Lecture 10: Molecular Machines and Enzymes Sources: Nelson – Biological Physics (CH.10) Julicher, Ajdari, Prost – Modeling molecular motors (RMP 1997) Tsvi Tlusty, [email protected] Outline I. Survey of molecular devices in cells. II. Mechanical machines. III. Molecular implementation of mechanical principles. IV. Kinetics of enzymes and molecular machines. Biological Q: How do molecular machines convert scalar chemical energy into directed motion, a vector? Physical idea: Mechano-chemical coupling by random walk of the motor on a free energy landscape defined by the geometry of the motor. I. Survey of molecular devices in cells Artificial and natural molecular machines • A molecular machine, or a nanomachine: a molecule or a an assembly of a few molecules, that performs mechanical movements as a consequence of specific stimuli. The most complex molecular machines are proteins found within cells. The Nobel Prize in Chemistry 2016: Sauvage, Stoddart and Feringa "for the design and synthesis of molecular machines". Classification of molecular devices in cells (A) Enzymes: biological catalysts that enhance reaction rate. (B) Machines: reverse the natural flow of a chemical or mechanical process by coupling it to another process. • “One-shot” machines: use free energy for a single motion. • Cyclic machines: repeat motion cycles. Process some source of free energy such as food molecules, absorbed sunlight, concentration difference across a membrane, electrostatic potential difference. • “One-shot” machines • Cyclic machines Cyclic machines are essential to Life • Motors: transduce free energy into directional motion (case study: kinesin) • Pumps: transduce free energy to create concentration gradients across membranes (ion channels). • Synthases: transduce free energy to drive a chemical reaction and then synthesize some products (ATP synthase). Let’s look at a few representative classes of the molecular devices. (A) Enzymes enhance chemical reactions • Even if ∆G<0, a reaction might still be very slow due energetic barriers. • Good for energy storage, but how to release the energy when needed? Enzymes display saturation kinetics • Most enzymes are made of proteins. speedup by 1012 • Ribozymes consist of RNA. 7 Vmax=10 molec/sec • Ribosome: complex of proteins with RNA. [푆] 푉 = 푉max [푆] + 퐾푀 (B) One-shot motors assist in cell locomotion and spatial organization ● ● Translocation Polymerization Actin polymerization from one end of an Several mechanisms can rectify (make actin bundle provides force to propel a unidirectional) the diffusive motion of Listeria bacterium through the cell surface protein through the pore. All eukaryotic cells contain cyclic motors • Molecular forces were measured by single molecule experiment. Some important cyclic machines II. Mechanical machines Macroscopic machines can be described by an energy landscape Uload w1 R load U moto = − w2 R U w R w R “motor” r perfect tot 12 axis Umotor w2 R The movement of macroscopic machines is totally determined by the energy landscape. Overdamping macroscopic machines cannot step past energy barriers imperfect axis (e.g. friction) Uload w1 R U moto = − w2 R r perfect UUUtot motor load axis load Umotor w2 R “motor” Some machines have more degrees of freedom • The angles are constrained to “valleys. • Slipping: imperfections (e.g. irregular tooth) may lead to hopping. Gears: the angular variables both decrease as w lifts w 2 1. n constraint: 2 • The machine stops if: N – w2 = w1 . – Slipping rate too large. Microscopic machines can step past energy barriers • A nanometric rod makes a one-way trip to the right driven by random thermal fluctuations. • Moving to the left is impossible by sliding bolts, which can move down to allow rightward motion; • It can move against external "load" f. • Where does the work against f comes from? • What happens if one makes the shaft into a circle? Can one extract work from thermal fluctuations? • Rod cannot move at all unless 휀~푘퐵 푇. • But then bolts will spontaneously retract from time to time… • Leftward thermal kick at just such a moment and rod steps leftwards. • Where does the work against f comes from? • What happens if one makes the shaft into a circle? Can we save the second law of thermodynamics? • Bolts are tied down on the left side, then released as they emerge on the right. • Where does the work against f comes from? • What happens if one makes the shaft into a circle? Can we save the second law of thermodynamics? • Where does the work against f comes from? -- Potential energy stored in the compressed springs on the left. • What happens if one makes the shaft into a circle? -- All springs are released and the motion stops. This is a toy model for model for molecular translocation Feynman’s ratchet and pawl U Pawl ratchet T2 φ Spring T1 Utotal U ratchet MR Ratchet MR Load φ Q: Can the small load be lifted if T1 =T2 =T ? A: May look quite possible, but 2nd law of thermodynamics: (Heat cannot be converted to work spontaneously) prohibits this. • load = 0: pawl pulled over tooth P=exp by vanes kBT1 • But also pawl jumps P=exp spontaneously kBT2 pawl is up and wheel enough energy Rate Rate can turn backwards freely to turn wheel forward Pawl no net rotations. T2 Spring φ T1 • Q: what if load > 0? Ratchet • A: work heat Load Ratchet can be made reversible if T1 >T2 MR • Forward rotation: Rforward 0 exp kTB 1 – Heat from bath 1: Q1 MR work MR pawl • Backward rotation: Rbackward 0 exp kTB 2 Utotal U ratchet MR – Heat from bath 2: Q2 work realeased MR , to the vanes MR QQ12 • Reversibility requires detailed balance RRforward backward MR TT12 • The efficiency is MR QQ 12 QQ11 T 1 2 maximal (Carnot) T1 2 φ Ratchet is irreversible if T1=T2=T MR • Forward rotation: Rforward 0 exp kTB 1 • Backward rotation: Rbackward 0 exp kTB 2 Ratchet moves backward φ Utotal U ratchet MR d MR RRforward backward 0 exp exp 1 dt k T k T BB MR ω ω ω T M MR1 k T 2 B Summary: necessary conditions for directional motion U • Asymmetric potential: • Break detailed balance: RRforward backward Pawl T2 TT12 Spring φ T1 Ratchet Load Modeling molecular machines: qualitative picture L x Low load: fL / High load: fL / forward backward Modeling molecular machines: qualitative picture L x • To move right need a thermal fluctuation of E f L • If kT B then the leftwards motion is negligible. • If f 0 rod diffuses freely between steps with LLL2 tvstep and velocity D tstep D • What happens when f 0? Mathematical framework: Smoluchowski equation • Think of large set of identical rods: L • What is the P(x,t) = prob. ratchet is in (x,x+dx)? • Ratchet diffuse according to Fick’s law: P x jD D x • Rods drift in the force field: DU jP v D kB T x kTB Einstein’s relation • The total current is: UDU j jDv j D P x kB T x Fokker-Planck and Smoluchowski equations L • Number conservation (continuity equation): jP 0 xt x • With current : Fokker-Planck equation PPPU D 0 t x x kB T x • At steady-state this is Smoluchowski’s equation UDU PPPUj j j D P Dv0 jD const. x kB T x t x kB T x Steady-state is solution to Smoluchowski equation L • No net current: PPU jD 0 x kB T x x • Gives equilibrium Boltzmann distribution: P U1 U ( x ) P( x , t ) exp . P kBB T Z k T The general steady state solution L • Steady-state PPU jD const. x kB T x x Ux() Define P( x ) P00 ( x ) p ( x ) where P( x ) exp kTB P P U p j p j U() x P0 exp x kBB T x x D x D k T jx U() y p( x ) C exp dy D0 kB T U()() x jx U y P( x ) exp C exp dy kBB T D0 k T Solving the molecular ratchet L • Steady-state x • Sawtooth potential U( x ) f x for 0 x L • “Perfect” ratchet – because S-S bond strong x jkTB U() y p( x ) C exp dy • Therefore: (1)Dj 0 0 protein kB T moves rightwards. (2) PLU(()() ) x 0 because jx it cannot U yjump back. P( x ) exp C exp dy kBB T D0 k T Solving the molecular ratchet L • Let’s scale the variables: f x f L uF;;; kBBB T k T k T x xu j U() y j Luu j L p( x ) C exp dy C e du C e 1 D00 kB T D F D F jL P( x ) euu p ( x ) C B e B where B DF • Boundary condition P( L ) C B eFF B 0 C B Be P( x ) B eFu 1 L • Normalization in [0,L] f x f L x LFuF;;; 2 k T kjL T Fu k T P( x ) dxBBB 1 e 1 du 1 00DF 2 2 jL F DF eF 11 j 2 F DF L e F 1 11L2 eF F • Time to move on bolt 2 j D F • Average speed v D/L LDF 2 v j L F L e F 1 fL F kTB III. Molecular implementation of mechanical principles What’s missing for molecular machines? • How can we apply these macroscopic ideas to single molecules? • What is the cyclic machine that eats chemical energy? • Where are the experiments? Reaction coordinate simplifies description of a chemical reactions • Chemical reactions can be regarded as transitions between different molecular configurations. To move between configurations atoms rearrange their relative positions. • The “states” are locally stable points in a multi-dimensional configuration space (local minima of free energy).