# Metabolism first by Dyson by AmnaKhan

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```									Metabolism first by Dyson
Seminar in Origin of Life
Yaki Setty Eran Keydar Gilad Doitsh

12.08.2008

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The lecture:
• Theory background
• Replication First (RNA WORLD) • Metabolism First (Oparin Theory)

• • • •

The meaning of Metabolism Dyson‟s Toy Model Consequences Open questions
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Oparin Theory Cells  Enzymes Genes
Dyson‟s model concentrated in the pass from cell to enzyme.

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Dyson‟s model
Introduce general (“real, complicated”) model Ask the question of “origin of life” on this model Build Toy Model (“reduced, simple”). Give the answer using the toy model.

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The general model
 Population of molecules within droplets.  There are chemical reactions within the droplet, causing the population to change.  In this case we will say the population moved from one state to second state.
State Chemical reaction State

A
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B
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The general model


The matrix M, represents the probability for chemical reactions.



The element Mi,j is the probability to move from state j to state i.

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The general model
Probability distribution is a vector
stating the probability to be in any state.
0.2 0.1 0 0.1 0.15 0.2 0 0

Probability to be at state 1

P(K) is the Probability distribution after k chemical reactions.
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The general model
P(k+1) can be calculated using P(k):

P(k+1) = M•P(k)
Therefore if P(0) is the initial distribution:

P(k) = Mk•P(0)
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Example…
We have 3 states S1 , S 2 and S 3 . 1 P(0)  0   0     0.1 0.2 0.3 M  0.3 0 0.7   0.6 0.8 0   

 0.1 0.2 0.3 1  0.1 P(1)  M  P (0)  0.3 0 0.7  0  0.3       0.6 0.8 0  0 0.6        0.1 0.2 0.3 P (100)  M 100 P(0)  0.3 0 0.7    0.6 0.8 0   
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100

1 0.22  0  0.36     0 0.42    
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The general model
A situation where the distribution of the population stay steady forever , is called

stationary distribution.
The population may get into

quasi-stationary distribution.
In this case the population will have this distribution for long time, but might get out of this distribution.
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The general model


Quasi-stationary distributions can have different levels of chemical activity.  In general, the system will get into a low level quasi stationary distribution.

Low level  disordered state High level  ordered state
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The main question
Can the population jump from disordered state to ordered state?

Let‟s figure out using Dyson‟s TOY MODEL

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What is a TOY MODEL?
 



Looking at the complex system in simple ways. Simplify the mathematics and make the model easily solvable. Allows us to check basic ideas, but not the whole system. The PHYSICS WAY!!!…
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The TOY MODEL

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Dyson‟s TOY MODEL
Dyson presents a toy model with two quasi stationary states, ordered and disordered, and determines the circumstances in which the jump will occur spontaneously.

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The goal of the TOY MODEL

disorder  order

Bad catalyst  Good Solid Liquid catalyst
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Genesis
How does the Toy model„s world look like?

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Confined volume of fluid

Surface

Fluid

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With monomers inside…

Monomers

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Chemical active surface with N sites exposed to the fluid

Surface Empty Site
Desorption of Monomer
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Occupied Site

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Polymers

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Monomers‟ classes
The monomers bound to the surface can be divided into two classes: – Active “1”

– Inactive “0”

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Activity
Active monomer – a monomer that happen to be of the right species at the right site, where it with its neighbors make a polymer that can act as an enzyme. Act as an enzyme – catalyze the adsorption of other monomers of the right species .

An active monomer helps other monomers to be active.
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Metabolism in the toy model
Population is metabolically active if the cycle shuffling maintains the active monomers at high level.

high activity  ordered state.
(metabolically active population)

low activity  disordered state.
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The dynamic of the model
Each adsorption or desorption of monomer can be regarded as an event. Each event can be regarded as reproducing changing of parent population into daughter population.
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Parent  daughter
Parent Event Daughter

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(x)
The “next generation”
number of sites with active monomers  Define x = N
 x is the activity of the population.  Define  (x) as the average activity in the daughter population.  At a steady state

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x = (x)

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1

y=x y = (x)


y



0

x

1


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1

y=x


y

y = (x)

 
0

x

1

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 - disordered (low activity) - stable state  - unstable state  - ordered (high activity) – stable state


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


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The origin of life
 the ordered stable state  “alive”  the disorder stable state  “dead”  
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death The origin of life
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Origin of life Revisiting the graph…
1

y=x y = (x)

“alive”
y
0

x

1

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Under which circumstances “Origin of life” occurs?

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Math assumptions
(For sake of simplicity)
The model is discrete, events happen one after the other. Each of the N sites on the surface, adsorbs and desorbs monomer with equal probability. HELP
lim


V

10

100  y  e n m

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(x) = x at a steady state
(x) = x =
active sites active sites active sites  inactive sites  empty sites all sites
Prob  be active    x site to   site to   site to  Prob    Prob be inactive   Prob be empty  be active     

b p

x  site to 

=

b p
x

n p

q p

bx: discrimination function

b e
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Ux kT

n+1: number of monomers‟ species.

P: probability of adsorption inactive monomer.

q: Desorption constant.
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bx - The discrimination function
W xU
W is the activation energy required for placing monomer into an empty site.
Perfect catalyst reduces W to W – U. In general, Catalyst will reduce W to W – xU, where x is the fraction of active monomers.
Reaction pathway

b e
x
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Ux kT

k is Boltzman constant.
T is absolute temperature (in Kelvin).
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(x) = x at a steady state
We define a = n + q,

( x)  (1  ab )
q 1 a 

 x 1

number of species of monomers

(x) has the desired S-shape graph.
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How a and b effect the “origin of life”?

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Parameters dependencies
(x) can have three types of steady states ,
depending on a and b.
1 0.9 0.8 1 1

Immortal

0.9 0.8 0.7 0.6 0.5

Transition

0.9 0.8 0.7 0.6 0.5

0.7
0.6 0.5

0.4
0.3 0.2 0.1 0 0 0.10.2 0.30.4 0.50.6 0.70.8 0.9 1

0.4
0.3 0.2

0.4
0.3 0.2

0.1
0 0 0.10.2 0.30.4 0.50.6 0.70.8 0.9 1

0.1
0 0 0.10.2 0.30.4 0.50.6 0.70.8 0.9 1

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g

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a , b dependencies
Immortal

Transition Region

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How long will the transition to “origin of life” take?

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Time estimation
We‟ve already estimated under which a,b Dyson‟s model would have two quasistationary states.  Two questions still open:

 Estimate the transition time from disordered

state into ordered state.  Whether it is reasonable time ?
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Transition time estimation


Solving the equations arising from the model, we get that the average transition time is:

T  e
(a,b) is the potential barrier the population should climb in order to jump to the ordered state.
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 ( a ,b )  N
 




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Reasonable Transition time
Let‟s assume , average desorption time of a
monomer, is equal to 1.

Reasonable transition time is: e30 1013
Critical size of population N = 30 / (a,b)

Critical population means that population larger than that, will not transit.
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When can “origin of life” occur?
a = 18 b = 324 N  314

Cusp
a=8 b = 64 N  22600

Immortal

Transition Region

a = 12 b = 144 N  886 a =10 b = 100 N  2070
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When can “origin of life” occur?
 a, from 8 to 10  b, from 60 to 100  N, from 2000 to 20000

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Are these results reasonable?

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Number of monomer species
 8  a  10  there should be 8 - 10 species of monomers. In Modern enzymes there are 20 species of amino acid. It is reasonable that primitive catalysts had only 10. The model fails to work with 3 or 4 monomers‟ species
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The discrimination factor
60  b  100  the primitive enzyme‟s discriminate factor is 60-100.

Modern enzymes have factor of 5000 to 10000, yet simple inorganic catalysts achieve factors of fifty.

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Number of monomers
 N range indicates that the population contains between 2000 to 20000 monomers.

N is large enough to display the chemical complexity characteristic of life. And small enough to allow the statistical jump from disorder to order.
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