Molecular Machines and Enzymes

Transcription

1 Molecular Machines and Enzymes Principles of functioning of molecular machines Enzymes and catalysis Molecular motors: kinesin 1

2 NB Queste diapositive sono state preparate per il corso di Biofisica tenuto dal Dr. Attilio V. Vargiu presso il Dipartimento di Fisica nell A.A. 2014/2015 Non sostituiscono il materiale didattico consigliato a piè del programma. 2

3 References Books and other sources Biological Physics (updated 1 st ed.), Philip Nelson, Chap. 10 Physical Biology of the Cell, R. Phillips et al., 2th ed., Chap. 16 Movies Exercise 3

4 Molecular machines Biological molecular machines are fueled by chemical forces Food is source of energy, and most useful work done in cells involves metabolizing nutrients (chemical synthesis). For general reaction involving m species (k reactants and m-k products): f ν 1 X ν k X k ν k+1 X k ν m X m free energy of reaction ΔG given by algebraic sum of m chemical potentials each multiplied by its stoichiometric coefficient ν k : ΔG ν 1 µ 1... ν k µ k +ν k+1 µ k ν m µ m ΔG net chemical force driving reaction: this will run forward if ΔG is negative, backward if it is positive. At equilibrium, ΔG and flows in both directions compensate. 4

5 Molecular machines Biological molecular machines are fueled by chemical forces Free energy is trapped in the chemical bonds, and described by chemical potential: µ = k B T ln( c c 0 ) + µ 0 ( T ) Rearrangement of general formula using relation above allows to express also ΔG as sum of concentration-dependent part and concentration-independent standard free energy change:, giving the Mass Action rule governing thermodynamics of chemical (not only) reactions: [ X k+1 ] ν k+1 [ ] ν 1 X 1 [ ] ν m [ ] ν k... X m... X k = K eq = e ΔG0 k B T where: [ X] c, c 0 =1M c 0 ΔG 0 ν 1 µ ν k µ 0 k +ν k+1 µ 0 k ν m µ 0 m 5

6 Molecular machines How do chemical free energy is harnessed by molecular machines and transformed into mechanical work? Molecules act as brokers between chemical and mechanical worlds, converting a scalar quantity (chemical energy) in a vectorial one (directed motion). Mechanochemical coupling arises from a free energy landscape with direction set by geometry of motor and track. Motor executes a biased random walk on this landscape. 6

7 Mechanical machines: macroscopic Macroscopic molecular machines work by moving on energy landscape One dimensional machines: one-shot cranking shaft Motor Load (external force) Biological Physics, P. Nelson, 1 st ed. updated, 2008 One-shot machine like cranking shaft can do work against external force (load) if torque τ exerted by motor larger than torque exerted by load Rw 1. If apparatus immersed in viscous fluid, angular velocity proportional to total torque: ω = dθ dt τ Rw 1 ( ) η 7

8 Mechanical machines: macroscopic Macroscopic molecular machines work by moving on energy landscape One dimensional machines: cycling shaft External drive (motor) External load Biological Physics, P. Nelson, 1 st ed. updated, 2008 Cyclic machine do work against load w 1 if external drive w 2 is larger than w 1. In viscous fluid, again kinetic energy (inertial term) can be neglected. Machine act as broker transducing potential energy drop of drive into potential energy gain of load. 8

9 Motor Mechanical machines: macroscopic One dimensional machines and energy landscape Load (external force) External load External drive (motor) Biological Physics, P. Nelson, 1 st ed. updated, 2008 Functioning of both machines involves sliding down an energy landscape U tot. 9

10 Motor Mechanical machines: macroscopic One dimensional machines and energy landscape Load (external force) External load External drive (motor) Biological Physics, P. Nelson, 1 st ed. updated, 2008 Functioning of both machines involves sliding down an energy landscape U tot. However, real machines are not perfect U(θ) not exactly linear, but bumps occur, which can give rise to metastable states needing further energy input to be overcome. Even if no bump is found in unloaded machine, net torque τ = -du tot /dθ gets small at bump compared to average value along curve. 10

11 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) External load External drive (motor) Biological Physics, P. Nelson, 1 st ed. updated, 2008 Gears link angles α and β rigidly. α increases clockwise, β counterclockwise. If N is number of teeth in each gear: α = β + 2πn N, n N In hypothesis of perfect coupling, retrieve one dimensional description U = U(α) 11

12 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) N=3 External load External drive (motor) Biological Physics, P. Nelson, 1 st ed. updated, 2008 If teeth are not rigid they could slip onto each other occasionally. Energy landscape becomes truly bidimensional U = U(α, β). Gears imperfections modeled as bumps on energy landscape. 12

13 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) N=3 External drive (motor) External load Biological Physics, P. Nelson, 1 st ed. updated, 2008 If teeth are not rigid they could slip onto each other occasionally. Preferred motions along parallel valleys, satisfying: α = β + 2πn/N. Slipping corresponds to jumps between adjacent valleys. Slipping more likely in presence of bumps lowering the energy barrier of hopping. 13

14 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) No load, no drive Loaded and driven (w 1 < w 2 ) External load External drive (motor) Biological Physics, P. Nelson, 1 st ed. updated, 2008 In presence of load and drive, energy landscape tilted: - Driving torque (Rw 1 ) tilts landscape downward as α decreases. - Load torque (Rw 2 ) tilts landscape downward as β increases. 14

15 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) No load, no drive Loaded and driven (w1 < w2) External load External drive (motor) Biological Physics, P. Nelson, 1st ed. updated, 2008 In presence of load and drive, energy landscape tilted: - Driving torque (Rw1) tilts landscape downward as α decreases. - Load torque (Rw2) tilts landscape downward as β increases. Machine slides down energy landscape, following one valley. 15

16 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) No load, no drive Loaded and driven (w1 < w2) External load External drive (motor) Biological Physics, P. Nelson, 1st ed. updated, 2008 Suppose a bump in energy landscape at (2,2): - Decreasing load will reduce surface slope along β, eventually allowing crossing the barrier. 16

17 Mechanical machines: macroscopic Bidimensional machines: coupled geared shafts (cyclic machine) No load, no drive Loaded and driven (w1 < w2) External load External drive (motor) Biological Physics, P. Nelson, 1st ed. updated, 2008 Suppose a bump in energy landscape at (2,2): - Decreasing load will reduce surface slope along β, eventually allowing crossing the barrier. - Increasing drive will increase slope along α, favoring hopping to neighbor valley at lower energy value α changes without any change in β (slipping). - No work done in slipping: energy is spent (α decreases) but load does not move (β unchanged). 17

18 Mechanical machines: microscopic Microscopic world could take advantage of thermal noise (random fluctuations)? Will a microscopic ratcheted rod move spontaneously right under action of entropic forces (temperature)? 18

19 Mechanical machines: microscopic Microscopic world could take advantage of thermal noise (random fluctuations)? Will a microscopic ratcheted rod move spontaneously right under action of entropic forces (temperature)? No, because this will violate second law of thermodynamics, continuously extracting for free mechanical work from thermal motion of surroundings: - Indeed, rod will move right if energy ε needed to lower beveled bolts is ~k B T. - But then bolts will spontaneously retract once in a while, allowing rod also to move left. - Actually, in the case of applied force to left, rod will move towards that direction. 19

20 Mechanical machines: microscopic Could microscopic world take advantage of thermal noise (random fluctuations)? Energy input required as to rectify motion to the right Suppose a latch keeps bolts down when they are attached to left side of wall. Suppose some mechanism (requiring energy!) unlatches bolts when they move right. Left movement of rod will push rightwards bolt towards wall, which will bounce off. Right movement allowed because bolts are retracted when pushed to left wall. 20

21 Mechanical machines: microscopic Could microscopic world take advantage of thermal noise (random fluctuations)? Net right movement Potential energy of bolt spring gets used up during the process (no free work!), as more and more bolts are released during rightwards movement of rod. Effective rectification of Brownian motion if springs stiff enough to lower spontaneous retraction probability (no more link to temperature only). Model above mechanical analog of protein translocation machine 21

22 Mechanical machines: microscopic Energy landscapes for ratchet under low and high load Ratchet under low load Ratchet under high load Only under low load ratchet will effectively exploit thermal fluctuations to move right and perform work against load. Condition is f < ε/l, with L distance between neighbor bolts. Drop in energy at x = nl due to release of potential energy of bolt spring. Rates of left and right movements reflect probabilities of getting thermal kick of energy k B T > fl and k B T > ε respectively. 22

23 Mechanical machines: microscopic Perfect ratchet: ε >> k B T Ratchet under low load Ratchet under high load Perfect ratchet are characterized by no spontaneous retractions of popped up bolts no back stepping. If f = 0, free diffusion between steps. Time from x = 0 to x = L inversely proportional to diffusion constant: t step L 2 /2D. Velocity of ratchet positive and proportional to D: v = L/t step 2D/L. 23

24 Mechanical machines: microscopic Perfect ratchet: ε >> k B T P(x) for ratchet under load If f 0, probability P(x, t) of staying at x at time t will depend on position along this axis, since energy landscape is tilted between steps due to constant action of load. Suppose releasing M copies of identical ratchets at position x = x 0. Suppose wrapping the road to realize a periodic ratchet. Quasi-steady equilibrium will be reached after some walking time, where P(x, t) = P(x) is periodical over distance L and time-independent. 24

25 Mechanical machines: microscopic Ratcheting flux Expression for flux over energy profile U tot can be found by applying diffusion law and considering effect of load force f. Suppose no mechanical forces, only diffusive terms remain (Fick s law): Including external force leads to quasi-steady equilibrium with drift velocity: v drift Summing up these contribution gives total flux, i.e. total number of systems crossing a given point per unit of time: j 1D 1D j Fick ( x) = f ς = du tot dx k B T D ( x) = MD dp dx = D k B T du tot dx j 1D drift " ( x) = MD$ dp dx + P # k B T du tot dx ( x) = MP( x)v( x) % ' & 25

26 Mechanical machines: microscopic Smoluchowski equation d dx! # " dp dx + P k B T du tot dx $ & = 0 % Derived from expression of flux by imposing that P(x, t) does not depend on time derivative of j 1D equal to zero everywhere, or constant flux. 26

27 Mechanical machines: microscopic Smoluchowski equation d dx! # " dp dx + P k B T du tot dx $ & = 0 % Equilibrium solution If U tot is itself periodic, equilibrium solutions of Smoluchowski equation is given by Boltzmann distribution: U tot P( x) = Ce U tot( x) k B T ( x) =U tot ( x + L) P( x) = P( x + L) Substituting into equation for flux gives j 1D = 0 No net motion if landscape has no average slope 27

28 Mechanical machines: microscopic Smoluchowski equation d dx! # " dp dx + P k B T du tot dx $ & = 0 % Non-equilibrium solution: perfect ratchet (ε >> k B T) Solution of Smoluchowski equation for perfect ratchet with U tot (x) = fx: ( ) = C e x L P x ( ( ) f k T B 1), x [0, L] 28

29 Mechanical machines: microscopic Smoluchowski equation d dx! # " dp dx + P k B T du tot dx $ & = 0 % Non-equilibrium solution: perfect ratchet (ε >> k B T) Solution of Smoluchowski equation for perfect ratchet with U tot (x) = fx: ( ) = C e x L P x ( ( ) f k T B 1), x [0, L] Substituting into expression for j 1D gives (constant and) positive flux: j 1D = CM D k B T f lowly loaded bolted ratchet can make net rightwards progress 29

30 Mechanical machines: microscopic Non-equilibrium solution: perfect ratchet (ε >> k B T) Average ratchet velocity found by calculating average time interval Δt needed for all ratchets initially in [0, L] to cross first rightwards step: Δt = v = L Δt = LMC D kt f = DL kt f $ v = & fl % k B T ' ) ( 2 D 0 L 0 L dxmp(x) j 1D L x L dxmc e 0 dx ( ( e x L) f k T B 1) L e fl k BT 1 fl k B T ( ( ) f k T B 1 ) = ( ) 1 Speed of loaded, bolted, perfect ratchet 30

31 Mechanical machines: microscopic Non-equilibrium solution: perfect ratchet (ε >> k B T) Average ratchet velocity found by calculating average time interval Δt needed for all ratchets initially in [0, L] to cross first rightwards step: Δt = # v = L Δt = % fl $ k B T 0 & ( ' L dxmp(x) 2 D If force very low, retrieve diffusion driven result If force very high (but fl << ε), velocity contains exponential activation energy factor:! v = # fl " k B T j 1D ( ) 1 L e fl k BT 1 fl k B T $ & % 2 D L e fl k BT Speed of loaded, bolted, perfect ratchet 31

32 Mechanical machines: microscopic Non-equilibrium solution: generic ratchet Probability given by: P( x) = C( be x f kbt 1), b = e ε kbt 1 ( ) k T B 1 e fl+ε Flux given by same expression for perfect bolted ratchet: j 1D = CM D k B T f Average velocity given by: v = L Δt = D L # % $ fl k B T & ( ' 2 ) fl k B T 1 e ε k B T + e fl kbt e ε k BT * + ( )( 1 e fl k BT ),

33 Microscopic machines: summary Microscopic machines move by random walks on their FREE ENERGY landscapes (no deterministic motion!). Average waiting time for crossing barriers given by exponential factor. Due to high viscous dissipation, they can store potential but not kinetic energy. More specific points from study of ratchets: Thermal machines can convert stored energy ε into directed motion if they are structurally asymmetrical AND if free energy landscape has a nonzero slope (system is out of equilibrium). Ratchet velocity does not increases indefinitely with increasing driving energy ε, but saturates at some limiting value. Smoluchoswki equation can be generalized to higher dimensions. 33