Statistical Physics of Self-Replication

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1 Statistical Physics of Self-Replication Review of paper by Jeremy England Mobolaji Williams Shakhnovich Journal Club Nov. 18,

2 Theoretical Physics of Living Systems Ambitious: Develop new mathematical/ principle based model of biological process. (e.g., Michaelis Menten, Hodgkin- Huxely) Conservative: Use our understanding of physics to place limits on biological processes. Can diffusion place limits on how fast bacteria should swim? Yes! They should swim faster than 30 microns/sec to outrun diffusion. Edward Purcell, Life at Low Reynolds Number,1976 2

3 Self-Replication is Irreversible and Thus Produces Entropy Self Replication One unit turns into two units over some period of time England s Argument Self replication is irreversible Irreversible processes are associated with increases in entropy Physics of entropy production (i.e., statistical physics) should place thermodynamic bounds on self replication. 3

4 Irreversibility and Entropy through Master Equation - Part 1 Transitions to and from a single state (i 1! j; t) (j, t) (i 1,t) (j! i 1 ; t) Transitions to and from all possible states (j, t). (i 1,t) (i 2,t). (i n,t). 4

5 Irreversibility and Entropy through Master Equation - Part 2 Transitions to and from all possible states (j, t). (i 1,t) (i 2,t). p(j, t) =X i. [ Master Equation ] h i (i! j; t)p(i, t) (j! i; t)p(j, t) Foundational equation of Non-Equilibrium Statistical Physics 5

6 Irreversibility and Entropy through Master Equation - Part 3 For long times the system goes into equilibrium and is time independent h i lim (i! j; t)p(i, t) (j! i; t)p(j, t) t!1 =0 By thermal physics and by definition Ei lim p(i, t) =e t!1 Z( ) lim (i! j, t) 1(i! j) t!1 [ Boltzmann Distributed Energies ] [ Time Independent Transition Amplitude ] Thus we have 1 (j! i) 1 (i! j) = e E i!j 6

7 Irreversibility and Entropy through Master Equation - Part 3 bath By conservation of Energy Q bath Q bath + E system = E universe =0, system and so we have 1 (j! i) 1 (i! j) = e Q i!j. E system The definition of (dimensionless) entropy in terms of heat: S = Q/k B T 1 (j! i) 1 (i! j) = e S i!j bath The more irreversible the reaction, the more entropy produced in the forward direction 7

8 Generalizations and Definitions Let i, j, k,.. define a microstate, and I, II, III, define the macrostate. Self-Replication Process (A single unit of I) (Two units of I) Macrostate of I, can be in any microstate i. (I! II) (II! I) Macrostate of II, can be in any microstate j. ln apple (II! I) (I! II) [Defines the irreversibility of a self replication process] 8

9 Second Law of Thermodynamics Part 1 Some probability theory and algebra leads to Heat Released into Bath Entropy Change of System h Qi bath, I!II +ln apple (II! I) (I! II) + S sys, I!II 0 Entropy Change Associated with Reaction- Irreversibility bath h Qi bath, I!II Conceptually Second Law of thermodynamics with macroscopic irreversibility correction system (I! II) S sys, I!II (II! I) This equation applies in a wide range of far from equilibrium scenarios! 9

10 Formal to Informal Dictionary Say we have a replicator which grows at a rate g and degrades at a rate n(t) (g )t n(t) =n 0 e We can then define (I! II) =g dt [Probability of one unit to duplicate in time dt] t (II! I) = dt [Probability of one unit to degrade in time dt] s int q [Entropy change over time dt] [Heat released over time dt] 10

11 Second Law of Thermodynamics Part 2 q + s int ln h g i [Main Equation of Paper] q + s int ln h g i Growth Rate Heat Released Internal Entropy Change Degradation Rate Two ways to look at this result: 1) Maximum Ratio: Given the amount of heat produced in a replication, what is the minimum ratio between growth and decay rate? 2) Minimum Heat Released: Given a growth rate and a decay rate, what is the minimum amount of heat released during replication/growth? 11

12 Maximum Net-Growth Rate (i.e., Fitness) We can also compute Net Growth Rate Degradation Rate g max = (exp[ q + s int ] 1) Maximum Fitness Heat Released Internal Entropy Change Replicator s maximum potential fitness is set by how it exploits energy in its environment to catalyze it s own reproduction. Reproductive Fitness depends on Efficient Metabolism 12

13 Application: Bacterial Cell Division Main Qualitative Prediction of Work g max = (exp[ q + s int ] 1) Low Durability of Replicator 1/ Large Heat Produced Large Entropy Change q s int } Efficient Metabolism Reproductive Fitness 13

14 Application: Self-Replicating Polynucleotides Self Replicating RNA Enzyme: + substrates Doubling Time: 1 hour Lincoln, Tracey A., and Gerald F. Joyce, Science (2009) Cleavage Rate (i.e. Half Life): 4 years Thompson, James E., et al. Bioorganic chemistry(1995) ln h g i =ln apple 4 years 1hr * Change in Internal Entropy: Negligible (Mixing Entropy of Reactants ~ Mixing Entropy of Products) h Qi RT ln(g/ ) = 7 kcal/mol Experimentally, the heat released is 10 kcal/mol Minetti, Conceição ASA, et al. "Proceedings of the National Academy of Sciences(2003) Replicating RNA enzyme operates close to thermodynamic efficiency 14

15 Application: Self-Replicating Polynucleotides (Hypothetical!) Self Replicating Single Stranded DNA: + substrates Doubling Time: 1 hour Cleavage Rate (i.e. Half Life): 3 X 10^7 years h g i ln =ln Schroeder, Gottfried K., et al. PNAS of the United States of America (2006) apple 4 years 1hr h Qi RT ln(g/ ) = 16 kcal/mol > enthalpy of ligation reaction DNA self-replication (of this kind) is forbidden thermodynamically! Does this relate to why RNA (and not DNA) was the selfreplicating nucleic acid on prebiotic earth? 15

Chapter 19. Chemical Thermodynamics

Chapter 19. Chemical Thermodynamics 19.1 Spontaneous Processes Chemical thermodynamics is concerned with energy relationships in chemical reactions. We consider enthalpy and we also consider entropy in