Physics of nano-motors: from cargo transport to gene expression Debashish Chowdhury Physics Department, Indian Institute of Technology, Kanpur Home page:

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 ratchet Power 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) KKTKDPKDK ATPPADP State 1State 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 ATP 2  m Nishinari, Okada, Schadschneider and Chowdhury, Phys. Rev. Lett. 95, (2005) Greulich, Garai, Nishinari, Schadschneider, Chowdhury, Phys. Rev. E, 77, (2007) Chowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, (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 bb ff dd aa 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, (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 regionHigh-density region Chowdhury, Garai and Wang, Phys. Rev. E (Rapid Commun.), 77, (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 Not all motors are cargo transporters Govindan, Gopalakrishnan and Chowdhury, Europhys. Lett. 83, (2008) Dependence of MT-length distribution on depolymerase concentration

Not all motors move on tracks made of filamentous proteins Track Filamentous ProteinNucleic Acid strand DNARNA 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, (2008). MicrotubuleF-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 assemblers Simultaneous Transcription and Translation in bacteria Rob Phillips and Stephen R. Quake, Phys. Today, May Many motors move on the same track; similarity with traffic

Initiation (Start),Elongation,Termination (Stop) Three Stages of transcription / translation  initiationtermination We model only elongation stage in detail. OPEN boundary conditions Av. 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, (2007) A. Garai, D. Chowdhury and T.V. Ramakrishnan, Phys. Rev. E (under review) (2008)

qq 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 mRNAProtein decodes genetic message, Ribosome polymerizes protein using mRNA as a template. A motor that moves along mRNA track,

The Ribosome The ribosome has two subunits: large and small 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 - 1i + 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, (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 - 1i + 1i a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 - 1i + 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 - 1i + 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 - 1i + 1i a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 EPA

β Theoretical model of ribosome traffic and protein synthesis A. Basu and D. Chowdhury, Phys. Rev. E 75, (2007) Termination Codon (Triplet of nucleotides on mRNA track) α Initiation EPAEPAEPA

dP 1 (i;t)/dt =  h2 P 5 (i-1;t) Q(i-1|i-1+l) +  p P 2 (i;t) –  a P 1 (i;t) dP 2 (i;t)/dt =  a P 1 (i;t) – [  p +  h1 ] P 2 (i;t) dP 3 (i;t)/dt =  h1 P 2 (i;t) – k 2 P 3 (i;t) dP 4 (i;t)/dt = k 2 P 3 (i;t) –  g P 4 (i;t) dP 5 (i;t)/dt =  g P 4 (i;t) –  h2 Q(i|i+ l ) P 5 (i;t) Master eqn. for ribosome traffic for arbitrary l > 1 Position 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 P 5 Q(i|i+l) =  h2 P 5 Q(1|1+l) P 5 = P/[1 + {  h2 k eff -1 (L-Nl )/(L+N-Nl -1)}] Where k eff -1 =  g -1 + k  h  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 /k eff.

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, (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.  PaPa Using Extremum current principle (Popkov and Schutz, 1999) Open Boundary Condition and Phase Diagrams aa-tRNA conc. PaPa PhPh 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 Time

RNAP traffic on DNA and transcriptional bursts Tripathi and Chowdhury, Phys. Rev. E, 77, (2008) Tripathi and Chowdhury, Europhys. Lett. (in press) (2008)

RNA polymerase: a mobile workshop DNARNA 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, (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 + RNA n → RNAP + RNA n + NTP → RNAP.RNA n+1.PPi → RNAP + RNA n+1

RNAP traffic and rate of RNA synthesis Flux= Total rate of RNA synthesis (No./second) Periodic Boundary conditions Open Boundary conditions Coverage densityNTP (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 D 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 t dur )  off exp(-  off t int ) Distr. Of burst duration and intervals depend only on the rates of switching

Burst Size P(n)  exp(-  off /k eff )] exp(-n  off /k eff ) where k eff =  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, (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 i i+1 t-RNA

i - 1i + 1i pp aa  h1 gg  h2 k2k2 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, (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, (2001) Technique:

where P jump = probability that, given a set of l empty sites, a ribosome jumps onto it in the next time step. We now identify P jump 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,ω a,ω h1, ω h2 ) = ρ * (α,ω a,ω h1, ω h2 )  PaPa J Extremum principle (Popkov and Schutz, 1999): j = max J(  ) if  - >  *

PhPh PaPa 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,ω a,ω h1, ω h2 ) = ρ * (α,ω a,ω h1, ω h2 ) hh J

Effect of sequence-inhomogeneity on translational noise Garai, Chowdhury and Ramakrishnan (2008) Homogeneous sequence Inhomogeneous sequence Time Headway Time series of translational events

ii+1i  b 22  b 11  f 22  f 11  f 21  b 12  21  12 Discrete state space of individual RNAP and the transitions Tripathi and Chowdhury, Phys. Rev. E, 77, (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 =  0 21 [PP i ]  f 21 =  f0 21 [NTP] ii+1i 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, (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a hh Nishinari, Okada, Schadschneider and Chowdhury, Phys. Rev. Lett. 95, (2005)

Binding site on MT track ii-1i+1 hh ss bb bb ff aa d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 dS i (t)/dt =  a (1-S i -W i ) +  f W i-1 (1-S i -W i ) +  s W i –  h S i –  d S i dW i (t)/dt =  h S i +  b W i-1 (1-S i -W i ) +  b W i+1 (1-S i -W i ) -  b W i {(1-S i+1 -W i+1 ) + (1-S i-1 -W i-1 )} –  s W i –  f W i (1-S i+1 -W i+1 ) S i = Probability of finding a motor in the Strongly-bound state. W i = Probability of finding a motor in the Weakly-bound state. i = 1,2,…,L GAIN termsLOSS 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, (2005)ATP(mM)∞

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

Master equations in the mean-field approximation dS i (j,t)/dt =  a [1-S i (j,t)-W i (j,t)] +  f W i-1 (j,t)[1-S i (j,t)-W i (j,t)] +  s W i (j,t) –  h S i (j,t) –  d S i (j,t) +  fl+ [W i (j-1,t)][1-S i (j,t)-W i (j,t)] +  fl- [W i (j+1,t)][1-S i (j,t)-W i (j,t)] dW i (t)/dt =  h S i (j,t) +  b [W i-1 (j,t) + W i+1 (j,t)] [1-S i (j,t)-W i (j,t)] -  b W i [2-S i+1 (j,t)-W i+1 (j,t) -S i-1 (j,t)-W i-1 (j,t)] –  s W i (j,t) –  f W i (j,t)[1- S i+1 (j,t)-W i+1 (j,t)] +  bl [W i (j-1,t) + W i (j+1,t)] [1-S i (j,t)-W i (j,t)]  bl W i (j,t)[2-S i (j+1,t)-W i (j+1,t)-S i (j-1,t)-W i (j-1,t)]  fl+ W i (j,t)[1-S i (j+1,t)-W i (j+1,t)] -  fl- W i (j,t)[1-S i (j-1,t)-W i (j-1,t)] i = 1,2,…,N; j = 1,2,…,13 S i (j,t) = Probability of finding a motor in the Strongly-bound state at site i on the protofilament j.

Chowdhury, Garai and Wang (2008)  fl  f Flux (per lane)  fl  f Density Non-monotonic variation with frequency of lane-changing!! New prediction: Flux can increase or decrease depending on the rate of fuel consumption.