from asymmetric exclusion processes to protein synthesis beate schmittmann physics department,...
TRANSCRIPT
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!