Presentation on theme: "Discussion topic for week 5 : Enzyme reactions"— Presentation transcript:
1 Discussion topic for week 5 : Enzyme reactions The lock in key hypothesis (Emil Fischer) asserts that both the enzyme and the substrate possess specific complementary geometric shapes that fit exactly into one another.What are the problems associated with this hypothesis?
2 Enzymes and Molecular Machines (Nelson, chap. 10) Enzymes are biological catalysts that enhance the rate of chem. reactions.Machines use free energy from an external source (e.g. ATP,concentration or potential difference) to do useful work. Examples:Motors: transduce free energy into linear or rotary motionmyosin on actin in muscles, kinesin on microtubules in cells.Pumps: create concentration differences across membranessodium-potassium pump transports 3 Na+ ions out of the cell and 2 K+ ions into the cell in one cycle.Synthases: drive chemical reactions to synthesize biomoleculesATP synthase synthesizes the ATP molecules that are used by most of the molecular machines in the cells.
3 EnzymesAn extreme example: cataleseConsider the decomposition of hydrogen peroxide: H2O2 H2O + ½ O2DG0 = -41 kT so the reaction is highly favoured but due to a highactivation barrier it proceeds very slowly:for 1 M solution the rate is 10-8 M/s (reaction velocity)Adding 1 mM catalese into the solution increases the rate by 1012 !10-3 NA catalese molecules perform 104 NA hydrolisis reactions per sec.So 1 catalese molecule catalyses 107 reactions per sec. (rate: 10-7 s)H2O2 is produced in cells while eliminating free radicals. Because it istoxic, its rapid breakdown is important.More typical rates for enzymes are around 103 s-1
4 Simple model of enzyme reactions: Chemical reactions involving biomolecules are extremely complex.Free energy surface typically involves thousands of coordinates.Nevertheless a reaction usually proceeds along the path of leastresistance (called reaction coordinate) which allows a simple description.transitionstateA simple reaction: H + H2 H2 + H
5 An enzyme facilitates a chemical reaction by binding to the transition state and thereby reducing the activation energy, DG‡ (but not DG)Free energy surface along the reaction coordinateDG‡DG‡DGDGSubstrate Enzyme + substrate
6 Direction of the reaction is controlled by DG. By changing DG, we can reverse the direction. The reverse reaction does not necessarily followthe same reaction coordinate.
7 A schematic picture of an enzyme E binding to a substrate S: E + S ES EP E + PE+S: The enzyme has a binding site that is a good match for the subst. SES: In order to bind, S must deform which stretches a bond to breaking pt.EP: Thermal fluctuations break the bond producing an EP complexE+P: The P state is not a good match to the binding site, hence it unbinds, leaving the enzyme free for binding of the next substrate.
9 Enzyme Kinetics:Consider an enzyme reaction with rate constants k1, k2 and k3Assume:For a single enzyme, the reaction simplifies toLet probability of E unoccupied be PE and occupied PES = (1- PE)The rate of change of PE is
10 Assuming quasi-steady state, the time derivative vanishes, yielding Rate of production of P per enzyme:Reaction velocity for a concentration cE of enzymesMichaelis-Menten (MM) rule
11 Experimental data for pancreatic carboxypeptidase vmax=0.085 mM/sKM=6.4 mM
12 MM rule displays saturation kinetics, which has very general validity The key idea is the processing time for S PAt low substrate concentrations, there are more enzymes than S so thatthere is no waiting and hence v is proportional to cSAs cS is increased beyond KM, there is competition among S for accessto an enzyme, and they have to queue for processing.Maximum velocity of the reaction is determined by the number ofenzymes available and the processing rate (the rate limiting step)Modulation of enzyme activity:Regulate the rate of enzyme productionCompetitive inhibition: direct binding of another moleculeNoncompetitive inhibition: binding of a molecule to a second site
13 Recent developments (Adenylate kinase) We know very little about the actual dynamical processes occurring in enzymes.There are only a few simple cases where the physical mechanism is understood, e.g. oxygen binding in myoglobin.Adenylate kinase catalyzes:ADP + ADP ATP + AMPRecent work indicates that the rate limiting step is the enzyme conformation, and not the chemistry.
14 Molecular motors in muscles: myosin and actin For structure of the myosin and actin filaments in a myofibril, seeExperiment with optical tweezers demonstrates how myosin pulls an actinfilament when 1 mM of ATP is added to the system.From Finer et al. “Single myosin molecule mechanics” Nature, 1994.
15 Translocation of proteins across membrane: Proteins produced in the cell are exported outside through proteins in themembrane that form pores. To pass through the pore, the protein has tounfold. The reverse motion is suppressed because the chemicalasymmetries between inside and outside of the cell leads to a morestable protein structure outside.Factors contributing toasymmetry:pHion concentrationdisulfide bondingbinding of sugarsprotein catalyzes translocation
16 Macroscopic machines are deterministic, there are no random fluctuations But molecular machines operate in a noisy environment with lots ofrandom fluctuations.Consider the ratchets below as possible models for molecular machines.In G-ratchet the spring retracts during the passage but pops back afterIn S-ratchet a latch releases the spring after the passage, which stays upCan either ratchet pull a load f towards right doing useful work?
17 Unloaded G-ratchet makes no net motion, the loaded one moves to the left S-ratchet moves to the right if e > f.L, and to the left if e < f.L(no net motion if e = f.L)
18 Simple model for a perfect Brownian ratchet: (e >> kT) In the absence of any forces, the ratchet diffuses freely until it travels adistance L. FromThus the average speed is:Next we introduce a load f that pulls the ratchet to the left.The potential energy increases asin the interval [0, L]From Boltzmann distribution, theequilibrium probability will be like
19 We need an equation to describe the nonequilibrium probability distribution of the ratchet’s position (cf. Fick’s law and Nerst-Planck Eq.)| | | | | x Dx >> La-Dx/2 a a+Dx/2 a+DxThe net flux from a a+Dx depends on (1) the probabilities at those pointsand (2) the external forces. If there are N ratchets in our ensemble,the bins at a and a+Dx have ratchets.Assuming they move randomly, the net migration from left to right is
20 Next consider the flux due to an external force, Drift velocity due to this force:The number of ratchets moving from left to right:Adding the two contributions and dividing by Dt, we obtain for the fluxSteady state: flux is constant, and from continuity eq. it is also uniform(Smoluchowski eq.)
21 Since the potential is periodic, the solutions must be periodic too.First consider the equilibrium case:A possible nonequilibrium solution for the perfect ratchet is1. vanishes at x = L2. yields a constant flux3. hence solves the Smoluchowski eq.(Boltzmann dist.)
22 Average speed of the perfect ratchet: The average number of ratchets in the interval [0, L]The time it takes for these ratchets move isand the speed is
23 Too complicated to make sense, so consider the limits: Plot of the ratchetspeed / (2D/L)as a function ofz = fL/kTActivation barrier kicksin around fL = 5 kT→ activation barrier
24 Estimate the speed for a typical molecular machine For small molecules, ions etc.: R 1-3 Å, D 10-9 m2/sFor macromolecules, proteins: R 1-3 nm, D m2/sTypical length scale: L = 1 nmAverage speed: v = 2D/L = 0.2 m/s, (e.g. to move 200 steps takes 1 ms)The perfect ratchet assumption is that backward rate vanishesWhen the forward and backward rates become equal and nonet motion is possible.In summary:Molecular machines move by random walk over free energy surfaceTheir speed is determined by the activation energy barrier (but not e)
25 Molecular Recognition Cells contain thousands of different proteins.Each protein performs a specific task that may require its interaction witha specific biomolecule, e.g. DNA, another protein or a ligand.How does a protein distinguish that biomolecule from the thousands ofothers that are floating around the cell?The lock and key hypothesis of Fischer (1894), namely, shapecomplementarity of the interacting parts, provided the first clues.Going beyond the descriptive accounts of protein interactions usingcartoons to a quantitative accounts that can make predictions has onlybecome possible in the last decade thanks to the advances inStructure determination of complexes and single molecule exp’sComputer power and simulation methods
26 Molecular recognition covers a vast area of research Enzyme functionProtein-ligand interactions: binding of a ligand changes theconformation of a protein enabling its function, e.g. ligand-gated ionchannels, oxygen binding to hemoglobin.Protein-protein interactions: e.g. formation of protein complexes(tertiary structure), signal transduction across membrane, protein transportand modificationProtein-DNA (or RNA) interactions: reading and duplication of DNA,protein manufacturingProtein interactions with non-native peptides: e.g. toxins from thevenomous animals (spiders, snakes, scorpions, snails)Protein interactions with chemical compounds: e.g. drugs
27 Experimental methods: Structure determination of complexes using x-ray diffraction or NMRMeasurement of dissociation (or binding) constants.(mM range: weak binding, μM range: intermediate, nM range: strong)High-throughput screening (automated testing of large number ofcompounds to discover new drugs)Theoretical methods:Docking methods (popular in “in silico” drug design)Monte Carlo methods: search for the free energy minimum using theMetropolis algorithmBrownian dynamics simulations: water is treated as continuum andprotein is rigid, but simulations are fast enough to observe dockingMolecular dynamics simulations: realistic representation but too slowto observe docking
28 Crystal structure of the barnase (blue) - barstar (green) complex The unbound conformations are superimposed in light blue and orange.
29 Close up view showing the side chain pairs in the hot spot. In the complex: barnase (blue) - barstar (green)Comparison ofthe two structuresshows theimportance of sidechain flexibility
30 Docking methodsThere are various docking methods that search for the free energyminimum of a protein-macromolecule system. The basic ingredients are:A phenomenological energy functional. Typically consists of:electrostatic, Lennard-Jones, hydrogen bond, solvation and entropicterms. It is parametrized using a training set.A search algorithm. Two common methods employed:1. Random search using the Monte Carlo method2. Systematic search using a grid over the active siteIn the current docking methods, ligand flexibility (mainly torsion angles) isalso taken into account (target protein is still rigid). Here geneticalgorithms provide a very efficient tool (different conformations correspondto mutations). AutoDock is the most popular method at present.
31 Computer simulation of protein interactions Protein association can be broadly divided into two stages:Diffusional motion until they form an encounter complexNon-diffusional rearrangement process leading to the final boundcomplex.The first stage could take quite a long time (ms), so it is neither possiblenor desirable to use molecular dynamics. Brownian dynamics (BD) is thenatural tool for this stage.The second stage involves conformational changes in the protein, andalso dehydration and rehydration of water molecules. Thus a microscopicdescription that treats all the atoms in the system is necessary at thisstage, which is provided by molecular dynamics (MD).The focus is, however, on the binding. Can we avoid the BD stage?
32 Molecular dynamics combined with docking Test study ingramicidin channel:1. Find the initialgramicidin channel-organic cationconfigurationfrom AutoDock2. Then employthis in MDsimulations
33 Organic cations that bind to the gramicidin channel
35 Calculation of free energy profiles for ions Potential of Mean Force (PMF) of a molecule is calculated using the channel axis (z) as the reaction coordinateThe PMF is obtained from the Boltzmann factor by measuring the z coordinates of the moleculeUmbrella samplingA harmonic potential is used to constrain the molecule at various points on the channel axis (typical interval, fraction of an Å), and its z coordinate is sampled during MD simulationsThe z distributions are unbiased and combined to obtain the PMF profile along the z axis.
36 Free energy profiles (potential of mean force, PMF) of cations determined from umbrella sampling calculations
37 Binding constantsBinding constant is obtained by integrating the free energy of the ligandin a volume around the binding sitewhere we have approximated the volume with a cylinder of radius R.Using the PMF’s, we can estimate the binding constants:Methylammonium: K = 4.1 M-1 (exp: 4.4 M-1)Ethylammonium : K = 0.2 M-1 (exp: ~ 0)Formamidinium: K = 0.6 M-1 (exp: 23 M-1) (there is a deeper site)
38 Drugs from toxins Development of new drugs is at an all time low. Major problem: finding new compounds with high specificity and affinity.High hopes from “in silico drug design” methods.k-conotoxin bound to K+ channelExample: Conotoxins as drug leadsConotoxins are small peptides found in the venom of cone snails thatselectively bind to specific ion channels with high affinity.It is estimated that there are over 50,000 different conotoxins.Already a few new drugs have been developed from conotoxins.The potential for development of further drugs is enormous.
39 Exp. structure of the KcsA*- charybdotoxin complex (NMR) Important pairs:Y78 (ABCD) – K27D80 (D) – R34D64, D80 (C) - R25D64 (B) - K11K27 is the poreinserting lysine –a common thread inscorpion and othertoxins.K11R34
40 Developing drugs from ShK toxin for autoimmune diseases ShK toxin binds to Kv1.3 channels with picomolar affinity, hence a goodcandidate for treatment of autoimmune diseases.ShK toxin has threedisulfide bonds and three other bonds:D5 – K30K18 – R24T6 – F27These bonds confer ShK toxin an extraordinary stability not seen in other toxinsNMR structure of ShK toxin
41 Kv1.3-ShK complex (Docking + MD) Monomers A and C Monomers B and D
42 Pair distances in the Kv1.3-ShK complex (in A) Kv1.3 ShK HADDOCK MD aver. Exp.D376–O1(C) R1–NS378–O(B) H19–N **Y400–O(ABD) K22–N **G401–O(B) S20–OH **G401–O(A) Y23–OH **D402–O(A) R11–N *H404-C(C) F27-C" *V406–C1(B) M21–C" *D376–O1(C) R29–N *** strong, * intermediate ints. (from alanine scanning Raucher, 1998)R24 (**) and T13 and L25 (*) are not seen in the complex (allosteric)
43 Convergence of the PMF for the Kv1.3-ShK complex
45 Comparison of binding free energies of ShK to Kv1.x Binding free energies are obtained from the PMF by integrating it along the z-axis.Complex DGwell DGb(PMF) DGb(exp)Kv1.1–ShK ± ± 0.1Kv1.2–ShK ± ± 0.1Kv1.3–ShK ± ± 0.1Excellent agreement with experiment for all three channels, which provides an independent test for the accuracy of the complex models.
46 Average pair distance as a function of window position ** denotes strong coupling and * intermediate coupling***********