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From Asymmetric Exclusion Processes to Protein Synthesis Beate Schmittmann Physics Department, Virginia Tech Workshop on Nonequilibrium dynamics of spatially extended interacting particle systems January 11-13, 2010 Funded by the Division of Materials Research, NSF with Jiajia Dong (Hamline U.) and Royce Zia (Virginia Tech), and many thanks to Leah Shaw (William & Mary).

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Outline: Basic facts about protein synthesis A simple model: TASEP with locally varying rates – Currents and density profiles for one and two slow codons – “point” particles – “extended” objects – Real genes Conclusions and open questions

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Protein synthesis Image courtesy of National Health Museum Two steps: Transcription: DNA RNA Translation: RNA Protein

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Shine-Dalgarno, Kozak A ribosome… starts at one end (initiation) goes to the other, “knitting” the amino acid chain (elongation) releases aa-chain at the end and falls off mRNA (termination) Before one falls off, another one starts! initiation elongation termination http://cellbio.utmb.edu/cellbio/rer4.jpg

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Knitting the aa into the polypeptide chain Left: http://www.emc.maricopa.edu/faculty/farabee/BIOBK/BioBookglossE.html Right: cellbio.utmb.edu/cellbio/ribosome.htm; also Alberts et al, 1994

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Some interesting features: In E. coli, 61 codons code for 20 amino acids, mediated by 46 tRNAs tRNA concentrations can vary by orders of magnitude Translation rate believed to be determined by tRNA concentrations “Fast” and “slow” codons Synonymous codons code for same amino acid; Degeneracy ranges from 1 to 6

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Example: Leucine in E. Coli H. Dong, L. Nilsson, and C.G. Kurland, J. Mol. Biol. 1996 tRNA codon

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Some interesting features: In E. coli, 61 codons code for 20 amino acids, mediated by 46 tRNAs tRNA concentrations can vary by orders of magnitude Translation rate believed to be determined by tRNA concentrations Codon bias: In highly expressed genes, “fast” codons appear more frequently than their “slower” synonymous counterparts “Fast” and “slow” codons Synonymous codons code for same amino acid; Degeneracy ranges from 1 to 6

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Towards a theoretical description: Translation is a one-dimensional, unidirectional process with excluded volume interactions Suggests modeling via a totally asymmetric exclusion process

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The model: TASEP of point particles Open chain: – sites are occupied or empty – particles hop with rate 1 to empty nearest-neighbor sites on the right – particles hop on (off) the chain with rate ( ) – random sequential dynamics (easily simulated!) Totally asymmetric simple exclusion process … Ring: much simpler The proto model: F. Spitzer, Adv. Math. 5, 246 (1970)

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Why study TASEP ? Mathematicians: “Consider… this stochastic process ” Biologists : simple minded model for protein synthesis Physicists: –Non-equilibrium statistical mechanics –Interacting systems with dynamics that violate detailed balance, time reversal –Novel states and stationary distributions –Many other potential applications

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(T)ASEP: Far from equilibrium ! Non-zero transport current – mass (energy, charge, …) Open boundaries Coupled to two reservoirs Simplest question: Properties of non-equilibrium steady state? Answer: Solve master equation! …

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TASEP of point particles: P*(C) can be found exactly: – density profiles, currents, dependence on system size – non-trivial phase transitions! … 1/2 1 1 High Low Max J Phase diagram: MacDonald et al, 1968; Derrida et al, 1992, 1993; Schütz and Domany 1993; many others High: Low: Max: Note on pbc

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Towards a theoretical description: Translation is a one-dimensional, unidirectional process with excluded volume interactions Suggests modeling via a totally asymmetric exclusion process Modifications: – Translation rates are spatially non-uniform; start with one or two slow codons, then consider a whole gene – Ribosomes are extended objects (cover about 10 – 12 codons); start with point- like objects, then consider different sizes Goal: Explore the effect of “bottle necks” (rates, location) and xxxribosome size (L.B. Shaw et al, 2003, 2004) (A.Kolomeisky, 1998; Chou & Lakatos, 2004)

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TASEP with bottle necks: To model the effects of one or two slow codons: – change hopping rates locally to q 1 – for simplicity, choose = = 1 q q x … y Measure current ( protein production rate) and density profile: – as a function of x, y and q

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One slow site: Without slow site: System is in max current phase: With slow site: Left/right segment in high/low density phase N = 1000 q = 0.2; centered Particles – holes : …except for q 0.7 Density profile: Simulations… Edge effect!

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Edge effect: Mean-field theory: Density profiles: Current: A.Kolomeisky, 1998 Simulations… N = 1000, q = 0.6 Maximized at q=0.49: 2.5% k=1: good results from FSMFT site

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Two slow sites: L = 1000; q 1 = q 2 = 0.2; separated by 500 sites Particles – holes: Typical density profiles: q 1 = q 2 = 0.2 q 1 = q 2 = 0.6 Simulations… … and extension of MFT

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Current is sensitive to separation: Current vs separation: q 1 = q 2 = 0.6 Current reduction vs q: q Significant effect! Chou and Lakatos, 2004

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Note: Two slow sites with q 1 q 2 : Slowest site determines current Fast site(s) : Significant effects on profiles; none on currents First set of conclusions: To maximize current, i.e., protein synthesis rate: – Slow codons should be spaced as far apart as possible! Check effect of particle size! Chou and Lakatos, PLR 2004; Dong, Schmittmann, Zia JSP 2007

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Effect of particle size, l … Entry: – only if first l sites are free; then, whole particle enters with rate Hopping: – left-most site is “reader”, determines local rate Exit: – hops out gradually, “reader” leaves with rate β Lakatos and Chou, JPA 36, 2027 (2003): Complete entry and incremental exit

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Phase diagram: 1 1 High Low Max J High: Low: Max: McDonald and Gibbs, 1969; Lakatos and Chou, 2003; Shaw et al., 2003 Results based on mean-field analysis or extremal principle; no longer exact but in good agreement with simulations.

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One slow site: Without slow site: System is in max current phase. With slow site: Left/right segment in high/low density phase Coverage density profile (all occupied sites) Reader density profile (only sites occupied by readers) Simulations… N = 1000, q = 0.2, x = 82 l = 0 1 l = 0 6 l = 12 Edge effect! Long tails!

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Edge effect: Simulations… Current reduction vs q:

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Two slow sites: Coverage density profile: Reader density profile: Simulations… N = 1000, q = 0.2 l = 0 1 l = 0 2 l = 0 6 l = 12 Shock still develops!

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Current is sensitive to separation: Current reduction vs q: Simulations…

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Second set of conclusions: The basic conclusion of the point particle study remains valid: – Currents are maximized if slow codons are spaced as far apart as possible. – Edge effect becomes more dramatic, as l increases Real genes?

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From TASEP to protein production: Lattice Site Particle Hopping rate γ i Current J mRNA template Codon Ribosome tRNA cellular concentration Protein production rate

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A real gene: dnaA in E. coli Protein required to initiate chromosome replication 467 codons, 138 (30%) are sub-optimal Raw tRNA abundances:

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Optimize: original (wild)optimalabysmal J 0.0114550.0175140.007115 Δ J + 53 % 38 % highest wild wild lowest ~ 1.5 ~ (138 replacements)(225 replacements)

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Optimize: original (wild)optimalabysmal J 0.0114550.0175140.007115 Δ J + 53 % 38 % 2.8% 2 slowest: 10 slowest: 17% Clustering!

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Clustering is important: Introduce “coarse-grained” rate: K 1 is time needed to traverse l consecutive sites Shaw, Zia, and Lee PRE 2003

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K 12 measure: originaloptimalabysmal J0.0114550.0175140.007115 Δ J+ 53 % 38 % Δmin { K 12 }+ 58 % 42 % K 12 min = 0.441 K 12 min = 0.699 K 12 min = 0.255

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Several sequences – same protein:

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Fully Optimized Wild (“original”) Totally Suppressed 700 other sequences Simulated current J MC vs. K 12 min Best linear fit through OWS Both fits provide tolerable and simple estimates for the J ’s Best linear fit through OWS and the origin

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Similar results for 10 other genes in E.coli Example of lacI : (with just 5 other randomly generated sequences) Slopes are ~10% of each other. J ~ const. K 12 min Simulated current J MC vs. K 12 min ??? DNA-binding transcriptional repressor

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Conclusions: Protein production can be increased significantly by a few xxtargeted removals of bottlenecks and clustered bottlenecks. K measure provides simple estimate of changes in production rates Extensions: Initiation-rate limited mRNA; finite ribosome xx supply; polycistronic mRNA; parallel translation of multiple xx mRNAs; and many other issues. J.J. Dong, B. Schmittmann, and R.K.P. Zia, J. Stat. Phys. 128, 21 (2007); Phys. Rev. E 76, 051113 (2007); J. Phys. A42, 015002 (2009) J.J. Dong, PhD thesis. Virginia Tech (May 2008) Experiments!

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