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Metabolism first by Dyson Seminar in Origin of Life Yaki Setty Eran Keydar Gilad Doitsh 12.08.2008 1 The lecture: • Theory background • Replication First (RNA WORLD) • Metabolism First (Oparin Theory) • • • • The meaning of Metabolism Dyson‟s Toy Model Consequences Open questions 2 12.08.2008 Oparin Theory Cells Enzymes Genes Dyson‟s model concentrated in the pass from cell to enzyme. 12.08.2008 3 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. 12.08.2008 4 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 12.08.2008 B 5 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. 12.08.2008 6 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. 12.08.2008 7 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) 12.08.2008 8 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 12.08.2008 100 1 0.22 0 0.36 0 0.42 9 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. 12.08.2008 10 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 12.08.2008 11 The main question Can the population jump from disordered state to ordered state? Let‟s figure out using Dyson‟s TOY MODEL 12.08.2008 12 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!!!… 13 12.08.2008 The TOY MODEL 12.08.2008 14 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. 12.08.2008 15 The goal of the TOY MODEL disorder order Bad catalyst Good Solid Liquid catalyst 12.08.2008 16 Genesis How does the Toy model„s world look like? 12.08.2008 17 Confined volume of fluid Surface Fluid 12.08.2008 18 With monomers inside… Monomers 12.08.2008 19 Chemical active surface with N sites exposed to the fluid Adsorption of Monomer Surface Empty Site Desorption of Monomer 12.08.2008 Occupied Site 20 Polymers Monomers adsorbed onto neighboring sites, will link to form a polymer 12.08.2008 21 Monomers‟ classes The monomers bound to the surface can be divided into two classes: – Active “1” – Inactive “0” 12.08.2008 22 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. 12.08.2008 23 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. 12.08.2008 24 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. 12.08.2008 25 Parent daughter Parent Event Daughter 12.08.2008 26 (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 12.08.2008 x = (x) 27 Can a steady state occur?? 12.08.2008 28 Steady states 1 y=x y = (x) y 0 x 1 29 12.08.2008 Steady states 1 y=x y y = (x) 0 x 1 12.08.2008 30 Steady states - disordered (low activity) - stable state - unstable state - ordered (high activity) – stable state 12.08.2008 31 The origin of life the ordered stable state “alive” the disorder stable state “dead” 12.08.2008 death The origin of life 32 Origin of life Revisiting the graph… 1 y=x y = (x) “alive” y “dead” 0 x 1 12.08.2008 33 Under which circumstances “Origin of life” occurs? 12.08.2008 34 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 12.08.2008 35 (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 12.08.2008 Ux kT n+1: number of monomers‟ species. P: probability of adsorption inactive monomer. q: Desorption constant. 36 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 12.08.2008 Ux kT k is Boltzman constant. T is absolute temperature (in Kelvin). 37 (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. 12.08.2008 38 How a and b effect the “origin of life”? 12.08.2008 39 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 Dead 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 12.08.2008 g 40 a , b dependencies Immortal Transition Region Dead 12.08.2008 41 How long will the transition to “origin of life” take? 12.08.2008 42 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 ? 12.08.2008 43 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. 12.08.2008 ( a ,b ) N 44 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. 12.08.2008 45 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 46 Dead 12.08.2008 When can “origin of life” occur? a, from 8 to 10 b, from 60 to 100 N, from 2000 to 20000 12.08.2008 47 Are these results reasonable? 12.08.2008 48 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 12.08.2008 49 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. 12.08.2008 50 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. 12.08.2008 51