Transcript Presentation

From Asymmetric Exclusion
Processes to Protein Synthesis
Beate Schmittmann
Physics Department, Virginia Tech
with Jiajia Dong (Hamline U.) and Royce Zia (Virginia Tech),
and many thanks to Leah Shaw (William & Mary).
Workshop on
Nonequilibrium dynamics of spatially
extended interacting particle systems
January 11-13, 2010
Funded by the Division of Materials Research, NSF
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
Two steps:
• Transcription:
DNA  RNA
• Translation:
RNA  Protein
Image courtesy of National Health Museum
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!
http://cellbio.utmb.edu/cellbio/rer4.jpg


initiation
elongation
Shine-Dalgarno, Kozak

termination
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
Some interesting features:
• In E. coli, 61 codons code for 20 amino acids, mediated by 46 tRNAs
Synonymous codons code for same amino acid;
Degeneracy ranges from 1 to 6
•
tRNA concentrations can vary by orders of magnitude
• Translation rate believed to be determined by tRNA concentrations
“Fast” and “slow” codons
tRNA cellular concentration [uM]
Example: Leucine in E. Coli
30
20
10
0
Leu2
Leu2
Leu3
Leu1,3
Leu5
Leu4,5
tRNA
CUU
CUC
CUA
CUG
UUA
UUG
codon
H. Dong, L. Nilsson, and C.G. Kurland, J. Mol. Biol. 1996
Some interesting features:
• In E. coli, 61 codons code for 20 amino acids, mediated by 46 tRNAs
Synonymous codons code for same amino acid;
Degeneracy ranges from 1 to 6
•
tRNA concentrations can vary by orders of magnitude
• Translation rate believed to be determined by tRNA concentrations
“Fast” and “slow” codons
• Codon bias: In highly expressed genes, “fast” codons appear more
frequently than their “slower” synonymous counterparts
Towards a theoretical description:
• Translation is a one-dimensional, unidirectional process with excluded
volume interactions
• Suggests modeling via a totally asymmetric exclusion process
Totally asymmetric simple 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!)


……
• Ring: much simpler
The 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?
lim P(C, t )  P* (C )  ??
t 
• Answer: Solve master equation!
 t P(C, t )   W (C '  C ) P(C ' , t )  W (C  C ' ) P(C, t )
C'
MacDonald et al, 1968;
Derrida et al, 1992, 1993;
Schütz and Domany 1993;
many others
TASEP of point particles:
• P*(C) can be found exactly:
Note on pbc
– density profiles, currents, dependence on system size
– non-trivial phase transitions!


……
• Phase diagram:
1
1/2

High: J   (1   )
High Max J
Low: J   (1   )
Low
1/2

1
1
Max: J  1 / 4  O( L )
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,
(A.Kolomeisky, 1998; Chou & Lakatos, 2004)
then consider a whole gene
– Ribosomes are extended objects (cover about 10 – 12 codons); start with point(L.B. Shaw et al, 2003, 2004)
like objects, then consider different sizes
• Goal: Explore the effect of “bottle necks” (rates, location) and
xxxribosome size
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
1
1
……
x
y
• Measure current ( protein production rate) and density
profile:
– as a function of x, y and q
Simulations…
One slow site:
• Without slow site: System is in max current phase: J  1 / 4  O( N 1 )
• With slow site: Left/right segment in high/low density phase
…except for q  0.7
Particles
N = 1000
– holes
:
q = 0.2; centered
1
0.8
Density profile:
Edge effect!
0.6
0.4
0.2
0
0
500
1000
Simulations…
Edge effect:
N = 1000, q = 0.6
Current:
Density profiles:
0.252
0.8
x=1
x=32
x=64
x=100
0.25
 2%
0.248
0.6
0.246
0.4
0.244
0
200
400
600
800
1000
position of the blockage
Mean-field theory:
0
50
100
150
site
A.Kolomeisky, 1998
J  q /(1  q) 2  0.234
Maximized at q=0.49: 2.5%
k=1: good results from FSMFT
200
Simulations…
Two slow sites:
… and extension of MFT
Particles – holes:
L = 1000; q1 = q2 = 0.2; separated by 500 sites
Typical density profiles:
q1 = q2 = 0.2
1
q1 = q2 = 0.6
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
0.2
0
200
400
600
800
1000
0
200
400
600
800
1000
Chou and Lakatos, 2004
Current is sensitive to separation:
Current vs separation:
Current reduction vs q:
q1 = q2 = 0.6
J (1) / J ()
1
0.25
0.9
0.24
0.8
 5%
0.23
0.7
0.6
0.22
q
0.5
0
100
200
300
0
0.25
0.5
0.75
1
separation
Significant effect!
Chou and Lakatos, PLR 2004;
Dong, Schmittmann, Zia JSP 2007
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!
Note:
• Two slow sites with q1  q2 : Slowest site determines current
• Fast site(s) : Significant effects on profiles; none on currents
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
McDonald and Gibbs, 1969;
Lakatos and Chou, 2003;
Shaw et al., 2003
Phase diagram:

1
Results based on mean-field analysis or
extremal principle; no longer exact but in
good agreement with simulations.
Max J
1 /(1   )
High

Low
1 /(1   )
1
• High:
J   (1   ) /[1   (  1)]
• Low:
J   (1   ) /[1   (  1)]
• Max:
J  1 /(1   ) 2
Simulations…
One slow site:
N = 1000, q = 0.2, x = 82
• Without slow site: System is in max current phase.
• With slow site: Left/right segment in high/low density phase
l = 01
l = 06
l = 12
Edge effect!
Long tails!
Coverage density profile
(all occupied sites)
Reader density profile
(only sites occupied by readers)
Simulations…
Edge effect:
Current reduction vs q:
1 (q) 
1 (q)
q
J (1)
J (center)
Two slow sites:
Shock still develops!
Coverage density profile:
Simulations…
N = 1000, q = 0.2
l = 01
l = 02
l = 06
l = 12
Reader density profile:
Simulations…
Current is sensitive to separation:
Current reduction vs q:
 2 (q)  J (1) / J ()
 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
mRNA template
Site
Codon
Particle
Ribosome
Hopping rate γi
tRNA cellular concentration
Current J
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:
J
ΔJ
original (wild)
optimal
abysmal
0.011455
0.017514
0.007115
+ 53 %
 38 %
(138 replacements) (225 replacements)
highest
wild
wild
~ 1.5 ~
lowest
Optimize:
original (wild)
optimal
abysmal
0.011455
0.017514
0.007115
+ 53 %
 38 %
J
ΔJ
2 slowest:
2.8%
10 slowest:
17%
Clustering!
Clustering is important:
• Introduce “coarse-grained” rate:

1


 
 k i  k 
i   1
K  ,i
1
• K 1 is time needed to traverse l consecutive sites
Shaw, Zia, and Lee PRE 2003
K12 measure:
K12 min = 0.441
K12 min = 0.699
K12 min = 0.255
original
optimal
abysmal
0.011455
0.017514
0.007115
ΔJ
+ 53 %
 38 %
Δmin { K12 }
+ 58 %
 42 %
J
Several sequences – same protein:
Simulated current JMC vs. K12 min
Both fits provide
tolerable and simple
estimates for the J ’s
700 other
sequences
Totally
Suppressed
Fully
Optimized
Best linear fit
Best linear fit through OWS
through OWS
Wild
and the origin
(“original”)
Simulated current JMC vs. K12 min
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
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.
• Experiments!
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)