B. Schmittmannstte/BPC-WG/APS_Schmittmann.pdf · Driven diffusive systems: • Simple lattice models, involving particles and holes – One or two species of particles – Excluded

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).