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Autocatalytic Sets of Catalytic Polymers. A Model by Stuart A. Kauffman. Goldschmidt Ya’ara Moore Shai. Proteins First – Some Facts About Proteins. Amino acids were present in the primordial soup (Miller - 1953)

A Model by Stuart A. Kauffman

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Autocatalytic Sets of Catalytic Polymers

A Model by

Stuart A. Kauffman

Goldschmidt Ya’ara

Moore Shai

- Amino acids were present in the primordial soup (Miller - 1953)
- Peptides and proteinlike polymers of amino acids (proteinoids) can be formed under primordial conditions (Fox & Dose - 1977)
- Peptides show a wide variety of catalytic activities

- Imagine a world of cell-like creatures
- containing peptides and proteinlike molecules,
- capable of living,
- i.e. undergo evolution (reproduction selection…)

- Some questions need to be answered:
- How can a protein replicate itself?
- Is it capable of evolution?How can this system undergo selective adaptation?
- Where would the fruit of selection be stored?
- How did ‘protein first’ invent translation and the genes?

…and some more …

Autocatalytic Set

- Think of a set of peptides having the property that each member has its formation catalyzed by one or more members, so that its own high concentration is maintained.
- It is the set of peptides which is collectively autocatalytic by virtue of reflexive catalysis among its members.

- 1. Ability to catalyze the formation and cleavage of peptide bonds

Trypsin

- 2. Abiogenetic formation was feasible in prebiotic evolution
- 3. Small volume
- 4. Anabolic flux

1. Ability to catalyze the formation and cleavage of peptide bonds

- An anabolic flux synthesizing larger peptides from some maintained “food set” (amino acid, small peptides or other molecules) must be thermodynamically feasible.

k1

k2/k1=10

k2

aa + aa aa-aa + H2O

- PN – concentration of specific polypeptide with N amino acidsC = molar concentration of a specie of amino acid.
- [PN] = CNK-(N-1)
- PN falls off as N increases:The concentration of a specific polypeptide is very low.

- PN = concentration of specific polypeptide with N amino acids
- C = molar concentration of a specie of amino acid
- [PN] = CNK-(N-1)
- A = number of different monomers available

If A is high enough then AC/K > 1,

at equilibrium most of the amino acids are bound up in long polymers.

(Although concentration of a specific long polymer stays very low).

- 2. Abiogenetic formation was feasible in prebiotic evolution
- 3. Small volume
- 4. Anabolic flux – without an energy source

1. Ability to catalyze the formation and cleavage of peptide bonds

5. Achieve autocatalytic closure

- Monomers join together to form Polymers.
- Polymers induce synthesis of each other by acting as catalysts.
- Catalysts speed up chemical reactions that would otherwise occur very slowly(by a factor of several magnitudes).

- An autocatalytic system is a set of molecules, which as a group catalyze their own synthesis. Thus if A catalyzes the conversion of Y to B, and B catalyzes the conversion of X to A, then A + B comprise an autocatalytic set .

- Peptides are oriented polymers: left and right differs (e.g. ABBA).
- A Peptides can be formed by two kinds of basic reactions : Condensation (Ligation) and Cleavage.

The ‘food set’ in this case consists of 0, 1, 00 and 11. The autocatalytic set eventually emerges contains all members of the original food set, which isn’t always the case

Central problem : achieving catalytic closure in a set of catalytic polymers?

Kauffman suggests following four steps :

- Consider the set of all possible polymers up to some maximum length M.
- Consider the set of all possible legitimate reactions by which these polymers can be synthesized from one another (condensation/ligation or cleavage).

- Consider a simple model for the distribution ofcatalytic capacities of different reactions among polymers .
- Consider the resulting probabilities that the set of polymers contains a subset which is reflexively (i.e. self repeating) autocatalytic.

- We need to show that :As thecomplexity of the polymer set increases past a sharp threshold, the probability that an autocatalytic closure (a subset in a catalytic set) exists jumps sharply to 1.

- Lets observe a simple model :
- Consider two different kinds of monomers (e.g. Alanine & Glycine)
- Consider polymers of up to a maximum length of M monomers each.
- Thus, N, the number of possible different polymers in the set, is equal to 2M+1 .

ba

a + b

ab

ABAAA

AABBA

ABABA

ABAA

AABB

ABAB

ABA

AABA

ABBA

AAB

AAABA

AB

ABB

ABBBA

AAAB

ABBB

AAAAA

AAAA

AAA

AA

A B

BB

BBB

BBBB

BAAAA

BBBBA

BBBA

BAAAB

BAAA

BA

BBA

BAA

BAABA

BBAB

BAAB

BAB

BAABB

BBAA

BABA

BABAA

BABB

BABAB

BABBA

BABBB

ba

a + b

ab

ABAAA

ABAA

AABB

ABAB

ABA

AABA

ABBA

AAB

AB

ABB

AAAB

ABBB

AAAA

AAA

AA

A B

BB

BBB

BBBB

BBBA

BAAA

BA

BBA

BAA

BBAB

BAAB

BAB

BBAA

BABA

BABB

- Graphs consist of dots, or ‘nodes’, connected to each other by lines, or ‘edges’.
- We will observe directed graphs.
- As the ratio of edges to nodes increases, the probability that any one node is part of a chain of connected nodes increases, and chains of connected nodes become longer.
- When this ratio reaches ~0.5 (percolation threshold), almost all these short segments become cross-connected to form one giant cluster .

Plotting the size of the largest cluster versus the ratio of edges to nodes yields a sigmoidal curve. We receive an upward monotone graph with a sharp and abrupt steep jump at ~0.5.

- Let R denote the number of possible reactions synthesizing polymers in an AC.
- R is calculated by the product of the number of polymers of a certain length times the number of bonds between the monomers they are build of, summed across all possible lengths:

R = 2M(M - 1) + 2M-1(M - 2) +...+2M-(M-2)(M - (M - 1))

- Any polymer has a constant probability of catalyzing any reaction.
- Assign (randomly) to each polymer those reactions it catalyses.
- Will there be a connected subgraph of polymers in which there exists at lease one cycle (reflexive AC)?
- We will see further on that the probability of this happening shifts from highly unlikely to highly likely (abruptly so) as R/N increases.

- As the maximum length of polymer M increases, the number of polymers increases exponentially but the number of reactions by which these polymers might interconvert increases yet faster, such that the ratio of reactions to polymers grows linearly, as M-2.
- Each member of the set must have its formation catalyzed by at least one member of the set.
- There must be connected catalysis pathways leading from the “food set” to all members of the autocatalytic set.

- Sufficient condition for a set of polymers to be reflexively autocatalytic:

M = longest polymer.

There are (M-1) ways to form a specific polymer by condensation of smaller polymers.

= the chance that none of the 2M+1 polymers in the set catalyzes any of these M-1 reactions is:

- p is the a-priori probability of catalysis of any specific reaction by one polymer species.

is the probability that no last step in the synthesis of a specific polymer of length M, is catalyzed by any other member of the set.

- p is the a-priori probability of catalysis of any specific reaction by one polymer species.

Almost all 2M+1 members of the set have a last step in their formation catalyzed by at least one other member of the set.

- Eventually, almost all polymers will have at least one last step in their formation catalyzed by some polymer within the system.
- Any sufficiently complex set of catalytic polymers will be expected to have a subset of collectively catalytic (i.e. forming Autocatalytic Closure).

- Assumed all reactions have the same kinetic properties.
- Reliance on mass flux to drive the system away from equilibrium.
- Catalytic properties for short polymers.
- Are they alive?
- What about DNA?

- Both parts are the same as original - live on.
- Different:
- sufficient to create new autocatalytic set - live on.
- Non sufficient:
- Merge with an existing autocatalytic set - live on.
- Die.

- Kauffman, "Autocatalytic sets of proteins", J.theor.Biol., 119, 1-24, 1986
- Kauffman, "Origin of Order", Oxford University Press, 1998
- R.J.Bagley and J.D.Farmer, "Spontaneous emergence of a metabolism", in Artificial Life II, SFI studies in the science of complexity, ed. by C.G.Langton et al., Addison-Wesley, 1991
- M. Huynen, P. F. Stadler, W. Fontana, “Smoothness within Ruggedness: The role of neutrality in adaptation”, PNAS, 93, 397-401 (1996)
- J.D. Farmer, S. Kauffman and N.H. Packard, “Autocatalytic Replication of Polymers”, Physica D, 22, 50-67 (1986)