Molecular Motors
Energy Sources: Protonmotive Force, ATP
Single Molecule Biophysical Techniques : Optical Tweezers, Atomic Force Microscopy, Single Molecule Fluorescence Microscopy
Physical Models : Brownian Ratchets, Smolunchowski Equation Molecular Motors
Molecular motors are proteins that are able to convert energy stored in ATP or ion gradients into mechanical work
• Cytoskeletal motor proteins: myosin, kinesin, dynein
• Polymerization motors: actin, microtubules, RecA
• Ion pumps: Na-K pump
• Rotary motors: ATP-synthase, bacterial flagellar motor
• DNA motors: RNA polymerase, helicases Molecular Motors Functions
Bacterial chemotaxis (flagellar motor)
Energy storage in the form of ATP (ATP synthase motor) Motor Protein Structures Molecular Motors in action
“The Inner Life of a Cell” by Cellular Visions and Harvard Molecular Motors
Energy Sources: Protonmotive Force, ATP
Single Molecule Biophysical Techniques : Optical Tweezers, Atomic Force Microscopy, Single Molecule Fluorescence Microscopy
Physical Models : Brownian Ratchets, Smolunchowski Equation Energy Flow In Living Systems
Life processes including:
Reproduction Food NADH e.g. glucose Growth
Photons pmf ATP Transport
Movement Protonmotive Force
Life processes including: Reproduction Food NADH e.g. Growth glucose pmf Photons ATP Transport
Movement ⎛ ⎞ Δµ kBT ci pmf = Vm + = Vm + ln⎜ ⎟ e e ⎝ co ⎠ pmf is the extra free energy per unit charge inside the cell versus outside - Sign convention inside minus outside - Units of volts
- Vm is the electrical potential, Δμ is the chemical potential - Typically pmf is in the range -150 mV to -200 mV - Usually both components negative : inside has lower voltage and lower [H+] - Powers FLAGELLAR MOTOR and ATP SYNTHASE ATP
Life processes including:
Reproduction Food NADH e.g. glucose Growth
Photons pmf ATP Transport
Movement
Hydration free energy is larger for the products. In part due to entropy, in part due to better hydrogen bonding and electrostatic screening.
Charge repulsion between phosphate groups
Resonance : partially delocalized electrons free to move between oxygens. The shared Hydrolysis of ATP : ΔG = -20 kBT to -30 kBT oxygen bond reduces the number of arrangements of electrons (entropy) compared to hydrolysis products Molecular Motors
Energy Sources: Protonmotive Force, ATP
Single Molecule Biophysical Techniques : Optical Tweezers, Atomic Force Microscopy, Single Molecule Fluorescence Microscopy
Physical Models : Brownian Ratchets, Smolunchowski Equation Optical Tweezers
➢ 10s – 100s pN ➢ Can be used to control beads specifically attached to proteins and cells ➢ Infrared light (low absorbance, cheap lasers) ➢ Acousto-optic deflectors enable beam ➢ Multiple traps made possible by time sharing ➢ Micro-Tetris with 1um glass beads Single Molecule Fluorescence Imaging : FRET
FRET a molecular ruler: Electronic excitation energy can be transferred between two chromophores in close proximity
Key Use : Protein-Protein interactions & Protein conformation changes at high spatial precision < 10 nm
Practical Limitations : Control of chromophore concentrations, Photobleaching, Signal Separation and Detection Fernandez et al Cell surface distribution of lectin receptors determined by resonance energy transfer. Nature 264, 411–415 (1976) Molecular Motors
Energy Sources: Protonmotive Force, ATP
Single Molecule Biophysical Techniques : Optical Tweezers, Atomic Force Microscopy, Single Molecule Fluorescence Microscopy
Physical Models : Brownian Ratchets, Smolunchowski Equation • Consider an idealised nanoscale device Thermal Ratchet operating in a thermal environment subject to random fluctuations : shaft with series of bevelled bolts mounted on springs • Thermal push > ε + fL will push shaft to right • Prevented from moving to left by tapered sliding bolts • Violates 2nd law (loop version) • If the energy for bolt retraction ε comparable to kT the machine can step to the left Thermal Ratchet with stored potential energy
• Consider a modification in which a latch keeps bolt down when it’s to left of wall but releases the bolt on right side • Movement to right doesn’t violate 2nd Law (Why?) • Rectified Brownian Motion • Stall force f = ε/L
Net motion ➔ Net motion ç Rectified Brownian Motion
This rectified Brownian ratchet is equivalent to a random walk with absorbing and reflecting boundary conditions.
The ratchet diffuses back and forth until it happens to move a distance L to the right, when the next spring engages. At this time, the ratchet starts at position x = 0.
If the free energy drop is very high, the ratchet cannot move to the left i.e. leftward flux is not allowed, so the LH boundary condition (at x = 0) is perfectly reflecting. If the free energy drop ε is finite, there is a probability of moving leftward, but it’s small if ε/kBT is large
If the ratchet ever reaches position x = L, the next spring is triggered and if the free energy drop ε is very high the position x = L acts like a perfectly absorbing boundary. If the free energy drop ε is finite, the walker simply falls off a very large cliff and is unlikely to be able to return.
This type of problem involves calculating the time it takes for a random walker to reach a specified target and is called a first passage time problem 26 334
displacements due to Brownian motion, or currents33t4hat do not vanish at equilibrium. In order to derive a numerical method of simulating random motions that does not have these problems, we have to consider an alternadistpivlaceemdeesntcripts due itoonBrofowntheian mmoolection,ularor currents that do not vanish at equilibrium. motion. In order to derive a numerical method of simulating random motions that does not have these problems, we have to consider an alternative description of the molecular motion. 12.2.3 The Smoluchowski model 12.2.3 The Smoluchowski model Consider a protein moving under the inuence of a constant external force, for example, Consider a protein moving under the inuence of a constant external force, for example, an electric eld. Because of Brownian motion, no tawnoeletcrtarijcecteoldr.ieBsecwauislel olof oBkrowthnieansammoteio.n, no two trajectories will look the same. 334 Moreover, even a detailed examination of the pathMcoarneonvoert, edviesntiangdeutiasihledwehxaemthineartioanpoaf rth-e path cannot distinguish whether a par- ticular displacement :step< was caused by a Browniaticunlarduicsptluaacemtioennt o:rstetph -$ ( Drift velocity ^ PBoltzmann distributionU ^ (, +e/ ) ( ) ! ^ ( ) ^ ( + , Sin)ce the n[1]umber of particles in the swarm remains constant, ( ) must obey a x ! ! , ! ! conse!rvation law. This is simply a balance on a small volume element, : At equilibrium the ux vanishes: (, e/ )(e/)+ ^ 0. Integrating with Diusion ux Drift ux respect to , shows that the concentration of particles at equilibrium in an external eld, () ^ Net Flux into ^ x(in) x(out) ^ x() x( + ) • At equilibrium the flux vanishes : Jx = 0. Integrating with ^ (, + ) (), is given by the Boltzmann distri4uti!on : ! ! respect to x gives the equilibrium concentration of particles in At equilibrium the ux vanishes: (, -$e/)(e/)+ ^ 0. Integrating with resexternal field pect to , shows thφa(x) : (Boltzmann Distribution)t the coen/ce^ntration of particles aPBt eoqlutizlimbriaunmnindainsterxitberunatlionelUd, (), is given by the Boltzmann distri4ution: Since the number of particles in the swarm remains constant, ( ) must obey a -$ ( 27 conservation law.e/T^hisis sim plPyBoaltzbmaalnanndciestroibnutaionsUmall volumeelem ent, : Since the number of particles in the swarm remains constant, ( ) must obey a conservation law. This is simply a balance on a small volume element, : () ^ Net Flux into ^ x(in) x(out) ^ x() x( + ) ! ! () ^ Net Flux into ^ (in) (out) ^ () ( + ) x ! x x ! x Smoluchowski Equation 335 or, taking the limit as 0: & • x 335 Reframe the problem in terms of the ^ Pconservation of particlesU probability of finding a single particle at (x,t). ! oRr,atthaekrintghatnhefolicmusistinags our att0e:ntion on the swarm of particles, we can rephrase • Normalize the concentration in the previous our discussion in termsof the&pro4a4ilityofnding a single particle at ( ). To do 335 this we normalize the concentrationinx (2.32) by dividing by the total population, 0 ^ Pconservation of particlesU ( ) ( )( ( )). Inserting (2.32) expressed in terms of ( ) into the slide : or, taking[2]th!e limit as 0: conserv"ation law ((2.34)) yields the Smoluchow&ski equation: Rather than focussing our attention on thex swarm of particles, we can rephrase ^ Pconservation of particlesU • Conservation of particles implies : [3] our discussion in t(erms, of) the pro4a4ility o!f nding a single particle at ( ). To do ^ P + U PSmoluchowski EquationU thisw e normalize theRcaothnecrenthtarantfioocnusisning(o2u.r32at)tebnytiodnivoniditnheg sbwyarmtheof tpoatratliclpeso,pwuelactainonr,ephrase • [1],[2] & [3] Combining gives the Smoluchowski 0our discussion in terms of the pro4a4ility of nding a single particle at ( ). To do ( ) ( )( D rif(t )).DInisuesrtiiong (2.32) expressed in terms of ( ) into the " this we normalize the concentration in (2.32) by dividing by the total population, equation. At steady state LHS is 0 (not necessarily conservation law ((2.34)) yields th0e Smoluchowski equation: Comparing this with the(La)ngev(in )eq(uatio(n )2.28). Inshseortiwsngtha(2t.3t2h)eexBroprewssnedianin forctermes iosf ( ) into the " conservation law ((2.34)) yields the Smoluchowski equation: equilibrium)replaced bythe diusion term(an,dth)e eectofthe deterministic forcing is captured by ^ P + U PSmoluchowski EquationU the drift term. (, ) We can nondimensionalize (^2.35P) by dening tim+e and spacUe sPcSamleosl.ucIhf otwhsekdi oEmquaaitnionU Drift Diusion 0 , the spatial variable can be normalizedas . A time scale can be de- ' ' Drift Diusion28 neCdobmypar^ingthis. IwnittrhodtuhceinLgatnhgeesvpinaceqaunadtitoinme2.28scaless,h(ow2.35s tha) ctanthbeeBrowrittenwnianin force is Comparing this with the Langevin equation 2.28 shows that the Brownian force is dimreenpsliaocneledssbyfotrhmeadsiusion term and the eect of the deterministic forcing is captured by replaced by the diusion term and the eect of the deterministic forcing is captured by the drift term. the drift term. ^ + We can nondimensWioencaalnizneo(nd2.35imen)sibonyadlizee(ni2.35ng t)ibmyedeanndinsgptaimcee sacnadlesps.acIef stchaelesd.oImf tahiendomain 0 , the sp0atial varia, bthlee scpaantiablevanroiarbmleacliaznedbeansormali.zeAd atsimes.cAaletimcae nscable cdaen- be de- where where and are n'ow 'dimensionless, and the potential, , is measured in units n'ed by' ^ .nIendtrboydu^cingth.eInsptraocdeucainngdthteimspeacsecaalneds,ti(m2.35e sca)lesc,an(2.35be )wcranittenbe winritten in of , . dimensionless form as d(im2.3en6s)iomnusletssbefoarumgmeasnted by appropriate boundary conditions specifying the value of ( ^ 0 ) ( ^ ), and ( ^0), where ( )^isdened o+n theinterval P0 U. ^ + These will depend on the system being modeled. where where and arenowdimensionless, and the potential, , is measured in units where where and of a,re .now dimensionless, and the potential, , is measured in units (2.36) must be augmented by appropriate boundary conditions specifying the value 12.2.4of , .First passage time of ( ^ 0 ) ( ^ ), and ( ^ 0), where ( ) is dened on the interval P0 U. (2.36) must beTahuegsemewinlltdedepbenydaponprothepsriateystembboeuingdamroydceloneddition. s specifying the value A voefry(use^fu0lq)uan(tity^inm)o, daenlidng(prote^in0)m, owtihoenrse is(the)aivsedraegenetidmoenitthteakinestefrovral P0 U. a dTihuessinegwpilrlodteeipnentod ornstthreeascyhstaenmabseoinrbginmgobdoeulendd.ary located at ^ , starting from position 0 12PB.2.4erg, 9First93, Wpeiassass, g96e7Ut.imeDenote the mean rst passage time (MFPT) to pos'ition' starting from position by ( ). The equation governing ( ) is derived as foAllovwersy. uAsefpual rqtuicalnetirteyleinasmedodaetlinpgospirtoioteninmcaotinondsiisutshee eaivthereargetotime it takes for the12rig.2.4ht or toFirstthe left.paassaAfditerusgianegtimeptriometein, itto corvsetrrseaacnh aavneraabgseordbinstganbocuenda,rysolotchaatetditatis ^ , starting located at with efrqoumalpporsiotbioanb0ility2. PTBheergM, F99P3T, Wtoeiss, from967U.theDennoteewthpeomsitionean srst passage time A very useful quan(tMitFyPTin) tmoopdoes'litiniogn'prosttaeritningmforotmionpsosiistiotnhe abvyera(get).imTeheiteqtuaaktieosn fgoorverning are ( + ' ) and ( ). The average value of ( ) is just ( ) ^ a diusing proteint(or)stis rdeearcivhedanas afoblsloowrsb.inAgpbarotuicnledarerlyeasleodcaattedposaittion ^ can, dstiaursteinegither to + (2)P ( + ) + (! )U. This equation can be re-written in the form: from position 0 the rig!hPBt eorrgto, th99e3l,efWt. eAfisters, a9time67U. D,eint octoevetrhs eanmaevaenragersdtisptaasncsae ge, timeso that it is (MFPT) to(po(s'it+iolno'cate)dstaatrt(in)g)frwo(imth(epq)ousailtipor(nobabibliyty))(2.Th)2e.MTFhPeTeqtouatifromon gtheovenrenwinpgositions are (! + ' ) and (! !). The ave+rage va^lu0e of ( ) is just ( ) ^ ( ) is derived as follows. A particle released at position can diuse either to +(2)P ( +! ) + (! )U. This equation can be re-written in the form: the right or to the left. After a time , it c!overs an average distance , so that it is ( ( + ) ()) ( () ( )) 2 located at with equal probability 2. The MFPT to from the n+ew p^os0itions ! ! ! are ( + ' ) and ( ). Theaverage v!alue of ( ) is just ( ) ^ + (2)P ( + ) + (! )U. This equation can be re-written in the form: ! ( ( + ) ()) ( () ( )) 2 ! ! ! + ^ 0 ! Molecular Ratchet : Smoluchowski Equation • Consider circular ratchet with bolts reset at (x = x+4L). Look at motion of ensemble M. • To find the speed of the ratchet we need to find the flux of particles Jx from which we can calculate the average time it takes for the ratchet to cross a ‘step’ in the potential energy landscape. The speed is the step length divided by this time. • The steady state probability distribution p(x) for the ‘perfect ratchet’ case where ε>KBT Speed of the Ratchet [Question Sheet Problem] • Verify P(x) = C(be-(x-L)F/kBT - 1) solves Smoluchowski equation for a constant force F. You can assume steady state. • Find the current flux : Expression in terms of p sub equation [2] in [1] • Show average speed of the loaded ratchet perfect ratchet is 2 1 2D FL D FL/kB T v = v = e 1 F L/KBT kBT L L ✓ ◆ ⇣ ⌘ F =0 F < 0 and FL/kT >>1 Forward stepping contains exponential activation barrier Polymerization Ratchet • Addition of monomers generates a force • When filament reaches the barrier (e.g. cell wall) fluctuations in position of cell wall or filament will allow another monomer to squeeze through and bind Polymerization Ratchet contd • How do we find the velocity (filament polymerization rate)? Driven diffusion equation • Want to find p(x) of finding a particle at x in width δ • Force p(x) to be 1 in (0,δ) interval • Mean rate at which particle reaches boundary at x = δ starting at x = 0 is ζ • We want to solve for p(x) in terms of j0 Velocity of Polymerization Ratchet The solution is the sum of the general solution of homogeneous Equation (absence of force) and a particular solution to inhomogeneous equation: ζ ζ ζ Low Force: Same as no force limit – diffusion – first passage problem High Force: Probability particle will find itself with energy Fδ Motion of Motor Proteins Kinesin walking along microtubules Organelles being carried along by carrying a lipid vesicle as cargo kinesin and dyenin motors How Proteins Work : Garland Science Example : Kinesin • Two headed molecular motor which walks along hollow protein structures called microtubules • Tightly coupled (one spatial step for each ATP molecule consumed) • Highly processive (many steps before detachment) • Nearly 100% duty cycle A tightly-coupled molecular motor with at least one irreversible step in kinetics should move at a speed according to MM kinetics (with parameters dependent on the load force) Models of Specific Motor Proteins • To build a model of motor movement need to consider : • (1) Range of possible conformational states of the motor • (2) Alterations to the energy levels of the conformational states by the biochemical reaction which powers the motor • (3) Ways in which motor can change from one conformational state to another 7 One-state & Two state models • Explore a class of problems where position and state are coupled and state transitions are driven by biochemical reactions • Calculate average speed v and diffusion D parameters in of the Smoluchowski equation • Incorporate details of internal states and transitions between then (two-state model). • Dependence of the transition rates on the applied force explicit - assumed to be in the -ve direction • Energy consumption factored in by having k- not equal to k+ even with zero load 8 Motor Stepping Real free energy isn’t flat between steps but more like this: In principle we could solve the Smoluchowski eqn for this landscape; in practice we don’t know its details Rate limiting step is diffusion over a free energy barrier L Transition rates can be related to barrier height using Arrhenius relation One-state ratchets Cartoon picture of e.g kinesin moving on a microtubule, 4 nm periodic protein structure ➢ Linear motors move along regular, periodic tracks ➢ One-state model : forward stepping with force dependent rate constant k+ backward stepping with rate constant k- all bound motors equivalent ➢ Free energy difference between steps: V([ATP]) predictions Two parameters : forwards and backwards stepping rate constants related by Thermodynamic arguments Either can change with [ATP] but v([ATP]) different depending on which one All dependence in backward rate Speed saturates with [ATP] All dependence in forward rate Speed is linear with [ATP] Kinesin : v([ATP]) • Kinesin data not consistent with pure k- or k+ modulation • Kinesin not a simple one state ratchet Saturates at high [ATP] Linear at low [ATP] One-state ratchets How do we know that cleverer forms for k–([ATP]) and k+([ATP]) wouldn’t fit the data for v(f, [ATP])? Dwell time distributions If transitions between states are controlled by activation barriers, the waiting time for each transition will be exponentially distributed (with mean time " = 1/k) One-state and two-state waiting time distributions look fundamentally different: t/ For transitions controlled by activation For a one-state ratchet we expect a dwell time distribution p(t) e barriers waiting time for transition Exponentially distributed with mean time A two-state ratchet moves when two sequential transitions (each a simple t/ t/ exponential) have occurred. The total dwell time distribution should look like p(t) e 1 e 2 ⇥ rel. number of event Can distinguish between one-state 1.0 0.8 0.6 and two-state waiting time distributions 0.4 0.2 time 0 2 4 6 8 10 Experimental Evidence <- Step data from individual kinesin molecules stepping in microtubules From which a distribution of Waiting times can be obtained | V Kinesin has a bi-exponential dwell time distribution: it’s at least a two-state ratchet Myosin V also at least two-state (not shown here) One-State Model Assumptions Two-state ratchets Most motor• Most motor free energy landscapes cannot be reduced to a single energy landscapes cannot be reduced to a single (distance) reaction coordinate.distance reaction coordinate We oversimplified• Single transition state assumption not accurate by assuming there was a single transition state: WRONG. Work and burning of ATP Work (motion against f) and burning of ATP occur concurrently transition statetransition state simultaneously one state modelone-state model simplified picture one-state cartoon Additional States • More detailed model with intermediate states required Two-state ratchets • One more reaction coordinate Make a more detailed model with intermediate state(s) • Decouple ATP hydrolysis from physical movement 856 CHAPTER 16. DYNAMICS OF MOLECULAR MOTORS • Complete decoupling ATP hydrolysis insensitive to load f, physical movement This adds (at least) another reaction coordinate This allows ATP burning (potentially 3 steps in itself) to be decoupled from physical movement. insensitive If theyto [ATP] are completely decoupled, the former will be insensitive to f and the latter will be insensitive to [ATP]. (A) + + kB kA ATP – – Δ x kB kA binding step 1 0 1 transitiontransition state state(s) n–1 n n+1 (B) + + kA kB – – kA kB ADP 0 1 0 transitiontransition state state(s) Release ΔG step n–1 n n+1 Figure 16.32: Two-state motor model. (A) The rates for the transitions that can occur to change the occupancy of internal state 0. The dark icon indicates multi intermediate states modelmulti-state model the current state of thesimplified picture two-statemotor head and cartoonthe two light icons indicate the two possible states in the next time step. In the two-state model, the motor head is constrained to convert from internal state 0 to internal state 1. This can occur either while the motor remains stationary with respect to the filament (with rate constant kA) or while the motor takes a single step backwards (with rate constant kB) (B) The rates for the transitions that can occur to change the occupancy of internal state 1. The motor head can convert to internal state 0, either while remaining in place, or while taking a single step forward. Two State Motor Models ⇒ Consider motor as two state system with states 0,1 ⇒ Find p0(n,t) and p1(n,t) protein being in state 0 on subunit n at time t (similarly state 1) ⇒ k values give rates of transitions to different internal or position states Rate equations: ⇒ Can reduce this to probability of being KA’s change state and n in state 0 or 1 irrespective KB’s change state of location n ⇒ Parameterized problem for one site is enough to capture dynamics ⇒ Assume : 0 -> 1 while remaining on same site gives movement δ 0->1 moving to neighbouring site gives movement a – δ (a subunit size kinesin 8 nm) The Limiting Speed of the Bacterial Flagellar Motor Jasmine A. Nirody1, Richard M. Berry2, and George Oster3 1Biophysics Graduate Group, University of California, Berkeley, Berkeley, CA 94720 USA 2Department of Physics, University of Oxford, Oxford OX1 3PU United Kingdom 3Department of Molecular and Cellular Biology, University of California, Berkeley, Flagellar Motors - Rotary Molecular Berkeley, CA 94720 USA NanomotorMay 25, 2015s Abstract➢ Assembly : Dynamic Self Assembly Recent experiments on the bacterial flagellar motor➢ haveProduction shown that : thefamine structure conditions of this nanomachine, which drives locomotion in a wide range of bacterial➢ species,Driving Force: is more dynamic H+ or than Na+ previously electrochemical believed. Specifically, the number of active torque-generating unitsgradient (stators) was shown to vary across applied loads. This finding invalidates the experimental evidence reporting that limiting (zero-torque) speed is independent of the number of active stators. Here, we propose➢ that,Gears: contrary clockwise/anticlockwise to previous assumptions, the maximumrotation speed of the motor is not universal, but rather increases➢ Switching as additional Control: torque-generators CheY are phosphorylation recruited. This result arises from our assumption that stators disengage➢ Torque at stall*: from the motor for~ a 4 significant x 10-18 portion Nm of their mechanochemical cycles at low-loads. We show that➢ this assumption is consistent with current experimental evidence and consolidate our predictions with argumentsMax that Speed*: a processive ~300 motor Hz must (H+) have (1700 a high Hz duty (Na+)) ratio at high loads. ➢ Max Power*: ~1 fW The bacterial flagellar motor (BFM) drives swim- NOVA productionsming in a wide variety of bacterial species, making it crucial for several fundamental processes includ- Flagellum ing chemotaxis and community formation [1, 2, 3, 4]. Accordingly, gaining a mechanistic understanding of Hook this motor’s function has been a fundamental chal- lenge in biophysics. The relative ease with which Composite electron the output of a single motor can be measured in real micrograph of the flagellum basal body time, by observing with light microscopy rotation of and hook, produced by a large label attached to the motor, has made it one stator ‘spokes’ rotational averaging of the best studied of all large biological molecular(MotA,MotB) { (FliG protein) (Francismachines. et al., 1994) However, because of its complexity and lo- calization to the membrane, atomic structures of the torque Rotor entire motor are not yet available. Still, relatively de- generating tailed models have been developed using a combina- surface tion of partial crystal structures [5, 6, 7], cross-linking and mutagenesis [8, 9, 10], and electron microscopic ~50 nm arXiv:1505.05966v1 [physics.bio-ph] 22 May 2015 Figure 1: The bacterial flagellar motor consists of a and cryo-electron tomography images [11, 12] (Fig 1). series of large concentric rings that attach to a flag- Arguably the most important physical probe into the ellar filament via a flexible hook. An active motor dynamics of a molecular motor is its torque-speed re- can have between 1 and 11 torque-generating stator lationship. For the BFM, this curve was shown to units. Stators interact with protein ‘spokes’ (FliG) have two distinct regimes (Fig. 2). This character- along the rotor’s edge to drive motor rotation. istic feature of the BFM was long held as the first ‘checkpoint’ for any theoretical model of the motor. 1 Experimental Measurements ⇒ Marker attached to flagella ⇒ Controlled voltage difference across inner and out segment ⇒ Speed of motor depends linearly on applied voltage for physiological range (up to -150 mW) ⇒ Motor powered by protons/sodium but what’s the mechanism? “Biological Physics”, Philip Nelson (2007) ⇒ Fluorescence bead (200 nm) Berry Group, Oxford : attached to flagellar stub of immobilised cell High-speed video of a 200 nm ⇒ Some experiments done with fluorescent bead attached to a chimera motors (parts of H+ and flagellar motor taking steps Na+ motors fused together) 30x slower than real time ⇒ Individual steps could be resolved 2400 fps position resolution ~5 nm for chimera motors where speed is slowed down Steps in Flagellar Rotation (organism : E. Coli) • Na+ driven chimaeric motor, low sodium-motive force. Small number of force generating units. • 26 steps per revolution (~ 14° per step) - depends on the number of force generating units • Step size matches periodicity of inner part of flagellar motors (ring of FliG protein) • Energetics: one ion crossing per step is not enough to explain this Sowa Y, Rowe AD, Leake MC, Yakushi T, Homma M, Ishijima A, Berry RM. (2005) Direct observation of steps in rotation of the bacterial flagellar motor. Nature. 437:916-919. 4 Operation of pmf driven Rotors 8 How Protein Motors Convert Chemical Energy into Mechanical Work 215 Periplasm Rotor 1. Rotor diffuses to the left, Periplasm Dielectric Barrier bringing the empty (negatively ∆ψ Membrane charged) site into attractive Cytoplasm Rotor Charges ∆µ field of the positive stator Input charge. k T Channel B + -2.3 ∆pNa Stator Stator charge e R227 2. Site captured, membrane Cytoplasm potential biases the thermal Figure 8.5. Operation of the F0 motor (Dim- is captured, the membrane potential biases the Oster, G. and Wang, H. (2002) How Protein Motors Convert Chemical Energy roth et al., 1999). (a) Simplified geometry of the thermal escape of the site to the left (by tilting escape of the site to the left sodium driven F0 motor showing the path of the potential and lowering the left edge). 3 p 4, into Mechanical Work, in Molecular Motors (ed M. Schliwa), Wiley-VCH ions through the stator. This is the same ar- the site quickly picks up an ion from the input 3. Site quickly picks up an ion Verlag GmbH & Co. KGaA, Weinheim, FRG. rangement as in Fig. 8.4c, but with the charge channel neutralizing the rotor. 4 p 5, an oc- array wrapped around a cylinder that is free to cupied site being nearly electrically neutral can from the input channel rotate in the plane of the membrane with re- pass through the dielectric barrier. If the occu- spect to the stator. In addition, a `blocking pied site diffuses to the right, the ion quickly neutralizing the rotor. charge' has been added to the stator to prevent dissociates back into the input channel as it the leakage of charge from the high concentra- approaches the stator charge. 5 p 6, upon ex- 4. Occupied site being nearly tion reservoir (periplasm) to the low concen- iting the stator the site quickly loses its ion. The tration reservoir (cytoplasm). (b) Free energy empty (charged) site binds solvent and cannot neutral can pass through the diagram of one rotor site as it passes through pass backwards into the low dielectric of the dielectric barrier. the rotor stator interface. Step 1 p 2, the rotor stator. The cycle decreases the free energy of À diffuses to the left, bringing the empty (nega- the system by an amount equal to the electro- 5. Upon exiting the stator the tively charged) site into the attractive field of motive force. the positive stator charge. 2 p 3, once the site site quickly loses its ion. The empty (charged) site binds depending on the organisms and/or conditions. Second, a `blocking charge' (R232) solvent. is present on the stator that prevents leakage of ions between the two reservoirs that would dissipateTorque–speed relationship of the bacterial flagellar motor, Xing the ion gradient unproductively. The presence of this blocking et al PNAS 2006 charge alters the picture of the rotation of the rotor with respect to the stator sub- stantially. Fig. 8.1b shows the potential experienced by one binding site of the F0 motor. The presence of the blocking charge creates an electrostatic potential 8 well that attracts the rotor binding site as soon as it diffuses into the polar channel. Once trapped in the well, the rotor depends on Brownian fluctuations to escape. However, the membrane potential biases escape by lowering the left side of the electrostatic well so that the rotor charge is much more likely to escape to the left than to the right. Once it escapes into the aqueous input channel, it is quickly neutralized by the positive ions so that it can move freely across the hydrophobic barrier. Note that the membrane potential only biases the Brownian fluctuations to the left, but does not actually drive a power stroke as in Fig. 8.4a. In summary, the F0 motor qualifies as a Brownian ratchet since it rectifies large thermal fluctuations (or long diffusions). The energy for rectification derives from the ion concentration gradient via the short range interactions between ions and Bacterial Chemo-taxis Overview E coli swims up a chemical gradient of aspartic acid (amino acid) Constraints: • only 1,000 receptors Receptors • only 1-10 s to evaluate concentration (due to Brownian rotary motion) Motor Performance: • can detect 0.2% change of occupancy • can operate over 5 orders of magnitude (2 nM – 1 mM aspartate) Does behavior require exquisite fine- tuning ...or are some features robust? Chemo-taxis in Action ➢ motor turns CCW for smooth swimming ➢ motor turns CW for tumbling ➢ rate of CW to CCW transition modulated by change in ligand concentration ➢ Comparing last 1 sec with prev. 3 sec ➢ If concentration does not change --> adaptation System Elements Termin CheA, CheB,... : Chemotaxis Fla (Fl) : Flagella D. Bray (1995) Protein molecules TarA,... : Taxis towards as computational elements in aspartate and away from living cells. Nature 376:307. repellent Trans-membrane Signalling Receptor is an allosteric protein CheA is kinase which phosphorylates CheY CheA activity depends on two things: ➢ when ligand bound ➢ when receptor methylated Model: assume 2 conformational states active : CheA phosphorylation inactive : no CheA phosphorylation Protein Networks Ligand L CH3 Pi Exact Adaptation Barkai and Leibler, Nature (1997), 387, 913 Exact Adaption : Experimental Evidence Protein Level Activity Methylation Time (s) Time (s) Cell Level U. Alon et al. (1999) Nature 397:168. Exact Adaption : Experiments Test: Vary expression level of methylating enzyme CheR Precision of adaptation is independent of [CheR] Steady-state tumble frequency and adaptation time depend on [CheR] U. Alon et al. (1999) Nature 397:168.