Random autocatalytic networks
Mike Steel
Allan Wilson Centre for
Molecular Ecology and Evolution
Biomathematics Research Centre
University of Canterbury,
Christchurch, New Zealand
MSRI April 22 2005
1
Random autocatalytic networks
•Origin of Life
many theories
(Oparin, Haldane, Eigen, Schuster,
Maynard-Smith, Dyson, Kauffman, du
Duvre, Wachtershauser, Morowitz,
Deamer, Lancet, Lindhal, Russel, Dyson,
Kauffman, Lifson, Joyce, Scheuring,
Szathmary, Poole, Penny …) – problems
with most of them
•Metabolism-first (protein/enzyme) vs
genetics-replication (RNA)-first
2
Sequence length vs error-correction
w
p
l
Demetrius, Schuster and Sigmund, 1985.
1− pl
1− w
3
L~ c/1-p
From Maynard-Smith and Szathmary, 1997
4
Random molecular networks
J.E. Cohen, 1988. Threshold phenomena in
random structures. Discr. Appl. Math. 19: 113—
128.
B. Bollobas and S. Rasmussen (1989)
(First cycles in random directed graph
processes, Discr. Math. 75: 55-68).
S. Kauffman (1986, 1993)
(Autocatalytic sets of proteins,
J. Theor. Biol. 119: 1-24).
Lifson (1997). “On the critical stages
in the origin of animate matter.
J. Mol. Evol. 44, 1-8.
5
Catalytic Reaction Graphs
X = {a , b , c , d , e , f , g }
d
r1 : a + b → c
a
R = {r1 , r2 , r3 , r4 }
r2 : b + c → d
C = {( d , r1 ), ( a , r2 ), ( f , r4 )}
r3 : c + d → e + f
F = {a , b}
f
r4 : a + e → g
6
Definition of a RAF (semi-formal)
„
Given a set of reactions R, a subset R’ is an
RAF if
‰
‰
(RA) Each reaction in R’ is catalysed by at least one
molecule that is involved in R’
(F-connected) Each molecule involved in R’ can be
constructed from F by repeated application of
reactions in R’
7
Example of RAF Set
F = {a,b}
8
Observation and questions
„
„
‘Contain an RAF’ is a monotone property
How much catalysis is needed? Without high
catalysis is the existence of an RAF rare
(require ‘fine tuning’) or expected?
9
Properties of RAF Sets
„
Let Q = ( X , R , C ) be a CRS
‰
If R1 , R 2 ⊆ R are RAF, then R1 ∪ R 2 is RAF
‰
If Q has an RAF set, then it has a maximal RAF set:
U ( R ' : R ' ⊆ R is RAF)
‰
“Irreducible RAF set”: no proper subset is RAF
10
Finding an RAF set
F = {a , b}
11
Unique irreducible RAF Set
12
Finding RAFs
R’
Theorem (Hordijk and S, 2004):
R
There is an algorithm for
determining if R contains a RAF
and if so constructing an
irreducible one in polynomial
time in |R|, |X|.
13
Random catalytic sets of sequences
„
„
Molecules:
Reactions:
‰
‰
„
„
R+(n) forward (ligation): a + b → ab
R-(n) backward (cleavage): ab → a + b
Example
0 + 10 → 010
100 → 1 + 00
Food set:F = {0,1,..., κ }≤ t
Assumptions:
‰
‰
„
X (n) = {0,1,..., κ − 1}≤ n
R ( n ) = R + ( n) ∪ R − ( n)
(R1)The events `x catalyses r’ are independent
(R2) Pr[x catalyses r] depends only on x
Note: R is given, the catalysis is random!
14
Quantities of interest:
Pn = Pr[system has a RAF],
µn(x):=
P∞ = lim Pn
n→∞
average # (forward) reactions molecule x catalyses
µn(x) = Pr[x catalyses r].|R+(n)|
15
A criticism (?)
„
Kauffman:
Lifson:
„
Theorem (S, 2000):
„
µ n ( x) ∝ 2 n ⇒ P∞ = 1
µ n ( x) = c ⇒ P∞ = 1
µ n ( x ) < 1 3 e −1 ⇒ P∞ = 0
µ n ( x ) ≥ cn 2 ⇒ P∞ = 1
16
Simulations
17
Proposition (2005, Mossel + S, J. Theor. Biol.)
max x∈X ( n )
min x∈X ( n )
Can replace
µ n ( x)
n
µ n ( x)
n
µ n ( x)
n
→ 0 ⇒ P∞ = 0
→ ∞ ⇒ P∞ = 1
by
µ n ( x)
| x|
18
„Two
‰
‰
further conditions on an RAF R’
Products of R’ are not all in F.
Given Ω ⊆ 2 X − F
possible minimal
requirements for ‘life’
If Ω ≠ ∅ then there is some ω in Ω all of
whose molecules is produced by R’.
19
Theorem (2005, Mossel + S, J. Theor. Biol.)
1
µ n ( x) ≤ λn ⇒ Pn (Ω) ≤ 1 − exp(−2λc(t ) (1 + O( ))
n
2
κ (κe − λc (t ) ) t
µ n ( x) ≥ λn ⇒ Pn (Ω) ≥ 1 −
1 − κe −λc (t )
c(t) = κ+κ2+...+κt
Example: κ=2, t=2, λ = 4, Pn(Ω)>0.99
20
Irreducible RAF Sets
21
Extension (I) Inhibition
R’
Theorem (Mossel and S, 2005):
R
Determining whether R
contains an RAF is NP-complete
22
Extension (II): RAFs vs CAFs
F = {a,b}
23
Theorem
(2005, Mossel + S, J. Theor. Biol.)
µ n ( x) ≤ λ ⋅ rn ⇒ Pn CAF (Ω) ≤ 2λc(t ) 2
µ n ( x) ≥ λ ⋅ rn ⇒ Pn
CAF
κ (κe − λc (t ) ) t
(Ω ) ≥ 1 −
1 − κe −λc (t )
24
Extensions
„
„
Inhibition, concentration
Other molecular networks
Other applications?
„
„
Spontaneous combustion?
Random neural networks?
25
Further details
E. Mossel and M. Steel. Random biochemical networks and the probability of selfsustaining autocatalysis. Journal of Theoretical Biology 233(3), 327--336.
W. Hordijk, and M. Steel. Detecting autocatalyctic, self-sustaining sets in chemical
reaction systems. Journal of Theoretical Biology 227(4): 451--461, 2004.
Steel, M. The emergence of a self-catalysing structure in abstract origin-of-life models,
Applied Mathematics Letters 3: 91-95 (2000).
26