What is special about autocatalysis? Peter Schuster Institut fr - - PowerPoint PPT Presentation

What is special about autocatalysis?

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA

Advances in Chemical Reaction Network Theory ESI Wien, 15.– 19.10.2018

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

Wilhelm Ostwald, 1853 – 1932

Definition of autocatalytic reactions: Reactions that show an acceleration

• f the rate as a function of time.

Wilhelm Ostwald, 1890

( )

( )t

x a k

e a x x x a t x

) ( ) (

) ( ) ( ) ( ) ( ) ( ) (

+ −

+ + =

x(0) = 0  x(t) = 0

( )x

x x a k x a k dt dx − + = = ) ( ) (

X X A 2 →  +

k

• D. Edelson, R.J. Field, R.M. Noyes. Mechanistic details of the Belousov-Zhabotinskii oscillations. Internat. J. Chem. Kinetics 7, 417-432 (1975)

• B. Rudovics, E. Dulos, P. De Kepper. Physica Scripta T67, 43-50, 1996
• V. K. Vanag, I. R. Epstein. Internat.J.Developmental Biology 53, 673-681

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

By Echis at English Wikipedia, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=29915305

batch reactor two basic features: (i) homogeneous medium achieved by stirring

(ii) temperature control

facilitates modeling enormously!

) exp( ) ( ) ( ) ( const ) ( ) ( , ) ( , ) (

2

t c k x h a k x h k x c k t x c t x t a x x a a dt da x h x a k dt dx − − + + = = = + = = − = − =

rate of reaction for a = const: 1 x – 2 x2

stationary states: (i) state of extinction S0: (ii) state of reproduction S1: .

= x

2 1

γ γ = x

The master equation of the autocatalytic reaction A + X  2 X

A + X  2 X

• E. Arslan, I.J. Laurenzi. J.Chem.Phys.128,e015101, 2008

A + X  2 X

The reflecting barrier of A + X  2 X at X(t) = 1

The reversible autocatalytic reaction A + X  2 X can‘t become extinct (X(t) = 0). A(t) + X(t) = M + L = C

autocatalysis first order: A + X  2 X, single trajectory

{ A , X }  { A – 1 , X + 1}

autocatalysis first order: A + X  2 X, single trajectory and deterministic solution

autocatalysis first order: A + X  2 X, bundle of trajectories

autocatalysis first order: A + X  2 X, bundle and deterministic solution

autocatalysis first order: A + X  2 X, expectation value and one  error band

autocatalysis first order: A + X  2 X, expectation value and deterministic solution

autocatalysis first order: A + X  2 X, measuring stochastic delay 

stochastic delay:  = Xmax X0 / N

A + X  2X

k

k = 0.01, 0.001, 0.0001; sample size: 10 000

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

stationary states: S0 = (c0 , 0) and S1 = ((c0 + r)/(1+K) , K(c0 + r)/(1+K) – r/h)

Approach of the reaction A+X2X towards the steady state in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1,2,3,…

example of a deterministic bifurcation

separatrix

Approach of the reaction A+X2X towards the steady state in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1,2,3,…

example of a stochastic bifurcation anomalous fluctuations

Approach of the reaction A+X2X towards the steady state in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1,2,3,…

Four phases of the autocatalytic process: (i) phase I: the empty reactor is filled with resource A, (ii) phase II: random events select the state towards which the trajectory converges, (iii) phase III: the trajectory approches the long-time state, and (iv) phase IV: the trajectory fluctuates around the long-time state.

Approach of the reaction A+X2X towards the steady state in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1, convergence towards S1

Approach of the reaction A+X2X towards the steady state in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1, convergence towards S0

Approach of the reaction A+X2X towards the steady states in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1

Approach of the reaction A+X2X towards the steady states in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1; deterministic solution dashed

The stochastic trajectory approaches the steady states S0 and S1 with probabilities that depend strongly on the initial condition X(0).

Approach of the reaction A+X2X towards the steady state S1 in the flow reactor initial condition empty reactor: A(0) = 0, X(0) = 1; k = 0.01, 0.02, 0.05, 0.10

S0: state of extinction, A = C, X = 0 S1: state of reproduction, A = r / k, X = C – r/k C = A + X

three classes of fluctuations with autocatalytic processes (i) thermal fluctuations all chemical reactions   N (ii) stochastic delay autocatalytic reactions (iii) anomalous fluctuations bistability

( )

const

max

= ≅ ∆ = α δ X N X ) , ( P X f ∆ ∆ = σ

Thermal fluctuations are universal in chemical kinetics in the sense that they

• ccur with every reaction.

Stochastic delay is special for autocatalytic process with very small initial concentrations of the autocatalyst. Anomalous fluctuations occur in systems with stochastic bifurcation points.

• F. de Pasquale, P. Tartaglia, P. Tombesi. Lettere al Nuovo Cimento 28, 141- 145, 1980.

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

Leonhard Euler, 1717 – 1783

geometric progression exponential function

Thomas Robert Malthus, 1766 – 1834

Pierre-François Verhulst, 1804-1849

the logistic equation: Verhulst 1838

the consequence of finite resources

) ( exp ) ( ) ( 1 t f x C x x C t x C x x f dt x d − − + = ⇒       − =

population:  = {X}

chemical models:

reversible autocatalytic reaction annihilation reaction

absorbing barrier: X = 0  dx/dt = 0

reversible autocatalytic reaction reflecting barrier

annihilation reaction

logistic growth: A + X  2 X, 2 X  , expectation value and deterministic solution

stochastic delay:  = Xmax X0 / N logistic equation:

) ( , ) ( ) ( X X e X C X X C t X

t f

= − + =

−

annihilation reaction: (A) + X  2 X , 2 X  

bistability in the logistic equation:

( )

) ( lim : extinct and ) ( lim : = =

∞ → ∞ →

t X C t X E

t t

X

state of reproduction, S1 and state of extinction S0

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

Darwin generalization of the logistic equation to n variables yields selection

( )

Φ f x x C Φ x f x f C x x f x C x x f x − = = ≡ − = ⇒       − = dt d : 1 , ) t ( dt d 1 dt d

[ ]

( )

( )

∑ ∑ ∑

= = =

= − = − = = = =

n i i i j j n i i i j j j n i i i i n

x f Φ Φ f x x f f x x C x x

1 1 1 2 1

; dt d 1 ; : , , , X X X X 

( )

{ }

var 2 2 dt d

2 2

≥ = > < − > < = f f f Φ

;

N(0) = (1,4,9,16,25) f = (1.10,1.08,1.06,1.04,1.02)

population:

 = {X1 , X2 , X3 , … , Xn}

selection in the flow reactor

probability of selection

n = 3: X1, f1 = f + f / 2f ; X2, f2 = f ; X3, f3 = f - f / 2f ; f = 0.1 initial particle numbers: X1(0) = X2(0) = X3(0) =1

new variables:

plus-minus replication

plus-minus replication in the flow reactor

stochastic trajectory deterministic trajectory k1 = 0.011, k2 = 0.09, r = 0.5, N = 400

, , , , , *

2 1

→  + →  + + →  + → 

r k k r c

Y X A Y X Y A Y X X A A

the logic of DNA (or RNA) replication and mutation

RNA replication by Q-replicase

• C. Weissmann, The making of a phage.

FEBS Letters 40 (1974), S10-S18

Charles Weissmann 1931-

kinetics of RNA replication

C.K. Biebricher, M. Eigen, W.C. Gardiner, Jr. Biochemistry 22:2544-2559, 1983

Christof K. Biebricher, 1941-2009

1. Autocatalysis in chemistry 2. Autocatalysis in the batch reactor 3. Autocatalysis in the flow reactor 4. Autocatalysis and the logistic equation 5. Natural selection 6. Concluding remarks

Fluctuations in autocatalytic processes consist of (i) stochastic delay and (ii) anomalous fluctuations besides the common thermal fluctuations. Autocatalysis is commonly not represented by a single elementary step but appears as the results of complex many-step reaction networks. Complex autocatalytic processes in reaction networks

• ften give rise to simple over-all kinetics under suitable

conditions.

Thank you for your attention!

Web-Page for further information: http://www.tbi.univie.ac.at/~pks