Effects of Spa-al Diffusion on a Model for Prebio-c Evolu-on Ben Intoy J. Woods Halley Aaron Wynveen U. of MN, Twin Ci-es School of Physics and Astronomy NASA grant: NNX14AQ05G Acknowledgements: Computa-onal resources of: • Minnesota Supercompu-ng Ins-tute. • A. Wynveen, I. Fedorov, and J. W. Halley Open Science Grid. • Physical Review E 89, 022725 (2014) UMN School of Physics and Astronomy • Condor Cluster. B.F. Intoy, A. Wynveen, and J. W. Halley Simon Schneider for discussions • Physical Review E 94, 042424 (2016) OSG All Hands Mee-ng March 7, 2017
Mo-va-ons • A protein first origin of life model might resolve Eigen’s paradox (the low probability of randomly construc-ng a starter “naked gene”). • Assume ini-a-ng event is the forma-on of a network of interac-ng molecules assumed to be polymers (but not necessarily proteins). • No genome, assume it comes much later. • Unlike previous similar models, we assume here that a necessary condi-on for a prebio-c chemical system is that it be a sta-onary state out of chemical equilibrium.
Kauffman-like Binary Polymer Model Network Forma,on 11 010 + 10 − 01010 Liga-on and Scission: * • − ) Given a maximum polymer length value (L max ) go through each possible • reac-on of the form: C A + B − AB − * ) and include it in the network with probability p. Dynamics Combine with reac-on rates to generate ar-ficial chemistry, then • stochas-cally simulate the following master equa-on: d n l X v l,l 0 ,m,e ( − k d n l n l 0 n e + k � 1 d n m n e ) + v m,l 0 ,l,e (+ k d n m n l 0 n e − k � 1 ⇥ ⇤ d t = d n l n e ) l 0 ,m,e Parameters in the model: p, L max , number of food par-cles, and maximum • number of par-cles. Kauffman, The Origins of Order (Ch. 7)
L max = 8 A network is considered viable if it is possible to go from the food set to an L max molecule via reac-ons. “The Origins of Order” – Stuart A. Kauffman
General Structure … p 1 p 2 p 3 For different p values: Network Forma-on: Generate mul-ple … networks (10 000) per p 1 Net 1 p 1 Net 2 p 1 Net 3 p value, check if they are viable. Dynamics: … p 1 Net 1 Run 1 p 1 Net 1 Run 2 p 1 Net 1 Run 3 • Do mul-ple dynamic simula-ons (50) with random ini-al condi-ons using a given viable network combined with reac-on rates un-l a steady state is reached. • Count the number of lifelike steady states by checking if the system is out of equilibrium. • We now have a measurement for the probability of forming a lifelike state for a value of p i , P lifelike (p i ).
How Close to Chemical Equilibrium? Use Entropy • Coarse-grain by polymer length, {N L }. • Given a macrostate {N L } the number of possible configura-ons is: L max ( N L + 2 L − 1)! Y W = N L !(2 L − 1)! L • Entropy is defined as S = k B Log W. • Chemical Equilibrium is reached when entropy is maximized (S eq ), with the constraint that there are N total molecules. • Simulate un-l steady state and consider it lifelike if the entropy is less than αS eq .
Where Kauffman and Our Group Differ Kauffman saw popula-on growth with increasing p. • System growing, but might be in chemical equilibrium. • 1 - 1 - Popula-on/Max Chem. Equil. Measure p=0.00320 0 - 0 - Simula-on Time Simula-on Time Same p value and ar-ficial chemistry, two different runs. One reaches chemical • equilibrium the other gets kine-cally trapped in a non-equilibrium steady state, Probability of forming lifelike state (%) which we postulate to be a necessary condi-on for life. 2.5 S max = 0.8 S eq S max = 0.5 S eq 2 S max = 0.3 S eq • The non-equilibrium constraint reduces the probability S max = 0.8 S eq , L max = 8 1.5 of lifelike systems at large p, giving a maximum 1 probability at a small value of p. 0.5 A. Wynveen, I. Fedorov, and J. W. Halley Physical Review E 89, 022725 (2014) 0 0 0.002 0.004 0.006 0.008 0.01 p
Extension to Include Diffusion Through Space • How might spa-al structure affect prebio-c evolu-on? • Mo-va-ons: – Can the non-equilibrium states of the model without diffusion survive interac-on with the environment through diffusion? – Are there collec-ve effects which might suggest the beginnings of mul-celluarity? – Space allows isola-on (if at low diffusion).
Spa-al Extension • We study M=64 sites arranged as an 8 x 8 2D periodic laoce. … … … … … … … … … … • Molecules are allowed to diffuse from site to site at a rate parameterized by η. • Due to computa-onal limita-ons we set L max = 6.
Simula-on General Structure … p 1 p 2 p 3 For different p values: Network Forma-on: Generate mul-ple … networks (10 000) per p 1 Net 1 p 1 Net 2 p 1 Net 3 p value, check if they Parameter are viable. sweep across η: … p 1 Net 1 η 1 p 1 Net 1 η 2 p 1 Net 1 η 3 Dynamics: … p 1 Net 1 η 1 p 1 Net 1 η 1 p 1 Net 1 η 1 Run 1 Run 2 Run 3 {N L,i } {N L,i } {N L,i } • Do mul-ple dynamic simula-ons with random ini-al condi-ons using a given viable network generated by parameter p combined with reac-on rates and diffusive value η. • A steady state is then reached with polymer length and spa-al distribu-on {N L,i }. • Analyze the {N L,i }’s to determine whether the run was lifelike or not.
Par-al and Complete Equilibra-on {N L,i =N L / M}= P d Diffusively Equilibrated (DDLA) {N L,i =g L N / (M G Lmax )} {N L,i }= P Totally Equilibrated (DEAD) {N L,i =g L N i / G Lmax }= P c Not equilibrated � Chemically Equilibrated at each site (DALD) Diffusively and Chemically (DALA) Distances From Par-al Equilibria: �� R d = ( N L,i − N L /M ) 2 . Macrospace with dimension= M L max = 384 L,i R d �� ( N L,i − g L N i /G l max ) 2 , System Point: P d R c = P L,i Hyperplane with fixed N L and dimension=(M-1) L max =378. P c Hyperplane with fixed N i R c and dimension=M (L max -1)=320. B.F. Intoy, A. Wynveen, and J. W. Halley Physical Review E 94, 042424 (2016)
Example of Results R c and R d in Simulated non- equilibrium Steady States DDLA DALA Diffusively Dead DALD Locally Alive (DDLA), Diffusively Alive Locally Dead (DALD), Diffusively Alive Locally Alive (DALA) ~20,000 Scarer points on this plot. B.F. Intoy, A. Wynveen, and J. W. Halley Physical Review E 94, 042424 (2016)
Probabili-es of DALD, DDLA, DALA states as a func-on of p and η DALD DALD DDLA Sample Probability (a) (b) 0 . 001 Sample Probability Sample Probability 0 . 02 0 . 0008 0 . 016 0 . 0006 0 . 012 0 . 0004 0 . 008 − 7 − 6 0 . 0002 − 5 0 . 004 − 1 − 4 log 10 η − 2 0 − 3 log 10 η − 3 0 − 2 0 . 01 − 4 0 . 008 0 . 01 − 1 log 10 η 0 . 006 − 5 log 10 η 0 . 008 0 . 004 p − 6 p 0 . 002 0 . 006 − 7 p 0 . 004 0 . 002 p DALA Sample Probability (c) 0 . 08 0 . 06 ~700,000 Simula-ons 0 . 04 were done to make 0 . 02 − 7 − 6 these plots. − 5 0 − 4 0 . 01 log 10 η − 3 log 10 η 0 . 008 − 2 0 . 006 − 1 p 0 . 004 p 0 . 002 B.F. Intoy, A. Wynveen, and J. W. Halley Physical Review E 94, 042424 (2016)
DALA States Display ‘cancer-like’ Explosions Before Jump (Green Dot) p=0.00452 η=10 -7 Ater Jump (Red X)
Collec-ve Effect With increasing η the explosion spreads: p=0.00452 , η=10 -1
Conclusions: With the inclusion of space we counted the likelihood, as func-ons of p • and η, of lifelike states characterized unequilibrated (DALA), diffusively but not chemically unequilibrated (DALD) and chemically but not diffusively unequilibrated (DDLA). DDLA states closely reproduce the states in the earlier, single site, model. • DALD are rare. • DALA exhibit explosive growth. • OSG Computa-onal Resources Used: ~1.4 Million computa-onal wall -me hours was used for the resul-ng • publica-on (Physical Review E 94, 042424 (2016)) . Current Work: Going back to single site (well mixed) simula-ons: • – Interested in the effects of bond energy and temperature on the model. – Exploring the sensi-vity of what is in the food set (have length one and two, but could be in different propor-ons). – Interested in the effects of increasing the number of monomer types (currently only have two, biologically DNA has 4 and proteins have 20). – S-ll using OSG to perform simula-ons! Thank you!
Entropy Calcula-ons and Misc � ( N L,i + 2 L − 1)! � M � S i ( { N L,i } ) = ln � S ( { N L,i } ) = S i ( { N L,i } ) . (2 L − 1)! N L,i ! L onditions that the i =1 � � ., N i = � N L N L,i , a = � N S global,eq ( N ) = ( MG l max − l max ) F N be � MG l max − l max � (3 L t N L = � i N L,i F ( x ) = (1 + x ) ln(1 + x ) − x ln x has a di ff erent value). In this case,
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