Fluctuations meet function: Molecular motors

Transcription

1 Fluctuations meet function: Molecular motors Diego Frezzato, July 2018 Part of the course Fluctuations, kinetic processes and single molecule experiments

2 A molecular machine is a device made of a single (complex) molecule, or a supramolecular complex, that transduces input energy into output energy; if the output energy is mechanical work, the machine is usually called motor. In cellular environment, these transductions are performed in accurate/precise way. Operations of molecular machines can be - cyclic (eg., motors and pumps) - one-shot (some examples?) Let us look here only at few traits of molecular machines. An excellent review: D. Chowdhury, Stochastic mechano-chemical kinetics of molecular motors: a multidisciplinary enterprise from a physicist s perspective, Physics Reports 529, (2013)

3 Forms of input energy - Chemical fuels. Energy is released by localized chemical reactions, mainly hydrolysis of nucleoside triphosphates (NTPs): ATP, Guanosine Triphosphate (GTP). Also inorganic pyrophosphate (PPi) generated from hydrolysis of ATP to AMP can be used. r G (25 C) 30.5 kj/mol - From the manipulated substrates themselves. For example, polymerases can extract energy from the substrates in creating the polymers. - From light absorption (photons) - From spatial gradients of chemicals concentration (eg., H + gradients), charge, etc.

4 Just a mention to some of the many machines/motors which operate in the cellular environment 1) Enzymes for synthesis/manipulation/degradation - degradation of macromolecules - template-dictated polymerisation - helicases, topoisomerases, etc. (unwrappers, unzippers, untanglers of DNA) - controllers (e.g. quality controllers of genome replication)

5 2) Translocation motor proteins - porters (intracellular cargo transport) - sliders acting as rowers (make relative sliding of two filaments) - depolymerases (kinesins which crash their track-filament from one end) -pistons, hooks, springs via polymerizing/depolymerizing cytoskeletal filaments (eg, dynamic filamentous proteins in prokaryotic cells) - translocases across membranes

6 (a) conventional kinesin (transport of organelles), (b) myosin V (transport of vesicles), (c) cytoplasmic dynein (transport of mrna) [figure taken from A. B. Kolomeisky et al, Annu. Rev. Phys. Chem. 58, 675 (2007)]. Details of the two kinesin heads [figure taken from S. M. Block, Cell 93, 5 (1998)].

7 3) Rotary motors 4) Ion pumps (active transport through membranes)

8 F 0 F 1 -ATPsynthase (for ATP production) ~ 10 nm inner side ~ 20 proteins ~ 500 kda mass outer side ~ 95% of ATP is our produced by this molecular machine In 75 years of life, 2000 tons of ATP are produced on average!

9 ~ 100% efficiency! 3 molecules of ATP per cycle, with a transit of 8 15 H + ~ 3 cycles/sec (~10 ATP molecules per second), depending on the difference of ph at the two sides of the membrane A nice animation: For insights: - Z. Ahmad, J. L. Cox, ATP synthase: the right size base model for nanomotors in nanomedicine, The Scientific World Journal, ID (2014) -Y. Q. Gao, W. Yang, M. Karplus, A structure-based model for the synthesis and hydrolysis of ATP by F 1 -ATPase, Cell, Vol. 193, pag. 193 (2005) - D. Okuno, R. Iino, H. Noji, Rotation and structure of F 0 F 1 -ATP synthase, J. Biochem., Vol. 140, pag. 655 (2011)

10 Such motor can operate in reverse (at low proton gradients): ATP hydrolysis! The F 1 unit sufficies to catalize ATP hydrolysis. Single-molecule observation of the F 1 rotation upon ATP hydrolysis D. Okuno, R. Iino, H. Noji, J. Biochem. 149, 655 (2011)

11 Cycle of ATP hydrolysis catalized by the F 1 D. Okuno, R. Iino, H. Noji, J. Biochem. 149, 655 (2011)

12 Bacteriorhodopsin In the archea of salty ambients, e.g. in Halobacterium salinarium Pump of H + ions towards the exterior of the cell membrane retinal hν λ max ~570 nm It can produce ph differences in-out up to 4 units! W. Kühlbrandt, Bacteriorhodopsin The movie, Nature, Vol. 406, pag. 569 (2000)

13 Coupling between bacteriorhodopsin and ATP-synthase! Radiant energy (hν) Chemical energy (ATP)

14 Bacterial flagella ~ 20 proteine Operates by exploiting gradients of H + or Na + ~ 100 μm length 45 nm Bacterium membranes In E. coli ~ 300 turns/sec Exerts a torque of ~ 550 pn nm Gives a speed of ~ 30 μm/sec Efficiency ~ 60% D. J. De Rosier, The turn of the screw: the bacterial flagellar motor, Cell, Vol. 93, pag. 17 (1998)

15 Cellular transporters R. D. Vale, The molecular motor toolbox for intracellular transport, Cell, Vol. 112, pag. 467 (2003) Kinesin I Myosin V Dynein move on microtubules moves on actin filaments (or it displaces actin filaments in the mechanism of muscle contraction) Transpsport/move vescicles, cell s nucleous, mrna, cytoskeleton filaments, signaling proteins, protein fragments, Energy is supplied by ATP hydrolisis.

16 α β tubulin dimer protofilaments α β - + (helical structure) 25 nm cytoskeleton Input energy: hydrolisis of ATP

17 hand-over-hand motion of kines ~ 100 steps before the detachment from the microtubule ~ 800 nm/sec (100 steps/sec) in vitro Efficiency ~ 60% To stop it ( stall ) it is required a force of ~6 pn in opposition

18 Hand over hand motion for kinesin on microtubules, as proved by means of FIONA technique ( Fluorescence Imaging One Nanometer Accuracy ). [Figure taken from A. Yildiz et al., Science 303, 677 (2004)]

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20 actin myosin Lymn-Taylor cycle for the muscle contraction

21 RNA polymerase J. Gelles, R. Landick, RNA polymerase as a molecular motor, Cell, Vol. 93, pag. 13 (1998) ~ 12 proteins From the DNA template, at need it produces the various forms of RNA that are involved in the cellular processes Very high accuracy: ~ 1 error every nucleotides! DNA RNA The energy is supplied by the polimerization itself! phosphorylated nucleotides (cytosine, uracil, guanine, adenine)

22 Richard P. Feynman Caltech (December 1959)

23 New trends: combination biologic-synthetic An example how to make the biologic molecular motors work for us! The F 1 -ATPase motor coupled to inorganic rod made of Nickel H. Hess, G. D. Bachand, V. Vogel, Chem. Eur. J. 10, 2110 (2004) M. G. L. van den Heuvel, C. Dekker, Science 317, 333 (2007)

24 Are molecular machines just a nanoscale version of man-made macroscopic machines? Typical scales of the molecular machines: Length: nm Time: ms Forces: pn Energy: k B T (= J involved per molecule at 25 C) Similarities with macroscopic machines are only apparent: it is not just a matter of length scale!

25 First argument: the Scallop theorem from hydrodynamics [E. Purcell, Am. J. Phys. 45, 3 (1977)] In hydrodynamics, the dimensionless Reynolds Number (Re) compares the magnitude of inertial and viscous forces for an object (or a fluid element itself) moving in a fluid: u Re c l c with u c the velocity of the object (and of the fluid sticked to it), l c a charateristic length of the object, ρ the fluid density, η the (shear) viscosity of the fluid. For water at 20 C, ρ = 10 3 kg m -3 and η = Pa s. Low Re means that viscous forces prevail. A scallop can move by opening and closing to expell water. Typical lengh is l c = 1 cm, and it moves by few times its length per second. It results Re 10 2 : high value, mechanical force prevails on viscous drag, hence propulsion occurs. A nano-scallop would have Re 10-10! It cannot propel itself: perceived viscous drag is so high that opening/closing make a balance. A nano-scallop just fluctuates. On the contrary, molecular machines are quite agile!

26 Second argument: look at the rate of energy exchanges - typical input energy-rate : J/s - typical energy-exchange rate with the thermal bath via collisions : 10-8 J/s There are 9 orders of magnitude of difference! Random perturbation from the environment is much intense than the detailed energy input. For molecules, moving in a straight line would seem to be as difficult as walking in a hurricane is for us. Nonetheless, molecular motors are able to move, and with almost deterministic precision [quotation taken from D. Astumian, P. Hänggi, Physics Today 55, 33 (2002)]. On the other hand, molecular machines find their way! so it is not only a matter of length scale: at the nanoscale one has a peculiar scenario. Let us look at the main features: what does a molecular machine feel?

27 Essential and ubiquitous traits of molecular machines Molecular machines operate under isothermal conditions (temperature gradients are not sustained at molecular scales...) Transduction from scalar energy into vectorial processes - Although the trajectory of the machine (in abstract sense) is stochastic, on average the motion is directed: there is a drift. Machines are able to receive the energy input in detailed way - Mechano-chemical coupling: a localized event (eg., a chemical reaction or photon absorption) generates a cascade of responses. How can we describe such a coupling? For cyclic machines, steady states far-from-equilibrium can be reached/maintained under energy input - A net drift is generated (for example, an average velocity of kinesins on microtubules, an average angular velocity of the F 1 -ATPase rotor, etc).

28 Sources of input energy - Chemical fuels, enrgy from the manipulated substrates themselves, photons, gradients of chemicals concentration or of electric charge. Reaction coordinate -The internal free-energy V(x) of the machine can be reduced (by means of proper averages over fast-fluctuating variables) to a low-dimensional free energy landscape on few essential degrees of freedom. One of these degrees of freedom is the peculiar reaction coordinate along which the specific action is performed. What distinguishes such a coordinate from the others? Accurate and precise operation - A single trajectory of a molecular machine (in its low-dimensional free-energy landscape) is stochastic. However, trajectories deviate little from the average, spread of cycles period is little, etc. How is it possible?

29 Release of waste products (waste chemicals and dissipated energy) - For example, release of hydrolysis products. - As for macroscopic finite-time (irreversible) transformations in isothermal conditions, the free-energy transduction into work cannot be complete (the Second Principle of Thermodynamics puts a limit!): part of the free-energy difference at disposal is wasted as heat exchange with the thermal bath (ultimately: global entropy production). Average energy dissipation rate for objects of different length-scale, operating under steady-state conditions. [figure taken from C. Bustamante et al, Physics Today 58, 43 (2005)] 3 10 k T / s B

30 In the essence: what do we need to let a dead molecule alive & working? 2) detailed energy input 1) fluctuations of structural variables x mechano-chemical coupling x fluid environment (also crawded) 3) Breaking forward-backward symmetry of the machine operation ( directionality ) All the three ingredients are necessary to let a machine working!

31 For example, fluctuations alone leave the molecule dead : it would fluctuate at thermal equilibrium without any net average drift (no directed action) The structural asymmetry [i.e., asymmetries in the energy landscape V ( x) ] is not sufficient to induce such a drift! For example, the polarity of an actin microtuble is not sufficient to make kinesins moving, on average, in one direction A ratchet model has been proposed as paradigm of browian motors: when the nano-ratched is in contact with the thermal bath (random noise from collisions), structural asymmetry of the teeth should rectify the fluctuations. That is not true! What is missing? Only a targeted energy input can keep the machine out-ofequilibrium to get a drift. The ratched idea must be revisited

32 activation Pictorial representation of the ratched model with energy input A. B. Kolomeisky, M. E. Fisher, Molecular Motors: A Theorist s Perspective, Annu. Rev. Phys. Chem. 58, 675 (2007) energized state state at rest The drift is originated by promotion to an energized state (eg., kinesin plus ATP on the head domains), followed by relaxation back to the state at rest: the two energy landscapes must be different (using the metaphor of the ratchet, the shape of the teeth must be different at rest and in the energized state )

33 Generalization (end of the story or the beginning for the modelling!) Fluctuations of x on an energy landscape which is, by itself, stochastically modulated by the energy input (chemical reactions, photon absorption, etc): V c (x) set of parameters that specify the istantaneous shape of the energy landscape configurational variables of the machine The energy input modulates (stochastically) c affects the fluctuations of x average drift along the reaction coordinate (directed action of the machine)

34 Stochastic trajectories of the operating machine could be generated by means of a generalized Langevin equation which couples: 1) Stochastic localized reactions ( chemical Langevin ) which modulate c 2) Brownian dynamics of x on the energy landscape ( x) V c Simulated trajectories could be compared with the real-time experimental observations of the single machine during operation! [ Recall the trajectories of the single F 1 domain of the ATPase ] Difficulties: too many variables to handle, and too many unknown parameters! Need to adopt a simpler approach: a discrete representation with the same kind of phenomenology. Full dynamics on a modulated energy landscape are replaced by a kinetic mechanism made of a few elementary steps,

35 Make a list of N s relevant sites (stable conformations) N s Sketch out a likely kinetic mechanism involving these sites as species. Some of the elementary steps must involve the energizing molecules (eg., ATP). 5 k + ATP k 2 6 k 3 3 k -1

36 x set of continuum variables DISCRETIZATION Finite number of relevat states (stable intermediates, or conformations individuated by an educated guess) x, x,... x Ns 1 2 reference configurations of the relevant states ( sites )

37 The objective Think to an ensemble of machines, describe the time-evolution of the populations of each site. Population of the site = probability of observing the machine in that given site Disadvantage - Full (small-steps) stochastic trajectories of the single machine, x(t), are not generated in such a coarse-grained perspective Advantages: - Direct pictorial representation of the chemical steps (see examples below); - More friendly for chemists! - Small number of parameters (kinetic constants) which could be measured experimentally - Simple calculations: under suitable conditions (eg., excess of chemical fuel, eg. high ATP concentration) the steps of the mechanism may become of the first-order; a master-equation is easily written and solved (see below)

38 Some steps are bimolecular (eg., those involving ATP). In the excess of energizing molecules, bimolecular steps can be reduced to pseudo uni-molecular steps (with kinetic constants dependent on the fixed concentration of the chemical fuels ) Master Equation for first-order kinetics: transitions amongst N s sites dp() t dt N s KP() t N s kinetic matrix P1 () t P2 () t P() t... PN s () t Constraint to assure conservation N ( s at any time): j 1 P( t) 1 j K k k j, j ' j, j ' j i j ' j i populations of the sites at time t (their sum is equal to 1) first-order kinetic constants

39 Given the initial populations, P(0), the populations P(t) at any subsequent time are obtained by applying standard numerical methods ( ) Fluctuations at thermal equilibrium (a dead molecule) P k P k i eq, i t j, eq j i i, eq i j lim P( t) detailed balance condition thermal equilibrium populations For activated fluctuations, detailed balance must be broken lim P( t) P P t i i, ss eq, i P steady-state populations If detailed balance is broken in proper way, a non-null current along the reaction coordinate can be present even at the steady-state: each machine experiences a drift along such a coordinate, i.e., the machine operates!

40 Abstract representation for a translocation motor track l-th segment on the track x, x, x,... x : specifies the conformations of sites 1, 2, 3,, N c 1, l 2, l 3, l N, l c for the l-th segment k diss, j : kinetic constant for the irreversible detachment from the track if the motor is on site j

41 r 2 D/ vd Randomness parameter: D = diffusion coefficient along the track d = length of the step v = mean velocity at steady-state experimentally achievable It can be demostrated theoretically that [R. D. Astumian, Science. 276, 917 (1997)] Nc 1/ r For kinesin at high ATP concentrations it is known that r 0.39 Nc 3 A likely scheme must consider at least 3 intermediates per segment of the track. A minimal scheme for Kinesin/Microtubule/ATP,ADP Involving 4 intermediates K M + ATP K M ATP K M ADP P K M ADP i K M ATP K M ADP P K M i K M ADP

42 A more elaborated scheme Pathways of ATP hydrolysis with kinesin. [from M. L. Moyer et al, Biochemistry 37, 800 (1998)]