Belousov-Zhabotinsky Reaction and Pattern Formation in the Distributed Systems

Lecture 22

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Pattern formation in nonlinear chemical systems

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Belousov-Zhabotinsky reaction – the history of discovery

• 1951 – Belousov discovers chemical reaction showing temporary oscillations of color – paper is rejected as non-scientific

• 1964 – the work is continued by A. Zhabotinsky who presents the reaction to the Western scientific community

• 1972 – Field, Körös and Noyes present first model of the reaction mechanism (FKN)

• 1980s – 1990s the BZ reaction is widely studied in CSTR and spatially distributed conditions. Waves, Turing patterns and various over nonlinear phenomena are discovered.

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Detailed mechanism – way too complex to model!

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FKN mechanism and Oregonator

1

2

3

4

5

2

2 2

2

0.5

k

k

k

k

k

A Y X P

X Y P

A X X Z

X A P

Z Y

+ → +

+ →

+ → +

→ +

→

42

3

, ,

,

X HBrO Y Br Z Ce

A BrO P HOBr

− += = =−= =

The simplified core of the FKN mechanism can be represented by five reactions where only concentrations of X,Y,Z are variables while A and P are parameters (fixed by the experimental conditions).

The system of kinetic equations for this system introduced by Field et al in 1972 is known is Oregonator :

21 2 3 4

*1 2 5

3 52

x k ay k xy k ax k x

y k ay k xy k z

z k ax k z

= − + −

= − − +

= −

&&&

After one of possible non-dimensionalization of the system it becomes:

(1 )x qy xy x xy qy xy z

z x z

εδ

= − + −= − − +

= −

&&

&

5

4

4

5 10

2 10

8 10q

ε

δ

−

−

−

= ⋅

= ⋅

= ⋅

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Reduction of Oregonator to 2D system with relaxation

0 (1 )qy xy x xy qy xy z

z x zδ

= − + −= − − +

= −

&&

The existence of small parameters at the time derivatives allows for the reduction of system dimensionality. Let us assume that x is in a quasi-stationary state and thus dx/dt = 0

[ ( ) ]( )

y z y x y qz x y zδ = − +

= −

&&

( ) ( )( )

y y z x x zz x z z

= ⇒ == −&

After first reduction we obtain 2D system with small parameter d. Analysis of this system demonstrates the phenomenon of relaxation oscillations.

Further reduction of the system will result in a single highly non-linear equation for z(y)

[ ( ) ]z y x y q= +

( )z x y=

z

y

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FHN model – prototype for excitable dynamics

( )x f x yy y ax bε = −

= − −

&&

Consider now again generic 2D system with cubic nonlinearity in the dx/dt nullcline but with small parameter at dx/dt, or generalized FitzHugh-Nagumo model:

Of particular interest is the combination of parameters resulting in a single stable state close to the “knee” of the nonlinear nullcline:

Such system, once perturbed over the unstable branch of the f(x) nullcline, will execute a single excursion before approaching a stable point:

t

( )y t( )x t

x

y

S

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Waves in excitable media

Provided some additional conditions are satisfied, the spatially distributed system with excitable dynamics will support waves:

Let us now assume variable y does not diffuse and we substitute time variable by “fast” time of the variable x:

With this transformation the system becomes:

( ) x

y

x f x y D x

y by ax D y

ε = − + ∆

= − + ∆

&&

, 0

' /y x yD D D

t t ε

≈

→

=

( )

( )xx f x y D x

y by axε= − + ∆= −

&&

Now we repeat the trick of transforming variables (x,y) to a single wave variable z=x-ct to find a potential wave solution:

0 ( )

( )zz z

z

f x y Dx cx

cy by axε

′′ ′= − + +′ = −

This system of two differential equations just like in the case of bistable system can be transformed into a system of first order equations:

1( ( ) )

( )

z

z

z

x w

w f x y cwD

y by axcε

′ =

′ = − − +

′ = − −

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Waves in excitable media

Just like in bistable case, transition to variables (x,y,w) changes the stability of the stationary point (0,0,0) from stable node to a saddle:

x

w

y

The waves in the original spatially distributed system will correspond to homoclinic trajectories in this system. In fact, there are two: stable and unstable (red).

Depending on the initial conditions, the wave will either develop into a propagating one or into a dying out pulse.

The initial condition is not the only requirement for propagation. It can be shown that speed of both stable and unstable solutions depend on e.

εcrε

c

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Excitable dynamics in BZ – target patterns and waves

BZ medium can be simultaneously oscillatory in well-stirred reactor and excitable in a thin 2D layer open to oxigen.

Excitable waves in the BZ system are mostly observed in the conjunction of so-called target patterns.

To induce these patterns, one needs to initiate oscillations at a point in the medium – excitable dynamics is not self-sustainable unless initiated in the periodic spatial domain.

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Spiral waves – phenomenological description

If excitable wave front is mechanically broken, the free end of the front will develop into a so-called spiral wave as shown in the figure.

Spiral waves have a number of characteristic properties:

• Their wavelength (or pitch) is considerably shorter than that of a target.

• Spiral waves tend to suppress other spatial patterns like targets in a homogeneous system – this has a very important consequence in cardiac arrhythmias (see later).

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The world of spiral waves in BZ reaction

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Spiral waves in biological systems –variety of scales

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What to take home

• Pattern formation in chemical systems provides phenomenologically and methodologically rich foundation for modeling of complex biological phenomena.

• Spatio-temporal patterns observed in distributed chemical systems are mainly defined by the internal properties of these media.

• A variety of complex dynamical patterns can be classified into three main categories according to the media properties: switch-like (multistability), excitable and oscillatory.

• The most relevant for biological applications is the excitable dynamics. It is observed as propagating pulses of excitation in the form of plane waves, circular waves and target patterns and spiral (scroll) waves.

• Spiral waves are widely occurring patterns of dynamics in biological systems. They can be identified on all scales of biological organization from intracellular to ecological population.