Macromolecular Interactions the equilibrium element Physical Reality Quantitative P + L PL K d,overall K d,overall = [P][L] [PL]
Driving force is difference in ground state free energies ΔG f o ΔG f o = RT lnk d,overall ΔG f o = ΔH f o - TΔS f o
Macromolecular Processes the time element Physical Reality Quantitative E + L k 1 k 1 EL k 1 k 1 Rates, e.g., k 2 E + S ES E + P d[el] dt d[p] dt
Enzymes This week - Enzymes as Individuals Next week - Enzymes in Complex Systems Susan Miller, smiller@cgl.ucsf.edu, GH S512B
Key Themes/Goals This Week - Free energy profiles - Rates what we measure & how we interpret - Observed or macroscopic constants - Relation to microscopic constants and energetics of models - Start with example of rates of ligand binding - Structural features/interactions that contribute to enzyme catalysis - Design how well do we really understand?
In wk1 often an energy barrier for molecular process K d = [E][L] [EL] ΔG f o = RT lnk d Rate constants defined by difference in free energy of GS and TS, ΔG k 1 = k B T/h exp{ ΔG 1 /RT) k 1 = k B T/h exp{ ΔG 1 /RT)
Equilibrium constant related to rate constants K d = [E][L] d[el] [EL] rate = = dt -d[e] dt At equilibrium 0 = d[el] dt = k 1 [E][L] k 1 [EL] k 1 [E][L] = k 1 [EL] [E][L] K d = [EL] = k 1 k 1
What is the predicted time-dependent behavior for E & EL for the simple 1-step binding reaction? d[el] rate = = dt -d[e] dt rate = d[el] dt = k 1 [E][L] k 1 [EL]
First, let s consider a 1 st order irreversible process: EL k -1 E + L rate = d[el] dt = k -1 [EL] = d[e] dt Rate units = M s 1 st order rate constant units = s -1 1 st order in [ES] & overall unimolecular
How does [EL] vary with time? rate = d[el] dt = k -1 [EL] = d[e] dt [EL] decays exponentially
How does [EL] vary with time? amplitude Half life: time pt when [EL] = [EL 0 ]/2 t 1/2 = (ln2)/k -1 constant over full reaction Time constant: τ = 1/k -1
k Now a 2 nd order irrev. process: E + L EL 2 nd order rate constant units = M -1 s -1 rate = d[e] - = k[e][l] dt = d[el] dt Second order overall, first order each in [E] and [L] In the limiting case where [E 0 ] = [L 0 ], the integrated rate equation is: 1 [E] = 1 [E 0 ] + kt 1/[E] 5 slope = 4 k (M -1 s -1 ) 3 2 1 intercept = 1/[E ] 0 0 0 1 2 3 4 5 time (s) Plotted in the normal sense of [E] vs time
k 2 nd order irrev. process: E + L EL we see that t 1/2 is not constant, but increases with time [E] (M) 1.0 10-5 8.0 10-6 6.0 10-6 4.0 10-6 2.0 10-6 t 1/2 [E 0 ] = [L 0 ] k = 10 7 M -1 s -1 t 3/4 2*t 1/2 t 7/8 3*t 1/2 [E] (M) 1.0 10-5 8.0 10-6 6.0 10-6 4.0 10-6 2.0 10-6 [L 0 ] = 2*[E 0 ] [L 0 ] = 5*[E 0 ] 0.0 10 0 0 0.04 0.08 0.12 0.16 0.2 time (s) 0.0 10 0 0 0.01 0.02 0.03 time (s) However, as [L 0 ] increases over [E 0 ], t 1/2 approaches a constant value and the reaction becomes pseudo-first order
2 nd order process under pseudo-first order conditions: E + L rate = k EL d[e] - = k[e][l] dt either [L 0 ] 10*[E 0 ] or [E 0 ] 10*[L 0 ] For [L 0 ] 10*[E 0 ] since [L 0 ] only decreases by 10% over the whole reaction, k obs = k[l 0 ] is ~ constant and rate = d[e] - = k dt obs [E] [E] = [E 0 ] e -(kobs)t Single exponential decay as we saw before
2 nd order process under pseudo-first order conditions: More importantly, k obs is linearly dependent on [L 0 ] 20 15 k obs (s -1 ) 10 slope = 5 k (µm -1 s -1 ) 0 0 2 4 6 8 10 [L 0 ] (µm)
Rate of approach to equilibrium: ES k f EP k r rate = d[es] - = k dt f [ES] -k r [EP] = d[ep] dt Assume initial [ES 0 ] is conserved, [EP] = [ES 0 ] - [ES] d[es] - = (k dt f + k r )[ES] - k r [ES 0 ]
Approach to equilibrium: ES k f EP k r which rearranges to: d[es] - = (k dt f + k r )[ES] - k r [ES 0 ] d[es] dt + (k f + k r )[ES] = k r [ES 0 ] And when multiplied by exp(k f + k r )t and rearranged to:
Approach to equilibrium: ES k f EP k r integrates to: [ES] = k r (k f + k r ) [ES 0 ] + k f (k f + k r ) -(kf + kr)t [ES 0 ] e equilibrium end point at t = amplitude A single exponential process with k obs = k f + k r i.e. monophasic 1 st order process
Approach to equilibrium: Comparing eqns for [ES] and [EP]: ES k f k r EP [ES] = [EP] = k r (k f + k r ) k f (k f + k r ) [ES 0 ] [ES 0 ] k r [ES] eq = [ES 0 ] (k f + k r ) k f [EP] eq = [ES 0 ] (k f + k r ) - + k f (k f + k r ) k f (k f + k r ) amplitudes Eq. endpoints reflect K eq = k f /k r = [EP] eq /[ES] eq -(kf + kr)t [ES 0 ] e -(kf + kr)t [ES 0 ] e To determine k f, k r and K eq = k f /k r, ES and EP must have a measurable property that differs, e.g. NMR chemical shift [ES] or [EP] 1.0 0.8 0.6 0.4 0.2 k obs and amplitudes same but k obs = 1 s -1 k f = 0.6 s -1 k r = 0.4 s -1 0.0 0 2 4 6 8 10 time (s)
Approach to equilibrium in a binding reaction: E + L k on k off EL Under pseudo-first order conditions, e.g. [L 0 ] 10*[E 0 ], the eqns are as above but with k f = k on [L 0 ] in place of k f k off k f' [E] = [E 0 ] + [E 0 ] e (k f' + k off ) (k f' + k off ) k f' k f' [EL] = [E 0 ] - [E 0 ] e (k f' + k off ) (k f' + k off ) -(kf' + koff)t -(kf' + koff)t Now, both k obs and the equil. end points vary with [L 0 ] [E] (nm) k on = 10 7 M -1 s -1 k off = 1 s -1 1.0 0.8 0.6 0.4 [L 0 ]= 10-8 M 3*10-8 M 10-7 M 0.2 3*10-7 M 10-6 M 0.0 0 1 2 3 4 5 6 7 8 time (s) k obs [EL] (nm) k = 10 7 M -1 s -1 on 1.0 k = 1 s -1 off [L ]= 0 10-6 M 0.8 3*10-7 M 0.6 10-7 M 0.4 0.2 3*10-8 M 10-8 M 0.0 0 1 2 3 4 5 6 7 8 time (s)
Approach to equilibrium in a binding reaction (con t): E + L k on k off EL Since k obs = k f + k off and k f = k on [L 0 ] k obs shows a linear dependence on [L 0 ], i.e.,k obs = k on [L 0 ] + k off 12.0 k obs (s -1 ) 10.0 8.0 6.0 4.0 slope = k on (µm -1 s -1 ) and the dissociation constant K d = k off /k on = s -1 /(M -1 s -1 ) = M 2.0 intercept = k off (s -1 ) 0.0 0 0.2 0.4 0.6 0.8 1 1.2 [L 0 ] (µm)
To summarize: Single elementary step processes exhibit monophasic, i.e., single exponential behavior defined by a macroscopic rate constant k obs (s -1 ): Irreversible Approach to equilibrium 1 st order reactions unimolecular steps ES k EP ES k f EP k r forward steps 2 nd order overall, 1 st order in both [E] and [L] E + L k EL E + L k on k off EL
What are the time-dependent and concentration dependent behaviors for 2-step reactions composed of a binding step followed by a chemical reaction or conformational change? E + S k 1 k 2 ES k 3 EP E + S k 1 k 2 ES k 3 k 4 EP
Biphasic reactions: 2 nd order + 1 st order E + S k 1 k 2 ES k 3 EP A 2-step process like this can exhibit 2 exponential phases IF there is a measureable signal for each phase or both steps are partially rate-limiting. However, in many cases, the only measureable signal occurs in one step typically the 2 nd step for ES -> EP. If pseudofirst order conditions are used, i.e., [S o ] 10*E o, then k obs shows a hyperbolic dependence on [S o ] k max k obs for appearance of EP 8.0 k obs = k 3 k 1 S 0 k 2 + k 3 + k 1 S 0 = k max S 0 K 1/2 + S 0 k obs (s -1 ) 6.0 4.0 When both k max = k 3 K 1/2 = (k 2 + k 3 )/k 1 2.0 0.0 0 0.004 0.008 K 1/2 [S 0 ] (M) steps are reversible, two new diagnostics arise
Biphasic reactions: 2nd order + 1st order k1 k2 E+S k3 k4 ES appearance of EP k max E decays biphasically -1 10.0 obs 6.0 10-6 (s ) E ES EP 8.0 10-6 k [species] (M) 1.0 10-5 4.0 10-6 k 3 K 1/2 0 0.1 0.2 0.3 time (s) 0.4 0.5 0.0 4 y-intercept = k 5.0 2.0 10-6 0.0 100 =k +k max 0 = (k + k )/k 2 0.004 K 1/2 4 3 0.008 [S ] (M) 0 all species reach equilbrium end pts kobs for formation of [EP] exhibits a hyperbolic dependence on [S0] but with a positive y-intercept 1 EP
Limiting Values of 2 nd order binding rate constants: k on E + L k on k off EL In some instances, the ΔG for association is 0. In this case, the process is diffusion limited, i.e. it is simply limited by how fast the two molecules can diffuse together. k on, diff lim = (M -1 s -1 ) k on, diff lim 10 10 M -1 s -1 4 π (r E + r L )(D E + D L ) N o 1000 cm 3 D E = k B T 6 π η r E r E, r L are radii of E & L in cm N o is Avogadro s # D E, D L are the Stokes-Einstein diffusion coefficients for E & L in cm 2 /s Test for diffusion limited: measure rate as f(η) η = viscosity in dyn-s/cm 2 = poise Models developed by Alberty, Hammes & Eigen (1958) and later by Chou (1970 s) see G. Zhou & W. Zhong (1982) Eur. J. Biochem. 128, 383 which compares the models and lists relevant references
Now a story about ligand binding the right design of an experiment reveals more complex mechanism Imatinib (Gleevec) binding to Abl and Src Imatinib Imatinib ATP-competitive inhibitor designed to mimic and bind in the extended ATP binding site of tyrosine kinases
Abl & Src are closely related tyrosine kinases involved in cellular signalling pathways - Src is the 1 st proto-oncogene - A BCR-Abl fusion underlies a type of leukemia (CML) Imatinib is a successful drug for CML in part because it is both potent and highly selective (~ 3000-fold), K d,overall ~10 nm for Abl vs K d,overall ~30 µm for Src DFG motif in ATP binding pocket initially postulated to prevent binding of imatinib to Src as in Abl, but recent structure shows same binding mode with most contact residues conserved
Seeliger et al, 2007 Structure 15, 299
Monitored decrease in protein Trp fluorescence to measure rates for Imatinib binding to Abl & Src under pseudo-first order conditions 50 nm Abl or Src + 12.5 µm Imatinib Linear concentration dependence consistent with 1-step binding model. Abl: k on = 0.146 µm -1 s -1 Abl: k off ~ 0.002 s -1 Src: k on = 0.003 µm -1 s -1 Src: k off ~ 0.11 s -1 Seeliger et al, 2007 Structure 15, 299
Propose Conformational Selection Model Difference in this K eq lowers binding affinity to Src Seeliger et al, 2007 Structure 15, 299
As you read the paper for Wed (2014 Agafonov et al, NSMB 21, 848) consider Only one kinetic trace is shown here, but several are shown in Agafonov. Consider the differences in presentation and what information is lost here. Consider the concentration ranges examined in both Consider the 2-step behaviors described above relative to the description in the paper
Switch Gears to Enzymes and their Energetic Profiles
Free energy determines fate: Applies to changes in chemical/ electronic structure, IF reaction path available ΔG o = RT lnk ΔG o = 1.4 logk (kcal/mol) ΔG o (std. states: 1 M, 298 K) thermodynamic driving force ΔG o = ΔH o TΔS o Changes in covalent bond enthalpies, internal entropy, molecularity, solvation
Reaction Paths Breaking Bonds - Transition States For a path involving a single elementary step: Transition State covalent bond(s) are partially broken/made Recall: Covalent bond enthalpies are large, e.g., C-H ~99 kcal/mol C-C ~83 kcal/mol C-O ~86 kcal/mol P-O ~90 kcal/mol O-H ~100 kcal/mol ΔG 1, ΔG -1 can be large
Reaction Paths Breaking Bonds - Transition States How large are ΔG values for uncatalyzed biological reactions? Again, experimentally determined rate constants defined by difference in free energy of GS and TS, ΔG k 1 = k B T/h exp{ ΔG 1 /RT) k 1 = k B T/h exp{ ΔG 1 /RT)
Reaction Paths Breaking Bonds - Transition States How large are ΔG values for uncatalyzed biological reactions? k 1 = k B T/h exp{ ΔG 1 /RT} ΔG non (kcal/mol) 39.7 33.3 29.3 25.4 24.3 19.1 Radzicka & Wolfenden 1995 Science Vol. 267
Reaction Paths Breaking Bonds - Transition States Enzymes catalyze specific reactions by providing a lower energy path - increased rate: larger k 1 & k -1, lower ΔG 1 & ΔG -1 - while ΔG o remains unchanged noncatalyzed catalyzed Profile drawn for standard state [A] = [Q] = 1 M
Reaction Paths Breaking Bonds - Transition States How large are ΔΔG values for biological reactions (ΔG non - ΔG cat )? Enzyme k non (s -1 ) k cat (s -1 ) ΔG non (kcal/mol) ΔG cat (kcal/mol) Rate constants from A. Radzicka & R. Wolfenden 1995 Science Vol. 267 ΔΔG (kcal/mol) OMP decarboxylase 2.8 x 10-16 39 39.7 15.7 24.0 Staphylococcal nuclease 1.7 x 10-13 95 35.8 15.1 20.6 Adenosine deaminase 1.8 x 10-10 370 31.6 14.3 17.2 AMP nucleosidase 1.0 x 10-11 60 33.3 15.4 17.9 Cytidine deaminase 3.2 x 10-10 299 31.2 14.4 16.8 Phosphotriesterase 7.5 x 10-9 2,100 29.3 13.3 16.0 Carboxypeptidase A 3.0 x 10-9 578 29.8 14.0 15.8 Ketosteroid isomerase 1.7 x 10-7 66,000 27.4 11.2 16.2 Triosephosphate isomerase 4.3 x 10-6 4,300 25.4 12.8 12.6 Chorismate mutase 2.8 x 10-5 50 24.3 15.5 8.8 Carbonic anhydrase 2.8 x 10-1 1 x 10 6 19.1 9.5 9.6 Cyclophilin 2.8 x 10-2 13,000 20.1 12.1 7.9
Reaction Paths Breaking Bonds - Transition States What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates? noncatalyzed catalyzed Profile drawn for standard state [A] = [Q] = 1 M
Reaction Paths Breaking Bonds - Transition States First note the differences in the free energy profiles Enzymes bind their substrates and products noncatalyzed catalyzed Profile drawn for standard state [A] = [Q] = 1 M
Reaction Paths Kinetics First consider kinetic impact of introducing binding steps For the noncatalyzed rxn, if Q init = 0 and Q removed - single elementary step - unimolecular 1 st order For the catalyzed rxn, If Q init = 0 and Q removed - 3 elementary steps - bimolecular binding 2 nd order - chemical step 1 st order - Q dissociation 1 st order 2 nd order 1st order d[a] rate = v = = k 1 [A] dt M/s s -1 rate =? Varies with [E] & [A] and depends on relative magnitudes of ΔG for each of the 3 steps
Reaction Paths Kinetics First consider kinetic impact of introducing binding steps Profile drawn for standard state [A] = [Q] = 1 M Profile with TS bind & TS off [A] 1 M, [Q] = 0
Reaction Paths Kinetics 2 nd order 1st order rate =? Varies with [E] & [A] and depends on relative magnitudes of ΔG for each of the 3 steps
Quantifying Overall flux or classic steady-state kinetics Common experimental design: A enzyme as catalyst Q Steady-state assumption: No change in concentration of any enzyme species during time of measurement So for: E T = E + EA + EQ d[e] dt = d[ea] dt = d[eq] dt = 0 Accomplished by: - setting [E T ] << [A T ] and [Q init ] = 0 - measuring initial velocity, v i = dq/dt = da/dt, i.e. <10% conversion where reaction remains ~ linear [A] [Q] v i time
*Most common* behavior of v i vs [A T ] & [E T ] at const. [E T ] << [A T ] V max (M/s) k cat (s -1 ) [E T ] (M) Hyperbolic dependence on [A] indicates minimal 2-step mechanism just as in 2-step single turnover k 1 k 2 k cat E + A EA E + Q v i = V max A T K M + A T = k cat E T A T K M + A T 2 nd order overall 1 st order in E, 1 st order in A 1st order in EA k obs = v i /E T = k cat A T K M + A T
Key Parameters at const. [E T ] << [A T ] v i = V max A T K M + A T = k 1 k 2 k cat E + A EA E + Q k cat E T A T K M + A T k obs = v i /E T = k cat A T K M + A T A T -> A T >> K M v i = V max - maximal velocity V max = k cat E T k cat - macroscopic 1st order rate constant (s -1 ) A T -> 0 A T << K M v i = (V max /K M ) A T = (k cat /K M )E T A T V max /K M - pseudofirst order rate const k cat /K M - macroscopic 2nd order rate constant (M -1 s -1 ) A T = K M v i = V max /2 K M = Michaelis constant [E] = E T /2 Σ[E bound ] = E T /2
Key Parameters are Macroscopic Constants de dea Applying the steady-state assumption: = = 0 dt dt to this minimal mechanism k 1 k 2 k cat E + A EA E + Q comparing v i /E T = where at face value k cat A T K M + A T = k cat A T k 2 + k cat + A T k 1 k cat appears to be a microscopic 1 st order rate constant (s -1 ) but both K M and k cat /K M are complex macroscopic constants k cat /K M = k 1 k cat k 2 + k cat K M = k 2 + k cat k 1 But further
the behavior of v i vs A T is macroscopic and consistent with many microscopic mechanisms, where k cat is also macroscopic, e.g. k 1 E + A EA EQ E + Q k 2 k 1 k 2 k 1 E + A EA EI EQ k 2 k cat /K M k 3 k 3 k 3 k 4 k cat E + A EA EQ k 4 k cat /K M k cat /K M k cat k 5 k 5 k cat k 5 k 6 E + Q k 7 E + Q The expressions for k cat and k cat /K M include microscopic constants for all steps within the brackets in each case. k cat /K M expressions include all steps from binding of A through the 1st irreversible step k cat expressions include all first order steps including chemistry, conformational changes, product dissociation k 2 k 3 k cat k 1 k 5 k 7 E + A EA E*Q E* + Q k cat /K M k 8 E K M expressions can be derived from ratio of k cat /(k cat /K M ) and are very complex
Key point: Without additional information, one cannot draw conclusions as to what types of processes (chemistry or binding, dissociation, conformational changes) limit the magnitudes of k cat and k cat /K M (i.e., are rate limiting) or how individual processes are altered by changes in protein structure (e.g., by mutation) k 1 E + A EA EQ E + Q k 2 k cat /K M k 3 k cat k 5 Consider the following energetic scenarios
Reaction Paths Breaking Bonds - Transition States First consider kinetic impact of introduction of binding step Recall, as [A] varies, the free energy of A varies, i.e. at [A] < K d, the binding equilibrium favors free E. Profile with TS bind & TS off [A] 1 M, [Q] = 0 A priori, cannot predict whether catalysis will have a higher energy TS than binding or dissociation, thus ΔG rev and ΔG off may be <, =, or > ΔG 1
What processes do k cat /K M and K M reflect? k cat /K M = k 1 k 3 k 2 + k 3 K M = k 2 + k 3 k 1 k = k B T/h exp{-δg /RT}, i.e. ln(1/k) ΔG k 1 k 2 k 3 k cat = k 3 E + A EA E + Q k cat /K M TS 3 ΔG TS 1,2 ΔG ΔG E+A EA E+Q E+A EA E+Q E+A EA E+Q Rxn Rxn Rxn k 2 >> k 3 k 2 ~ k 3 k 2 << k 3 k cat /K M = K M = K D k 3 k 2 /k 1 = k 3 K D both k cat & k cat /K M limited by TS 3, i.e., chemistry K M = k cat /K M = k 2 + k 3 k 1 k 1 k 3 k 2 + k 3 = 2K D k cat /K M only partially limited by TS 3 (chemistry) K M = k cat /K M = k 1 k 3 k 1 = k cat k 1 > K D k cat /K M limited only by TS 1,2, i.e, diffusion
and what happens if EP accumulates? k cat Full expressions: k cat /K M = k 1 k 3 k 5 k 1 k 2 k 3 E + A EA EQ k 4 k 5 E + Q k 2 (k 4 + k 5 ) + k 3 k 5 k cat /K M k cat = k 3 k 5 K M = k 2 (k 4 + k 5 ) + k 3 k 5 k 3 + k 4 + k 5 k 1 (k 3 + k 4 + k 5 ) TS 3,4 TS 5 k 5 << k 3, k 4 k cat = ( ) k 3 k 3 + k 4 k 5 fraction of bound enzyme that accumulates as EQ k cat reflects TS 5, i.e., product dissoc. ΔG TS 1,2 k cat /K M ~ k 5 k 2 k 4 /k 1 k 3 EQ forms in equil with E & EA E+A EA EQ Rxn E+Q k cat /K M also reflects TS 5 for product dissoc. K M = ( ) k 2 (k 4 ) k 4 k 1 (k 3 + k 4 ) = K D k 3 + k 4 In this case, K M < K D
Summary of Steady-State Points k cat, k cat /K M and K M are all macroscopic constants k cat may be limited or partially limited by any 1st order process including chemistry, conformational changes, product dissociation - chemistry often not fully rate limiting k cat /K M may be limited by any step reversibly connected to substrate binding. Diffusion limited means chemistry is faster than substrate dissociation K M is a Kinetic constant = [A] that gives half maximal velocity. It may be equal to K D, the dissociation constant for A, but often is not and does not a priori reflect the binding affinity of A for E K M = Σ (net flux from EA to E: back + fwd) k on
Next up Structures how do they contribute to catalysis
Long standing concepts [TS ] in the TS Noncovalent binding interactions maximized at TS E+A obs ΔG bind A EA additional favorable ΔG bind felt here and Q EQ unfavorable ΔG destab felt only in GS E+Q Environmental changes that make A more reactive than in solution