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

UwRload  1 

load U moto = − w2 R Uw Rw R  “motor” r perfect tot  12 axis

Uwmotor R  2 

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)

UwRload  1 

U moto = − w2 R r perfect UUUtotmotorload axis load Uwmotor R  2  “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 UUMRtotalratchet

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 RateRate  can turn backwards freelyto turn wheel f orward

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: Rforward0 exp kTB 1

– Heat from bath 1: QMRMR1  work pawl  • Backward rotation: Rbackward0 exp kTB 2

– Heat from bath 2: QMRMR2  work realeased , to the vanes

QQ12 • Reversibility requires detailed balance RRforward backward   TT12

• The efficiency is Utotal U ratchet MR MR QQ  12 QQ11 T MR  1 2  maximal (Carnot) T1  2 φ Ratchet is irreversible if T1=T2=T

 MR UUMRtotalratchet • Forward rotation: Rforward 0 exp kTB 1

 MR • Backward rotation: Rbackward0 exp kTB 2  Ratchet moves backward  2 φ

dMR    RRforward backward0 expexp1  dtk Tk T  BB ω ω ω T

M MR1 k T  B Summary: necessary conditions for directional motion U • Asymmetric potential: 

• Break detailed balance: RRforwardbackward

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 Tx    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 : UDU jjjDPDv   xk TxB  Fokker-Planck equation

PPPU    D    0 t  x  x kB T  x

• At steady-state this is Smoluchowski’s equation

PPPU   0 jD     const. t  x kB T  x Steady-state is solution to Smoluchowski equation

L • No net current:

PPU jD 0 xk TxB x

• Gives equilibrium Boltzmann distribution:

PUU x 1( )    P( x , t )exp.  Pk TZkBB T  The general steady state solution

L • Steady-state PPU jD const. xk TxB x Ux() Define P ( xP )( )x00 p( )xP where 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

jUx y () p( xCdy )exp  Dk T 0 B x U( xjU )( ) y x P( xCdy )expexp   kBB TDk T 0

• Sawtooth potential U( xf )  for xL 0x

• “Perfect” ratchet – because S-S bond strong

 kTB • Therefore: (1)j  0 protein moves rightwards. (2) PL ( ) 0 because it cannot jump back. Solving the molecular ratchet

L • Let’s scale the variables: fxfL uF ;;;  kBBB Tk Tk T x xu jU yj Lj() L uu p( xCdy )expe1 CduCe    Dk00 TD FDB F jL P( xp ) xC e(uu ) B where eB B  DF

• Boundary condition

P( L ) C  B eFF  B  0  C  B  Be

P( x )  B eFu  1 fxfL uF ;;;  kTkTkTBBB L

• Normalization in [0,L] x LF2 jL Fu P( x )111 dxedu   00DF 2 2 jL F DF  eF11    j  2 F DF LeF 1

11LeF2 F  • Time to move on bolt   2 jDF

• 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).

• Thermal motion can push molecules to transit between states. Chemical reactions are random walks on free energy landscape in space of molecular configurations

 HHHH22

• There exists a path in configuration space that joins two minimums while climbing the free energy landscape as little as possible.

• Chemical reaction is approximately 1D walk along this path, which is called the reaction coordinate. Even big macromolecules can be described by one or two reaction coordinates

• Rate of transition through a barrier GETS‡  ‡ G‡ Rateexp kTB E‡ Rateexp kTB Enzyme catalysis: Conceptual model of enzyme activity (Haldane 1930)

(1) Binding site of enzyme E matches substrate S.

(2) E and S deform to match perfectly and form ES state.

One bond in S (“spring” in S) is easily broken by deformation.

(3) Thermal fluctuation break bond and get EP state.

A new bond forms (“upper spring”), stabilizing product P.

(4): P does not perfectly fit to binding site, so it readily unbinds, returning E to its original state. Enzymes reduce activation energy by binding tightly to the substrate's transition state.

• Interaction free energy with large binding free energy (dip) with entropic cost aligning S and E.

• Binding free energy partly offset by deformation of E, but net effect is to reduce the barrier

• Net free energy landscape (sum of three curves): Enzyme reduced ∆G‡ but not total ∆G Enzymes are simple cyclic machines

• Enzyme work by random-walking down 1D free energy landscape.

Barrier: G‡  f  L

Net descent each work cycle: G  f  L  Mechanochemical motors move by random walking on a 2D landscape

• E catalyzes the reaction S at high chemical

potential μS into P with low μP .

• E has another binding site which can attach to a periodic “track”. e.g., kinesin which converts ATP to ADP+Pi and can bind to a microtubule.

• 2D free energy landscape with 2 variables:

– Reaction coordinate: # of remaining S.

– location of machine on track. Potential energy is periodic in reaction coordinate and position, reflecting cyclic nature of enzymatic turnovers and motor cycles.

• tilted along chemical axis by free energy of reaction. • load force tilts of the surface along position. • Furrow couples chemical energy to mechanical motion. System random walks in furrow.

• Correspondence between potential energy surface and kinetic mechanism of motor. Breaking the symmetry induces motion

• Mechanochemical cycle: landscape with 2 directions, reaction coordinate and displacement.

• Landscape asymmetric in displacement +

concentrations of S and P out of equilibrium

= directed net motion.

• Deterministic free energy landscape: coupling of variables is tight and we can follow a single reaction coordinate one valley. VI. Kinetics of enzymes and molecular machines Michaelis-Menten rule describes the kinetics of simple enzymes

[푆] 푉 = 푉max [푆] + 퐾푀 Free energy E + S • MM kinetics originate from the energy originatefromthe MM kinetics

ipiidcci rcs: . Simplified cyclic process: Different Different Initial to and final starting states returns initial but state. enzyme landscape of the enzyme landscape ofthe R ES eac t i o n coo ∆ EP G r di ' n a te E E + + P P a a t t l h o i w g h P P

SEP E ES S E

   



  •  •

kS

k

1



1 ∆ shortlyneglect lived EP.state low G ' >> P: EP ES o th → e → EP is r barriers r barriers and

k E+P is quick

2 irreversible. process. k B T MM kinetics originate from the energy landscape of the enzyme

kS1 k ESESEP 2 .

k1 y g ' r ∆G

e Steady state: n e E+S e dE[] e ES  k112[ S ][ EkkES ] ()[] 0 Fr dt [SE ][ ] EP [],ES  KSM [] Reaction coordinate kk12 KM  MM constant k1

dP[] k2[ S ][ E ] v  k2[] ES  dt KM [] S []S v vmax with v max k 2 [ E ] KSM [] Experimental test: Lineweaver-Burk graph

11KM 1 v vmax [] S Two-headed kinesin is a tightly coupled, perfect ratchet

• tightly coupled molecular motor ~ enzyme:

1D random walk along a valley, a step per reaction.

• P kept low motion is irreversible.

• A tightly coupled molecular motor with an irreversible step

move at a speed determined by MM kinetics.

• Kinesin is bound to microtubule ~ 100% duty ratio.

• Kinesin is highly processive: makes many steps before falling Kinesin is highly processive, even with load

• kinesin motility assay: optical tweezers apparatus pulls a bead backwards.

• Feedback circuit continuously moves the trap following the kinesin at constant force 6.5 pN, with 2 mM ATP.

[Viscscher et al. 1999] Kinesin obeys MM kinetics with load-dependent parameters

[Data from Visscher et al. 1999]

• Kinesin rarely moves backward  “perfect ratchet” limit.

• Kinesin is tightly coupled: exactly one step per one ATP molecule (even under load). Structural clues for kinesin as perfect ratchet

• Microtubule:

– β-subunit has a binding site for kinesin.

– Regular space at 8 nm intervals. + - – Microtubule is polar (+/- ends).

• Kinesin binding is stereospecific:

 bound kinesin points in one direction.

Kinesin heads: neck

motor domain • ADP at nucleotide-binding site: neck linker

neck flops between two different conformations. linker • ATP at site: neck linker binds tightly to kinesin Nucleotide binding site head and points to the + end. Structural clues for kinesin as perfect ratchet

• Kinesin binds ADP strongly.

• Kinesin without bound nucleotide binds microtubules strongly.

• Complex Microtubule·Kinesin·ADP is only weakly bound.

• Complex Microtubule·Kinesin·ATP is strongly bound; Only one head could bind microtubule when only ADP exists;

• Binding ATP to one head stimulates another head to release its ADP Cyclic model for kinesin stepping Summary U • Necessary conditions for directional motion:  • Break spatial inversion symmetry • Break thermal equilibrium. RRforwardbackward

• Smoluchowski equation for micro-machines: PPPU  0const. jD txk Tx B

• Enzymes are simple cyclic machines; work by random walking on 1D free energy landscape.

• Saturation kinetics: Michaelis-Menten rule []S vv max KSM []

• Mechanochemical motors move by random-walking on 2D landscape reaction coordinate vs. spatial displacement. • Two-headed kinesin is a tightly coupled motor.