia.org/

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

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● 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



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



Igor Khovanov




● 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

http://www.ks.uiuc.edu/Research/smd_imd/kcsa/

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


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


-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


Basin of attraction


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

dynamics of large fluctuations: from chaotic attractors to ...1 dynamics of large fluctuations: from...

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