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Autocatalytic Sets and Models of Early Life

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Autocatalytic Sets and Models of Early Life Autocatalytic sets and models of early life Wim Filipa Sousa Hordijk ! Mike Steel Joint work with… Elchanan Mossel Stuart Kauffman Oxford, 2016 1 2 It is often said that all the conditions for the first production of Some ‘formal’ models a living organism are now present, which could ever have been present.— But if (& oh what a big if) we could conceive ! (1940s-1980s) in some warm little pond with all sorts of ammonia & phosphoric salts,—light, heat, electricity &c present, that a ! Self-reproducing automata (von Neumann) protein compound was chemically formed, ready to undergo ! Chemoton model (Gánti) still more complex changes, at the present day such matter would be instantly devoured, or absorbed, which would not ! ‘Hypercycles’ (Eigen and Schuster) have been the case before living creatures were formed. ! Collectively autocatalytic systems (Kauffman, Farmer, Bagley) Letter to J. D. Hooker, 1 Feb [1871] ! First cycles in directed graphs (Bollobás and Rasmussen) ! Many ideas/theories re. origin of life ! (M,R)-systems (Rosen) ( ‘RNA world’, genetic first/vs metabolism first, hydrothermal vents ! More recently (1990s-) etc). ! Petri Nets (Sharov) ! Chemical organisation theory (COT) ! Current DNA/RNA/protein molecular " (Contreras et al. 2011; Kreyssig et al. 2012) machinery too complex to have arisen ! RAF theory spontaneously all at once. 3 4 Two features of catalysis ! Key early steps require the emergence (and evolution) of self-sustaining and autocatalytic Accelerates the production of networks of reactions. molecules in the network so they accumulate spatially in concentrations sufficient to sustain further reactions and fight diffusion. Not only much faster rates, but also Vaidya et al., Nature, 2012 tightly ‘coordinated’ Wolfenden,)Snider,)Acc.)Chem.)Res,) 2001) 5 6 Catalytic Reaction System (CRS) Another way to view a CRS =(X, R0,C,F) A directed (and bipartite) graph with Q two types of vertices (molecule types, reactions) and two types of arrows Molecule Catalysis “Food” set types (reactants + products, catalysis). Reactions C X R0 F X ✓ ⇥ ✓ f X X (x,r) 1 R0 2 2 ✓ ⇥ r p 1 r =( reactants , products )=(⇢(r), ⇡(r)) 4 { } { } p p 3 1 f r p r f 3 3 2 2 2 7 8 Simple example: polymer model Early claim: “The formation of autocatalytic sets of polypeptide catalysts is an expected emergent collective property of sufficiently complex A set of molecules represented by strings over an alphabet (e.g. 0, 1) sets of polypeptides, amino acids, and other small molecules.” up to length n, with food molecules up to length t (with t << n). (Kauffman, 1986) A set of reactions of two types: ligation: 000+111000111 Basic idea: Given a fixed probability of cleavage: 01010100101+010 catalysis p and increasing n, at some point there is a phase transition where the entire reaction graph becomes an Randomly assigned catalysis: autocatalytic set, similar to Pr[x catalyzes r] = p(x,r) giant connected components appearing Uniform model p(x,r)=p in random graphs. 9 10 Main Criticisms Our approach ● Argument requires an exponential growth rate in Use mathematics (and simulations) to study the level of catalysis (Lifson, 1996). polymer model and its extensions ● Autocatalytic sets lack evolvability (Vasas, Szathmáry & Santos, 2010). ● Binary polymer model is not realistic enough (Wills & Henderson, 1997). First we need to formalize some notions…. We will consider all these issues…. 11 12 Definitions: Closure Definition: F-generated ! Given any subset R of R , the closure of F (relative to R) 0 R is F –generated if clR(F ) contains clR(F ) every reactant of every reaction in R is the set of molecule types that can be constructed from F by applying just reactions from R (whether they are catalysed or ⇒ each reactant of any reaction in R is either in F not). o r is a product of some other reaction in R ! Formally, clR(F) is the unique (minimal) subset W of X that contains F and satisfies: ⇢(r) W ⇡(r) W ⇐ ? ✓ ) ✓ ! clR(F) is computable in polynomial time in |Q| 13 14 Definition: RAF (Reflexively Earlier example Autocatalytic network over F) A subset R of R0 is an RAF if R ≠ ∅, and it satisfies the two properties: (RA): each reaction r in R, is catalysed by a product of some other reaction (or by an element of F), (F): R is F-generated 15 16 f1 f4 Equivalent definition p4 Two nice properties of RAFs f5 p1 p3 ! The union of any collections of RAFs is itself an RAF " So if Q has an RAF then it contains a unique maximal RAF. " Denote this by maxRAF(Q) f3 f2 p2 ! There is a simple algorithm to determine whether or not Q has an RAF, and if so to compute maxRAF(Q) (polynomial time in A subset R of R0 is a RAF if |Q|). R = , and for each reaction r R 6 ; 2 all of the reactants and at least one catalyst of r are present in clR(F ) 17 18 f1 f4 Related notions: p4 RAF Algorithm f5 p3 p1 R0,R1,... (nested decreasing sequence) with limit R 1 RAF f3 Ri+1 = reactions in Ri that have all their reactants f2 f1 p p4 and at least one catalyst in cl (F ) f1 f4 2 Ri p4 f5 p3 p1 If R = ,then has no RAF, else R = maxRAF( ). p3 p1 1 ; Q 1 Q f3 f2 f3 f2 p 2 p2 CAF pseudo-RAF (Constructively Autocatalytic network over F) 20 Quantities of Interest Early results ! p constant (i.e. independent of n): Q = (X, R0, C, F ) full binary polymer model (on all n+1 n+1 [Kau↵man; 1986, 1993] Pn 1 as n sequences of length up to n).|X|~2 ; |R0|~n2 . ! !1 ! But this requires f to grow exponentially with n which is biochemically unrealistic (Lifson ‘96) ● Average number of reactions catalyzed by any molecule type: f = p·|R | 0 ! What if f grows more slowly? " [S: 2000] 1 1 ● Probability P = Pr(R contains an RAF) that an n 0 If f<3 e− then Pn 0 as n instance of the binary polymer model contains an ! !1 2 If f>cn then Pn 1 as n RAF set. ! !1 [Conjecture: sub-quadratic] 21 22 Probability of RAF Sets Main theoretical results I (Mossel+S, 2005) ! Theorem 1: Linear transition for RAFs 1 δ If f n − then P 0 as n / n ! !1 If f n1+δ then P 1 as n / n ! !1 ! Theorem 2: Exponential transition for CAFs If f (2 δ)n then P 0 as n / − n ! !1 If f (2 + δ)n then P 1 as n / n ! !1 [Hordijk+S, 2004] 23 24 The actual bounds (Mossel+S, 2005) Small RAFs λ t (e− ) ! If f λn then Pn 1 λ Finding a smallest RAF is NP-hard. 1 e− ≥ ≥ − − = #states t = length of sequences in F ! Definition: An RAF of Q is irreducible if it contains no proper sub-RAF t =2, =2 Generates all of X but not all of R0 (this ! Finding an irrRAFs is easy… requires f to grow quadratically with n) λ ! But there can be exponentially many Infact, P10 = 0.5 when f = 1.3 (not f = 17). of them If f λn then P Mλ n 25 26 Small RAFs Main theoretical result (S+Hordijk+Smith 2013) ! irrRAFs can be of different sizes: Theorem 3 2000 There are no small RAFs 1500 when they first appear # reactions 1000 If f = λn then as n : !1 cn 500 (R0 contains an RAF of size < 2 ) 0 P maxRAF ! irrRAF 0 1.15 1.20 1.25 1.30 1.35 1.40 1.45 f = p | R| ! When first cycles first appear in a random directed graph, they are of all sizes (Bollobás and Rasmussen, 1989) ! There are tiny RA sets, and tiny F-generated sets but none that are both. Two instances of binary polymer model with n=10 (|R0|= 16,388) at 27 28 f =1.2 where Pn~0.5. Structure of RAFs Poset of RAFs ! In general maxRAF(Q) may contain (many) other sub- RAFs Dynamics (Gillespie algorithm) An instance of binary polymer model Movie by Wim Hordijk (KLI institute) 2016 with n=5 (|R0|= 196), f=073, where Pn~0.5. 29 30 Computing: It’s easy to…. Application to a real experimental system Find all the maximal proper subRAFs of the maxRAF (polynomial time in |Q|). Construct the poset P of subRAFs of Q. RNA ribozyme replicator system: 16 reactions, 18 molecules, |F|=2, 64 catalysation pairs (x,r). (polynomial time in |P|x|Q|). +Dynamics of subRAFs via Gillespie algorithm Forms an RAF (but not a CAF!). Contains many subRAFs. 31 32 Application to a living organism Extensions beyond the uniform model ! Allowing p(x,r) to vary ! Methanopyrus Template-based catalysis kandleri E coli ! p(x,r) depends on length of x # CRS has 1826 reactions, 1199 molecule types (42 ! Partitioned polymers system catalysts), |F|=438. # RAF set of 1787 reactions out of 1826 = 98%. Not a CAF # Complex subRAF structure # min F for a RAF = 123 molecules ! Extension of transition theory to non-polymer systems [n replaced by ratio of # reactions to # molecules] 33 34 JGAA, 0(0) 0–0 (0) 11 M(1 q ), and so, from Eqn. (13), Pn is less than or equal to M(λ + o(1)), which− converges∗ to zero as λ 0. → ! • Hordijk, W. and Steel, M. (2016). Autocatalytic sets in Fig. 3 shows the behavior of Pn as a function of f for the binary polymer Recent work model, across our four exemplar distributions (for n = 10 and n = 16).
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