b. schmittmannstte/bpc-wg/aps_schmittmann.pdf · driven diffusive systems: • simple lattice...
TRANSCRIPT
Teaching at the edge of knowledge:
B. SchmittmannPhysics Department, Virginia Tech
Funded by the Division of Materials Research, NSF
Non-equilibrium statistical physics
With many thanks to:
Uwe C. Täuber and Royce K.P. Zia
Outline:
• Non-equilibrium statistical physics: The challenges
• At Virginia Tech: Our course in NESM
• An application: Protein synthesis
• The future
Challenges:
• There is a huge, fundamentally unsolved problem out there:
xxxStatistical physics for systems far from equilibrium
– Many (102 - 1023) interacting components
– Open boundaries: fluxes in/out of system are defining feature
– External driving forces: chemical potential gradients, temperature xxxgradients, external fields, etc.
– Outside the domain of usual thermodynamics (Gibbs free energy, xxxentropy, etc., not defined)
NAS study “CMMP 2010”:
• Complex systems
• Physics far from equilibrium
• The physics of life
…and impacts:
• Applications arise in many fields of science and engineering
• Biology is complex, but simple models from non-equilibriumphysics can be very useful.
• Many examples for systems far from equilibrium:
– Biological processes– Weather patterns– Traffic problems– Ripples on water, sand; granular materials; gel electrophoresis, etc.
The Virginia Tech environment:
• 24 faculty members, 14 in Condensed Matter, 7 in CondensedMatter Theory
– Kulkarni, Park, Pleimling, Schmittmann, Slawny, Täuber, Zia
– Statistical physics of far-from-equilibrium model systems, biological physics, density functional theory, mathematical physics, high-performance simulations
• 55 – 60 graduate students
The Virginia Tech environment:
• Outside physics: Interested faculty in Chemistry, Mechanical Engineering, Engineering Science and Mechanics,Chemical Engineering
The Virginia Tech course:
• Two semesters of “Statistical Mechanics”
– First semester covers standard equilibrium SM.
– Offered every fall, part of graduate “core curriculum”.
– Second semester is offered upon student demand.
– Offered roughly every 2 – 3 years. Elective.
The Virginia Tech course:
• Structure:
– Two lectures per week (75 min each)
– Optional problem sets
– Students choose between presentation or research project
Topics:
• Review of equilibrium statistical mechanics
• Basics: Random variables, probability distributions, stochastic processes
• Markov processes and master equations:
− Stationary solutions P*(C)
− Detailed balance, probability currents K(C,C’), topology of configuration space and probability currents
− Monte Carlo simulation techniques
− Time-dependence
− Examples: Random walks and exclusion processes (biased and unbiased), xxxbirth-death processes … revisited throughout the course
R.K.P. Zia, Session B22.00005;R.K.P. Zia and BS, J. Phys. A: Math. Gen. 39, L407, (2006);R.K.P. Zia and BS, cond-mat/0701763.
{P*, K*}
Topics:
• Fokker-Planck equations, Kramers-Moyal expansion
• Stochastic equations of motion:
− Brownian motion, via Langevin: Einstein, Kubo, …
− Brownian motion, via Fokker-Planck
− Brownian motion in external potentials: Langevin & Fokker-Planck
− Kramers’ escape rate, approach to equilibrium in bistable potentials
− Other issues: inertia, general noise correlations, reversible forces
Topics:
• Infinitely many degrees of freedom:
− Dynamics near equilibrium critical points: Models A, B, C, …
− Static and dynamic scaling
− Field theory formulation: Response functions, FDT, and all that
− Dynamics near non-equilibrium critical points: DDS, DP, …
− Gaussian approximation and the beginnings of perturbation theory
… and a nice application
TASEPs and protein synthesis:
• Totally asymmetric simple exclusion process: – Paradigmatic model for transport far from equilibrium – Exactly soluble in d =1; nontrivial phase diagram
• Protein synthesis – Fundamental component of cell activity
J.J. Dong, Session V30.00007 Thu 12:27 CCC 304;J.J. Dong, BS, and R.K.P. Zia, J. Stat. Phys. (Online First 2006)
Protein synthesis (“translation”)
Left: Image courtesy of National Health MuseumRight, top: http://www.emc.maricopa.edu/faculty/farabee/BIOBK/BioBookglossE.htmlRight, bottom: 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 41 tRNAs
• Translation rate believed to be determined by tRNA concentrations
Synonymous codons code for same amino acid
“Fast” and “slow” codons
… opportunities for optimization!
Example: Leucine in E. Coli
Amino acid codon anticodon tRNA %
Leucine CUU, CUC GAG 1.75
Leucine CUA * (UAG?) 0.58
Leucine CUG CAG 5.83
Leucine UUA, UUG AAA (UAA?) 1.46
Solomovici et al., 1997
Some interesting features:
Synonymous codons code for same amino acid
• Codon bias: In highly expressed genes, “fast” codons appear more frequently than their “slower” synonymous counterparts
• In E. coli, 61 codons code for 20 amino acids, mediated by 41 tRNAs
• Translation rate believed to be determined by tRNA concentrations“Fast” and “slow” codons
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 α (β)
– random sequential dynamics (easily simulated!)
Totally asymmetric simple exclusion process
… …βα
(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:
• Steady state can be solved exactly: – density profiles, currents, dependence on system size
– non-trivial phase transitions!
… …βα
1/2 1
α
β
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( ββ −=J
)1( αα −=J
)(4/1 1−+= LOJ
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 (L.B. Shaw et al, 2003, 2004)
(A. Kolomeisky, 1998; Chou & Lakatos, 2004)
Questions:
• Localized “bottlenecks”?
• Extended objects?
• Real genes?
• Optimizing protein production rates?
• Abundance constraints? Correlations?
• Multi-mRNA mixtures – effective interactions?
• …
A. Kolomeisky, 1998; Chou and Lakatos, 2004
McDonald and Gibbs, 1969; Lakatos and Chou, 2003; Shaw et al., 2003
Dong, BS, and Zia, 2006; Dong, BS, and Zia, to appear
J.J. Dong, Session V30 – today at 12:27 in 304
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
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 10000.2
0.4
0.6
0.8
0 200 400 600 800 1000
q1 = q2 = 0.2 q1 = q2 = 0.6
Conclusions:
• Protein production can be increased significantly by a few xxtargeted removals of bottlenecks and clustered bottlenecks.
• Overexpress tRNA, rather than modifying mRNA structure: xxAgain, target selected tRNA for overexpression.
• Extensions: Initiation-rate limited mRNA; finite ribosome xxsupply; polycistronic mRNA; parallel translation of multiple xxmRNAs; and many other issues.
The future:
• Generalized fluctuation theorems
• Thermostatted molecular dynamics and phase space contraction
• Soft matter far from equilibrium; granular and glassy systems;aging phenomena; and much, much more!
Evans, Jarzinsky, Seifert, …Gallavotti & Cohen, Lebowitz & Spohn, …
Driven diffusive systems:
• Simple lattice models, involving particles and holes– One or two species of particles– Excluded volume constraint – Biased hopping rules– Applications: FICs, colloidal systems, traffic, …
• … with some rather counterintuitive properties
BS and R.K.P. Zia, Vol. 17 of “Phase Transitions and Critical Phenomena” (1995);BS, J. Krometis and R.K.P. Zia, Europhys. Lett. 70, 299 (2005); I.T. Georgiev, BS, and R.K.P. Zia, PRL 94, 115701 (2005);D. A. Adams, BS, and R.K.P. Zia, submitted to PRE (2007).