From Asymmetric Exclusion Processes to Protein Synthesis

Beate SchmittmannPhysics 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).

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

Protein synthesis

Image courtesy of National Health Museum

Two steps:

• Transcription: DNA RNA

• Translation: RNA Protein

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

Knitting the aa into the polypeptide chain

Left: http://www.emc.maricopa.edu/faculty/farabee/BIOBK/BioBookglossE.htmlRight: cellbio.utmb.edu/cellbio/ribosome.htm; also Alberts et al, 1994

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

Example: Leucine in E. Coli

0

10

20

30

Leu2 Leu2 Leu3 Leu1,3 Leu5 Leu4,5

CUU CUC CUA CUG UUA UUG

tRN

A c

ellu

lar

con

cen

trati

on

[u

M]

H. Dong, L. Nilsson, and C.G. Kurland, J. Mol. Biol. 1996

tRNA

codon

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

Towards a theoretical description:

• Translation is a one-dimensional, unidirectional process with excluded volume interactions

• Suggests modeling via a totally asymmetric exclusion process

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 simplerThe proto model: F. Spitzer, Adv. Math. 5, 246 (1970)

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

(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!

… …

??)(),(lim *

CPtCPt

'

),()'(),'()'(),(C

t tCPCCWtCPCCWtCP

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

1/2High

Low

Max J

• Phase diagram:

MacDonald et al, 1968; Derrida et al, 1992, 1993; Schütz and Domany 1993; many others

High:

Low:

Max:

)1( J

)1( J

)(4/1 1 LOJ

Note on pbc

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)

TASEP with bottle necks:• To model the effects of one or two slow codons:

– change hopping rates locally to q 1

– for simplicity, choose = = 1q q

x

… …11

y

• Measure current ( protein production rate) and density profile:

– as a function of x, y and q

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

)(4/1 1 NOJ

Density profile:

0

0.2

0.4

0.6

0.8

1

0 500 1000

Simulations…

Edge effect!

Edge effect:

0.4

0.6

0.8

0 50 100 150 200

x=1

x=32

x=64

x=100

0.244

0.246

0.248

0.25

0.252

0 200 400 600 800 1000

position of the blockage

%2

Mean-field theory:

Density profiles:

234.0)1/( 2 qqJ

Current:

A.Kolomeisky, 1998

Simulations…

N = 1000, q = 0.6

Maximized at q=0.49: 2.5%k=1: good results from FSMFT

site

Two slow sites:

L = 1000; q1 = q2 = 0.2; separated by 500 sites

Particles – holes:

Typical density profiles:

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

0.2

0.4

0.6

0.8

0 200 400 600 800 1000

q1 = q2 = 0.2 q1 = q2 = 0.6

Simulations…

… and extension of MFT

Current is sensitive to separation:

0.22

0.23

0.24

0.25

0 100 200 300

separation

%5

Current vs separation:

q1 = q2 = 0.6

Current reduction vs q:

0.5

0.6

0.7

0.8

0.9

1

0 0.25 0.5 0.75 1

q

)(/)1( JJ

Significant effect!

Chou and Lakatos, 2004

Note:

• Two slow sites with q1 q2 : 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

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

Phase diagram:

1

1

High

Low

Max J

• High:

• Low:

• Max:

)]1(1/[)1( J

)]1(1/[)1( J

2)1/(1 J

McDonald and Gibbs, 1969; Lakatos and Chou, 2003; Shaw et al., 2003

)1/(1

)1/(1

Results based on mean-field analysis or extremal principle; no longer exact but in

good agreement with simulations.

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 = 01

l = 06

l = 12

Edge effect!

Long tails!

Edge effect: Simulations…

Current reduction vs q: )(

)1()(1 centerJ

Jq

)(1 q

q

Two slow sites:

Coverage density profile: Reader density profile:

Simulations…

N = 1000, q = 0.2

l = 01

l = 02

l = 06

l = 12

Shock still develops!

Current is sensitive to separation:

Current reduction vs q: )(/)1()(2 JJq

Simulations…

)(2 q

q

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?

From TASEP to protein production:

Lattice

Site

Particle

Hopping rate γi

Current J

mRNA template

Codon

Ribosome

tRNA cellular concentration

Protein production rate

A real gene: dnaA in E. coli• Protein required to initiate chromosome replication

• 467 codons, 138 (30%) are sub-optimal

Raw tRNA abundances:

Optimize:

original (wild) optimal abysmal

J 0.011455 0.017514 0.007115

Δ J + 53 % 38 %

highest wild

wild lowest

~ 1.5 ~

(138 replacements) (225 replacements)

Optimize:

original (wild) optimal abysmal

J 0.011455 0.017514 0.007115

Δ J + 53 % 38 %

2.8%2 slowest:

10 slowest: 17%

Clustering!

Clustering is important:

• Introduce “coarse-grained” rate:

11

,

1

i

ik kiK

• K 1 is time needed to traverse l consecutive sites

Shaw, Zia, and Lee PRE 2003

K12 measure:

original optimal abysmal

J 0.011455 0.017514 0.007115

Δ J + 53 % 38 %

Δmin { K12 } + 58 % 42 %

K12 min = 0.441

K12 min = 0.699

K12 min = 0.255

Several sequences – same protein:

Fully Optimized

Wild (“original”)Totally

Suppressed

700 other sequences

Simulated current JMC vs. K12 min

Best linear fitthrough OWS

Both fits provide tolerable and simple estimates for the J ’s

Best linear fitthrough OWS and the origin

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. K12 min

Simulated current JMC vs. K12 min

???DNA-binding transcriptional repressor

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 xxsupply; polycistronic mRNA; parallel translation of multiple xxmRNAs; 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!