from asymmetric exclusion processes to protein synthesis

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!