Physics of nano-motors: Physics of nano-motors:

from cargo transport to gene from cargo transport to gene expressionexpression

Debashish Chowdhury

Physics Department,

Indian Institute of Technology,

Kanpur

Home page: http://home.iitk.ac.in/~debch/profile_DC.html

Motor Transport System = Motor + Track + Fuel

(A) Properties of single-motor:

(i) Composition and structure (inventory of parts and architectural design)

Fundamental questions:

(B) Collective properties:

(i) Machines within machines, e.g., replisome (DNA replication factory):

Helicase + primase + polymerase + ligase + clamp & clamp loader

(ii) Collective phenomena: coordination, cooperation and competition

(iii) Control systems and regulators of operation.

(ii) Structural/conformational and bio-chemical dynamics (operational mechanism driven by mechano-chemical cycles):

power-stroke or Brownian ratchet?

(iii) Effects of steric interactions on the spatio-temporal organization

Power-stroke versus Brownian ratchet

Joe Howard, Curr. Biol. 16, R517 (2006).

The operational mechanism of a real molecular motor may involve a combination of power stroke and Brownian ratchet

Brownian ratchetPower Stroke

Input energy drives the motor forward

Random Brownian force tends to move motor both forward and backward.

Input energy merely rectifies backward movements.

Mechanisms of energy transduction by molecular motors

A Brownian motor operates by converting random thermal energy of the surrounding medium into mechanical work!! Such systems are far from thermodynamic equilibrium and, therefore, do NOT violate second law of thermodynamics.

Simplest Model of Interacting Self-Driven Particles in 1-d

A particle moves forward, with probability q, iff the target site is empty.

q

Totally Asymmetric Simple Exclusion Process (TASEP)

Discretized position, discrete velocity (0 or 1) and discrete time

Steric interactions of the motors are often captured in the theoretical models by appropriate extensions of

We plot phase diagrams in planes spanned by exprimentally accessible parameters.

2. Brief overview of the motors of our current interest

4. Ribosome traffic on mRNA track

5. RNA polymerase traffic on DNA

1. Introduction

6. Summary and conclusion

Outline of the talk

3. Single-headed motor traffic on microtubule track

Brief overview of molecular motors of our

current interest

Cytoskeletal Molecular Motors: Cargo transport

Porters

Animated cartoon: MCRI, U.K.

Kinesin-1 on Microtubule

Myosin-V on F-actin

Ribbon diagram of the two heads of kinesin-1 (also called conventional kinesin)

Distribution of Step sizes of KIF1A Okada et al. Nature (2003)

(1) +Ve and –Ve steps sizes, i.e., both forward and backward steps.

(2) Step sizes are distributed around multiples of 8 nm

Not all kinesins have two-heads.

KIF1A kinesins are single-headed (“lame” porters);

These motors are physical realizations of Brownian ratchets

Okada and Hirokawa, PNAS (2000)

Experiments on a series of KIF1A mutants with different number of lysines in the K-loop and with E-hook digested microtubules

Molecular mechanism of processivity of KIF1A

Processivity depends on the K-loop; the larger the number of lysines, the higher is the processivity.

KIF1A becomes practically non-processive on E-hook-digested MT

Both +vely charged K-loop of KIF1A and the –vely charged E-hook of MT

are essential for the processive movement of KIF1A.

(Diffusive)

K KT KDP KD K

ATP P ADP

State 1 State 2

Strongly Attached to MT

Weakly Attached to MT

Brownian ratchet mechanism of movement of single KIF1A

In the weakly-attached state, because of the electrostatic attraction between E-hook of the microtubule and the K-loop of the kinesin, the motor remains tethered while executing Brownian motion along its track. This corresponds to the diffusive part of the dynamics of a Brownian ratchet.

KIF1A (Red) MT (Green)

10 pM

100 pM

1000pM

2 mM of ATP2 mNishinari, Okada, Schadschneider and Chowdhury, Phys. Rev. Lett. 95, 118101 (2005)

Greulich, Garai, Nishinari, Schadschneider, Chowdhury, Phys. Rev. E, 77, 041905 (2007)Chowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, 050902(R) (2008)

Many motors are moving simultaneously on the same track;

similarity with traffic

Model of interacting KIF1A on a single MT protofilament

Current occupation Occupation at next time step

b b

fd a

1 2 2 21 2 21

Greulich, Garai, Nishinari, Okada, Schadschneider, Chowdhury

KIF1A traffic on MT = TASEP for particles with “internal” states

+ Attachments & Detachments

MT track = 1-d lattice; motor-binding site on MT = lattice site

Greulich, Garai, Nishinari, Schadschneider, Chowdhury, Phys. Rev. E, 77, 041905 (2007)

Co-existence of high-density and low-density regions, separated by a fluctuating domain wall (or, shock): Molecular motor traffic jam !!

Position

Density

Low-density region High-density region

Chowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, 050902(R) (2008)

Non-trivial effects of lane changing on the flux of the KIF1A motors

Mean-field theory versus computer simulations

A new “probe-particle” method developed for locating the domain wall

MCAK, KLP10A and KLP59C :

members of kinesin-13 family

Kip3p:

a member of kinesin-8 family

SHREDDERS: walk/diffuse and depolymerize the track

www.nature.com/.../v7/n3/thumbs/ncb1222-F7.gifwww.nature.com/.../n9/thumbs/ncb0906-903-f1.jpg

Not all motors are cargo transporters

Govindan, Gopalakrishnan and Chowdhury, Europhys. Lett. 83, 40006 (2008)

Dependence of MT-length distribution on depolymerase concentration

Not all motors move on tracks made of filamentous proteins

Track

Filamentous Protein Nucleic Acid strand

DNA RNA

Example: DNA helicase that unzips a double-stranded DNA and translocates on one of the single strands.

Garai, Chowdhury and Betterton, Phys. Rev. E 77, 061910 (2008).

Microtubule F-actin

But, today I’ll talk about the “real engines of creation”, the motors which also polymerize the macromecules of life (e.g., RNA and proteins), from the respective templates which also serve as the corresponding tracks.

Motor traffic on Nucleic Acid Tracks

(RNA polymerase)

Translation

(Ribosome)

DNA

RNA

Protein

Transcription

Central dogma of Molecular Biology and assemblersSimultaneous Transcription and Translation in bacteria

Rob Phillips and Stephen R. Quake, Phys. Today, May 2006.

Many motors move on the same track; similarity with traffic

Initiation (Start), Elongation, Termination (Stop)

Three Stages of transcription / translation

initiation terminationWe model only elongation stage in detail.

OPEN boundary conditionsAv. speed of a ribosome = Av. speed of synthesis of a single protein

Flux = No. of ribosomes detected at the stop codon per unit time

= Total no. of proteins synthesized per unit time

RNAP/Ribosome traffic = TASEP for RODS with “internal states”

Ribosome traffic on mRNA track;

pause-and-translocation of ribosomes

A. Basu and D. Chowdhury, Phys. Rev. E 75, 021902 (2007)

A. Garai, D. Chowdhury and T.V. Ramakrishnan, Phys. Rev. E (under review) (2008)

q q

mRNA track = lattice; codon (triplet of nucleotides) = a lattice site.

Ribosome = a hard rod that covers L lattice sites; moves by one site.

Entire mechano-chemical cycle is captured by the single hopping parameter q.

MacDonald and Gibbs (1969); Lakatos and Chou (2003); Shaw, Zia and Lee (2003);

Shaw, Sethna and Lee (2004), Shaw, Kolomeisky and Lee (2004),

Dong, Schmittmann and Zia (2007)

TASEP-like models of ribosome traffic

L = 2

BUT,

a ribosome is not a “particle”;

it’s mechanical movement

is

coupled to its biochemical cycle

Ribosome: a mobile workshop

http://www.molgen.mpg.de/~ag_ribo/ag_franceschi/

mRNA Protein

decodes genetic message,

Ribosome

polymerizes protein using mRNA as a template.

A motor that moves along mRNA track,

http://www.mpasmb-hamburg.mpg.de/

The Ribosome

• The ribosome has two subunits: large and small

www.cancerquest.org

B Alberts et al Mol. Biol of the Cell

The small subunit binds with the mRNA track

The synthesis of protein takes place in the larger subunit

Processes in the two subunit are well coordinated

tRNA, an adapter molecule,

helps in the coordination of the operations of the two subunits

(Monomer of protein)

Correct codon-anticodon matching guarantees correct amino acid species

Codon = Triplet of nucleotides on mRNA

Amino acid

Anti-codon

Interacts with SMALLER s.u.

Interacts with LARGER s.u.

i - 1 i + 1i

mRNA track

Large subunit

Small subunit

Codon (Triplet of nucleotides) Ribosome

Three main stages in the mechano-chemical cycle of a ribosome

Cryo-electron microscopy: Frank et al. PNAS, 104, 19671 (2007).A toy model: Basu and Chowdhury, Amer. J. Phys. (2007)

For simplicity, I explain the process schematically assuming L = 1

Basu and Chowdhury, Amer. J. Phys. (2007)

A toy model of Ribosome Traffic on a mRNA template during protein synthesis

i - 1 i + 1i

a

mRNA track

Arrival of cognate tRNA

Basu and Chowdhury, Amer. J. Phys. (2007)

A toy model of Ribosome Traffic on a mRNA template during protein synthesis

i - 1 i + 1i

fl

mRNA track

Peptide bond forms and Larger s.u. moves forward

Basu and Chowdhury, Amer. J. Phys. (2007)

A toy model of Ribosome Traffic on a mRNA template during protein synthesis

i - 1 i + 1i

fs

mRNA track

Smaller s.u. pulled forward

Basu and Chowdhury, Amer. J. Phys. (2007)

A toy model of Ribosome Traffic on a mRNA template during protein synthesis

i - 1 i + 1i

a

fl

fs

mRNA track

Large subunit

Small subunit

Peptide bond forms and Larger s.u. moves forward

Smaller s.u. pulled forward

Arrival of cognate tRNA

But, a ribosome is not simply two pieces of rods connected by a spring

Three binding sites for tRNA: E, P, A

Two GTPases (engines which hydrolyze “fuel” molecules GTP) control movement of tRNA from one binding site to the next:

Elongation-factor (EF)-Tu and Elongation-factor (EF)-G

E P A

β

Theoretical model of ribosome traffic and protein synthesisA. Basu and D. Chowdhury, Phys. Rev. E 75, 021902 (2007)

Termination

Codon

(Triplet of nucleotides on mRNA track)

αInitiation

E P A E P A E P A

dP1(i;t)/dt = h2 P5(i-1;t) Q(i-1|i-1+l) + p P2(i;t) – a P1(i;t)

dP2(i;t)/dt = a P1(i;t) – [ p + h1] P2(i;t)

dP3(i;t)/dt = h1 P2(i;t) – k2 P3(i;t)

dP4(i;t)/dt = k2 P3(i;t) – g P4(i;t)

dP5(i;t)/dt = g P4(i;t) – h2 Q(i|i+l) P5(i;t)

Master eqn. for ribosome traffic for arbitrary l > 1Position of a ribosome indicated by that of the LEFTmost site.

P(i|j) = Conditional prob. that, given a ribosome at site i, there is another ribosome at site j = 1 - Q(i|j)

Steady-state solution with periodic boundary conditions

J = h2 P5 Q(i|i+l) = h2 P5 Q(1|1+l)

P5 = P/[1 + {h2 keff -1 (L-Nl )/(L+N-Nl -1)}]

Where keff -1 = g-1 + k2

-1 + h1-1 + a

-1 + p a-1 h1

-1

P(1|1+l) = Z(L-2l,N-2, l)/Z(L-l,N-1, l) = (N-1)/(L+N-Nl-1)

Where Z(L,N, l) = (N+L-Nl)!/[N! (L-Nl)!]

= No. of ways of arranging N ribosomes and N-Nl gaps.

J = {h2 (1-l)}/{(1+-l) + h2(1-l)},

Where h2 = h2/keff.

Effects of sequence inhomogeneity of real mRNA

Genes crr and cysK of E-coli (bacteria) K-12 strain MG1655

“Hungry codons” are the bottlenecks

Basu and Chowdhury, Phys. Rev. E 75, 021902 (2007)

LD: j is independent of HD: j is independent of MC: j is independent of both

q

Phase diag. for q=1 and RSU

Phase diagram of TASEP with Open B. C.

ribosome conc.

Pa

Using Extremum current principle (Popkov and Schutz, 1999)

Open Boundary Condition and Phase Diagrams

aa-tRNA conc.

Pa

aa-tRNA conc.

Ph GTP conc.

A novel way of creating high-density phase: reduce fuel supply to the motors!!

TASEP

= 1)

Wen,…, Noller, Bustamante and Tinoco Jr., Nature (March, 2008)

Manipulation of translation by a single ribosome

Dwell Time

Translocation Time

Comparison between theory and experiment

SIMUL.: Garai, Chowdhury and Ramakrishnan, PRE (under review) (2008)

EXPT.: Wen,…, Noller, Bustamante and Tinoco Jr., Nature (March, 2008)

Dwell TimeDwell Time

RNAP traffic on DNA and transcriptional bursts

Tripathi and Chowdhury, Phys. Rev. E, 77, 011921 (2008)

Tripathi and Chowdhury, Europhys. Lett. (in press) (2008)

RNA polymerase: a mobile workshop

DNA RNA

decodes genetic message,

RNA polymerase

polymerizes RNA using DNA as a template.

A motor that moves along DNA track,

Roger Kornberg

Nobel prize in Chemistry (2006)

T. Tripathi and D. Chowdhury, Phys. Rev. E 77, 011921 (2008)

Theoretical model of RNAP and RNA synthesis

Transcription-elongation complex (TEC)

= RNAP + DNA template + mRNA transcript

Mechano-chemistry of each RNAP + Steric interactions

RNAP + RNAn → RNAP + RNAn + NTP → RNAP.RNAn+1.PPi → RNAP + RNAn+1

RNAP traffic and rate of RNA synthesis

Flux= Total rate of RNA synthesis (No./second)

Periodic Boundary conditions Open Boundary conditions

Coverage density NTP (RNA subunit concentration)

Conclusions from single-cell experiments on transcription in-vivo:

Relatively long periods of transcriptional inactivity are interspersed with brief periods of transcriptional bursts.

Golding et al. Cell 123, 1025 (2005): prokaryotes (E-coli bacteria)

Chubb et al. Curr. Biol. 16, 1018 (2006): eukaryotes (amoeba Dictyostelium)

Raj et al. PLoS Biol. 4(10): e309 (2006): eukaryotes (chinese hamster ovary) Agents responsible (speculation):

In prokaryotes, unbinding and binding of transcription

repressor molecules

In eukaryotes, chromatin remodeling enzymes

Such a universal feature indicates a generic mechanism

Tripathi and Chowdhury, Europhys. Lett. (in press) (2008)

A Generic model: Transcriptional burst caused by gene switching

“ON”

“OFF”

A typical time series in our model

Sort the events into separate bursts: members of the same burst are separated from the immediate preceding and succeeding events by time gaps smaller than t while the time gap between any pair of successive bursts is at least t. Two choices: t = 0.5 min. and 2.5 min.

Experiment: Chubb, Trcek, Shenoy and Singer, Curr. Biol. 16, 1018 (2006)

Burst DURATION

Burst INTERVAL

Theory: Tripathi and Chowdhury, EPL (2008)

Burst INTERVAL

Burst DURATION

on exp(- on tdur)

off exp(- off tint)

Distr. Of burst duration and intervals depend only on the rates of switching

Burst Size

P(n)exp(- off /keff)] exp(-noff/keff)wherekeff = eff/l, and eff = 12 21

f/(12 + 21

f)

Burst-size Distribution

Burst-size distribution depends on the rate constants in the elongation cycle.

Summary and Conclusion

(1)We have developed models for template-dictated polymerization of macromolecules of life by incorporating

mechano-chemistry of individual machines + steric interactions

between the machines. These efforts go beyond the earlier works on single-machine modeling and models of “ribosome traffic” (TASEP for hard rods).

(2) We have not only calculated the average rate of polymerization and

average density profile, but also studied

transcriptional and translational noise.

Our models account for transcriptional “bursts” observed in single-cell experiments. These models go beyond the earlier models of noise in gene expression (at the single gene level) as the roles of the machinery are captured explicitly.

Thank You

Acknowledgements

Collaborators (Last 4 years):

On Ribosome: Aakash Basu*, Ashok Garai, T.V. Ramakrishnan (IITK/IISc/BHU).

On RNA Polymerase: Tripti Tripathi, Prasanjit Prakash.

On Helicase: Ashok Garai, Meredith D. Betterton (Phys., Colorado).

On Chromatin-remodeling enzymes: Ashok Garai, Jesrael Mani.

On KIF1A: Ashok Garai, Philip Greulich (Th. Phys., Univ. of Koln), Andreas Schadschneider (Th. Phys., Univ. of Koln), Katsuhiro Nishinari (Engg, Univ. of Tokyo), Yasushi Okada (Med., Univ. of Tokyo), Jian-Sheng Wang (Phys., NUS).

On MCAK & Kip3p: Manoj Gopalakrishnan (HRI), Bindu Govindan (HRI).

Funding: CSIR (India), MPI-PKS (Germany).

Now at Stanford University

Support: IITK-TIFR MoU, IITK-NUS MoU.

Shaw, Zia, Lee, PRE (2003)

Coverage density = N l/L

Main steps of ribosome in the mechano-chemical cycle in the elongation stage

tRNA selection

Peptide bond formation

translocation

Mechano-chemical cycle of ribosome during polypeptide elongation

Basu and Chowdhury, Phys. Rev. E 75, 021902 (2007)E P A

t-RNA t-RNA t-RNA-EF-Tu (GTP) t-RNA t-RNA-EF-Tu (GDP+P)

t-RNA t-RNA-EF-Tu (GDP) t-RNA t-RNA EF-G (GTP)t-RNA t-RNA

t-RNA t-RNA

i

i+1

t-RNA

i - 1 i + 1i

1

2

3

4

5

1

2

3

4

5 5

4

3

2

1

pa

h1

g

h2

k2

Naturally discretized positions of a ribosome:

separation between successive codons (triplets of nucleotides)

Steady-state flux with periodic boundary conditions: mean-field theory versus computer simulations

Basu and Chowdhury, Phys. Rev. E 75, 021902 (2007)

Flux = Total rate of protein synthesis

Number density

l = 3

l = 12

Flux of ribosomes = Total rate of protein synthesis (No./second)

-

+

= -

= 1--

Open Boundary Condition and Phase Diagrams

Imagine that the left and right ends of the system are connected to two reservoirs with appropriate number densities ρ- and ρ+ ,

respectively, so that, assuming the same jumping rates as in the bulk, effects of and can be incorporated

Popkov and Schutz, Europhys. Lett. 48, 257 (1999)

Antal and Schutz, Phys. Rev. E 62, 83 (2000)

Popkov and Peschel, Phys. Rev. E 64, 026126 (2001)

Technique:

where

Pjump = probability that, given a set of l empty sites, a ribosome jumps onto it in the next time step.

We now identify Pjump as the parameter α.

Evaluation of ρ- and ρ+

i.e.,

+ = 0

and

Phase diagram of the open system (for = 1, i.e., += 0) in the ribosome conc. – aminoacyl-tRNA conc. plane

The Phase boundary is the solution to:ρ-(α,ωa,ωh1, ωh2) = ρ*(α,ωa,ωh1, ωh2)

PaJ

Extremum principle (Popkov and Schutz, 1999): j = max J() if - > *

Ph

Pa

Phase diagram of the open system (for = 1, i.e., += 0) in the GTP conc. – aminoacyl-tRNA conc. plane

Extremum principle (Popkov and Schutz, 1999): j = max J() if - > *

The Phase boundary is the solution to:ρ-(α,ωa,ωh1, ωh2) = ρ*(α,ωa,ωh1, ωh2)

h

J

Effect of sequence-inhomogeneity on translational noise

Garai, Chowdhury and Ramakrishnan (2008)Homogeneous sequence Inhomogeneous sequence

Time Headway Time Headway

Time series of translational events Time series of translational events

i i+1i-1

22

1

2 2

1 1

bb2222

bb1111

ff2222

ff1111

ff2121

bb1212

2121

1212

Discrete state space of individual RNAP and the transitions

Tripathi and Chowdhury, Phys. Rev. E, 77, 011921 (2008);

adapted from Wang, Elston, Mogilner, Oster (1998)

NO PPi is bound to the RNAP catalytic site

PPi is bound to the RNAP catalytic site

12 Release of PPi (the rate-limiting step)

Dominant pathway in each cycle of individual RNAP

21 = 021 [PPi]f

21 = f021 [NTP]

i i+1i-1

22

1

2 2

1 1

PPi is bound to the RNAP catalytic site

NO PPi is bound to the RNAP catalytic site

12 Release of PPi

Tripathi and Chowdhury, Phys. Rev. E, 77, 011921 (2008);

adapted from Wang, Elston, Mogilner, Oster (1998)

= a , 1,2 = d,

1,2 = = 0

Phase-diagram in the h- a plane

Blue = 1,

Red = 2

Very low h: almost all in state 1 and Homogeneously distributed

a

h

Nishinari, Okada, Schadschneider and Chowdhury, Phys. Rev. Lett. 95, 118101 (2005)

Binding site on MT track

ii-1 i+1

h

s

1 11

2 2 2bb

f

a

d

1,2 Two “chemical” states

Discrete “State-space” of a single KIF1A and the transitions

Spatial position

Chemical state

Master eqns. for KIF1A traffic in mean-field approximation:

continuous time

dSi(t)/dt = a(1-Si-Wi) + f Wi-1(1-Si-Wi) + s Wi – h Si – d Si

dWi(t)/dt = h Si + b Wi-1 (1-Si-Wi) + b Wi+1 (1-Si-Wi)

- b Wi {(1-Si+1-Wi+1) + (1-Si-1-Wi-1)}

– s Wi – f Wi(1-Si+1-Wi+1)

Si = Probability of finding a motor in the Strongly-bound state.

Wi = Probability of finding a motor in the Weakly-bound state.

i = 1,2,…,L

GAIN terms LOSS terms

Validation of the model of interacting KIF1A

Excellent agreement with qualitative trends and quantitative data

obtained from single-molecule experiments.

Low-density limit

Nishinari, Okada, Schadschneider and Chowdhury, Phys. Rev. Lett. 95, 118101 (2005)

ATP(mM)ATP(mM)

∞∞0.90.9

0.33750.3375

0.150.15

X

Y

W(x,y) → W(x,y+1) with bl+

W(x,y) → W(x,y-1) with bl-

W(x,y) → S(x,y+1) with fl+

W(x,y) → S(x,y-1) with fl-

Lane-changing by single-headed kinesin KIF1A motorsChowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, 050902(R) (2008)

Lane = Protofilament

Lane-change allowed from weakly-bound state

Discrete State-Space of a KIF1A motor and mechano-chemical transitions (including lane-changing)

Chowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, 050902 (R) (2008)

Master equations in the mean-field approximation

dSi(j,t)/dt = a[1-Si(j,t)-Wi(j,t)] + f Wi-1(j,t)[1-Si(j,t)-Wi(j,t)] + s Wi(j,t) – h Si(j,t) – d Si(j,t)

+ fl+[Wi(j-1,t)][1-Si(j,t)-Wi(j,t)] + fl-[Wi(j+1,t)][1-Si(j,t)-Wi(j,t)]

dWi(t)/dt = h Si (j,t) + b [Wi-1(j,t) + Wi+1(j,t)] [1-Si(j,t)-Wi(j,t)]

- b Wi [2-Si+1(j,t)-Wi+1(j,t) -Si-1(j,t)-Wi-1(j,t)] – s Wi(j,t) – f Wi(j,t)[1-Si+1(j,t)-Wi+1(j,t)]

+ bl[Wi(j-1,t) + Wi(j+1,t)] [1-Si(j,t)-Wi(j,t)]

bl Wi(j,t)[2-Si(j+1,t)-Wi(j+1,t)-Si(j-1,t)-Wi(j-1,t)]

fl+Wi(j,t)[1-Si(j+1,t)-Wi(j+1,t)]

- fl- Wi(j,t)[1-Si(j-1,t)-Wi(j-1,t)]i = 1,2,…,N; j = 1,2,…,13

Si(j,t) = Probability of finding a motor in the Strongly-bound state at site i on the protofilament j.

Chowdhury, Garai and Wang (2008)

flf

Flux

(per lane)

flf

Density

Non-monotonic variation with frequency of lane-changing!!

New prediction:

Flux can increase or decrease depending on the rate of fuel consumption.