Sedimentation Velocity Analysis of Interacting Systems using c(s) Peter Schuck

Transcription

1 Sedimentation Velocity Analysis of Interacting Systems using c(s) Peter Schuck

2 simulated sedimentation of rapidly self-association systems monomer/dimer monomer/trimer 5 concentration (Kd) 4 2 concentration (Kd) radius (cm) radius (cm) concentration (Kd) radius (cm) concentration (Kd) radius (cm) large amount of data for fast reactions: provides information on complete isotherm of s w (c) for range of c < c load

3 Boundary shapes for a heterogeneous reaction with different binding kinetics effect of protein size A+B AB, concentration equimolar at K D with koff = e-2, e-, e-4, e-5 A = 0 kda, B = 50 kda A = 90 kda, B = 50 kda bimodal boundary for 2component mixture smaller size diminishes characteristic features from reaction kinetics

4 Boundary shapes for a heterogeneous reaction with different binding kinetics c(s) curves from slow to fast reactions log(k off ) instant. c(s) s-value (S) c(s) curves can be used to diagnose kinetic regime J. Dam, C.A. Velikovsky, R.A. Mariuzza, C. Urbanke, P. Schuck (2005) Sedimenetation velocity analysis of heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient distributions c(s). Biophysical Journal 89:69-64

5 Boundary shapes for a heterogeneous reaction with different binding kinetics limitations diffusion does reduce information on kinetics, reaction scheme, therefore small proteins will be more difficult interpretation of boundary broadening is exquisitely sensitive to everything that made already the determination of molar mass difficult: (micro)-heterogeneity, multiple conformations, glycosylation sensitivity only over a narrow range of time-constants, given by the sedimentation time: best for ~ /sec describing kinetic models possible, but depend on parameters that are frequently ill-defined by the experiment need less detailed models, which permit to quantitatively analyze interactions in SV based on the strength of SV: separation in mixtures J. Dam, C.A. Velikovsky, R.A. Mariuzza, C. Urbanke, P. Schuck (2005) Sedimenetation velocity analysis of heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient distributions c(s). Biophysical Journal 89:69-64

6 c(s) curves from analysis of self-association and hetero-association slow ( /sec) fast (0.005/sec) self-association hetero-association c(s) analysis of reaction boundary may result in broad distribution rms error can go up apparent f/f 0 may be smaller than real f/f 0 (sometimes < ) for slow systems, c(s) gives peak positions of species for fast systems, c(s) gives undisturbed boundary and approximations of asymptotic reaction boundary

7 Boundary shapes for a heterogeneous reaction with different binding kinetics c(s) curves from slow to fast reactions log(k off ) instant. c(s) s-value (S) c(s) curves can be used to diagnose kinetic regime c(s) shows either populations of species, or something in between how does this work? peak s-values do not coincide well with species s-values why? what do these curves mean, and how can they be used quantitatively?

8 Constant Bath Approximation: Krauss, Pingoud, Boehme, Riesner, Peters, Maass. Eur. J. Biochem. (975) Example: A + B AB ( ) a a = D r s ω r a + q t r r r r r 2 2 a a a ( ) b b = D r s ω r b + q t r r r r r 2 2 b b b ( ) c c = D r s ω r c + q t r r r r r 2 2 c c c coupled with reaction fluxes q a =q b = q c =k on ab k off c K D =k on /k off (K D = ab/c for instant. reaction) system of Lamm equations what if A was small and would not exhibit a spatial gradient? a r = 0 fast reaction c r = Ka b r add Lamm eq. for b + c ( ) 2 2 ( ) ( ) b c b + = D + KaD r s + Kas ω r b + q + q t t r r r r r b c b c b c introduce new variable for b+c β β = b+ c * β * 2 = D r s ω ( r 2 β) t r r r r r non-interacting single-species Lamm equation D D + KaD s + Kas =, s = + Ka + Ka * b c * b c J. Dam, C.A. Velikovsky, R.A. Mariuzza, C. Urbanke, P. Schuck (2005) Sedimenetation velocity analysis of heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient distributions c(s). Biophysical Journal 89:69-64

9 Constant Bath Approximation: cannot be strictly fulfilled, since A, B, and C are always in mass action law equilibrium there will be a gradient of A. how large is this gradient? can this be a realistic and useful limiting case for protein-protein interactions? address this questions using Lamm equation solutions for instantaneous reaction of proteins A = 00 kda, B = 200 kda s A = 7 S, s B = 0 S, s AB = S equimolar concentration at Kd = 0 um two boundaries: free A at 7 S reaction boundary signal (OD) fit with 2 species model: s = 7.02 S (expect 7.0 S) s 2 =.5 S (s* =.4 S) M A = 8.4 kda ( 7 % ) M B = 98 kda ( 25%) rmsd = residuals radius (cm) diffusion coefficients quantitatively not very precise, but s-values very good excellent description why?

10 indeed the first boundary is of free A there is free A in the fast boundary there is some gradient of free A however: free A changes from 76 to 97 % where free B goes from 0 to 90% mass action law causes the fractional occupation of B (i.e. C/(B+C)) to vary only between 0 to 5% this means that the gradient in A does not lead to a big change in the pseudo-species B+C constituting the fast boundary in the the constant bath theory. Therefore, overall description good. (the dispersion in s caused by the finite range of C/(B+C) does show up in a slight increase in D; how to treat the dispersion in terms of range of s-values will be topic of Gilbert-Jenkins theory later) the big result here is that, in a first approximation, diffusion of the reaction boundary is very similar to that of a single species. consistent s-values of the fast boundary with s* from constant bath theory

11 mass action law: ab = ck D time-average τ a τ b = τ c K D concentration concentration concentration concentration concentration concentration concentration radius dc/ds A + B AB with s < s < s Schematics of diffusion-free sedimentation of an rapidly reacting system sedimentation coefficient

12 Gilbert & Jenkins, Nature, 956 linear geometry and radial-independent force 2 i i i = Di v 2 i + ji c c c t x x The reaction fluxes j i follow mass conservation with j A = j B = j C = j, and it is assumed that all species are in instantaneous equilibrium following mass action law c c K = c A B C A change of variables from spatial and time coordinates x and t to the velocity v = x/t and the inverse time w = /t 2 c i ci c i j ( v vi) + w + Di v w 2 = v w the limit of infinite time (w 0) ca cb cc ( v va) = ( v vb) = ( v vc) v v v which can be solved for c A (v), c B (v), c C (v) stable boundaries with asymptotic Schlieren patterns ds/dv j 0 with w 0 no net reaction at infinite time (Gilbert & Jenkins, Proc. Royal Soc. London A, 959)

13 summarized by Fujita (975) s A s B s C s A s B s C quantitative application in AUC has been largely dormant: difficulty in measuring asymptotic boundary; computational limitation at the time

14 c(s) approximately deconvolutes diffusion from reaction boundaries D 0 Gilbert-Jenkins theory predicts diffusion-free asymptotic boundaries at D = 0 should be equivalent! J. Dam, P. Schuck (2005) Sedimentation velocity analysis of heterogeneous protein-protein interactions: Sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory. Biophysical Journal 89:65-666

15 equimolar mixture c(s) c (K D ) A B s-value (S) 0.25 absorbance [OD]

16 equimolar mixture c(s) c (K D ) 0. A B s-value (S). absorbance [OD]

17 equimolar mixture c(s) c (K D ) A B s-value (S) absorbance [OD]

18 5absorbance [OD] equimolar mixture c(s) c (K D ) A B s-value (S)

19 equimolar mixture c(s) c (K D ) A B s-value (S) 5 absorbance [OD]

20 larger species constant at K D c(s) c (K D ) A B s-value (S).5 absorbance [OD]

21 larger species constant at K D c(s) c (K D ) A B s-value (S).5 absorbance [OD]

22 larger species constant at K D c(s) c (K D ) A B s-value (S) absorbance [OD]

23 0.2 c(s) absorbance [OD] larger species constant at K D 0 c (K D ) A B s-value (S)

24 larger species constant at K D c(s) c (K D ) A B s-value (S) 5absorbance [OD]

25 smaller species constant at K D c(s) c (K D ) A B s-value (S).5 absorbance [OD]

26 smaller species constant at K D c(s) c (K D ) A B s-value (S).5 absorbance [OD]

27 smaller species constant at K D c(s) c (K D ) A B s-value (S) absorbance [OD]

28 4absorbance [OD] smaller species constant at K D c(s) c (K D ) A B s-value (S)

29 smaller species constant at K D c(s) c (K D ) A B s-value (S) 0 absorbance [OD]

30 pretty consistent, enough to convince conceptually OK but not 00% identical (maximum entropy regularization, linear Lamm equation model) how to utilize that quantitatively? INTEGRATE weight-average s-value of of boundary components, and amplitudes compare with theoretical values from GJT, also integrated corresponds to a BOUNDARY SHAPE analysis without requirements of Lamm equation modeling in global model with weight-average in global model with different signals/wavelengths J. Dam, P. Schuck (2005) Sedimentation velocity analysis of heterogeneous protein-protein interactions: Sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory. Biophysical Journal 89:65-666

31 what do we do with c(s) curves? slow multi-signal analysis concentration series s w s w,fast a slow a fast

32 THEORY - what to expect? - sw is well-known s w approaches s AB only for precise stoichiometry, otherwise far away

33 THEORY - this is s fast from GJT s fast approaches s AB in any case with c A or c B >> K D

34 c/k D sw (S) sfast (S) sw (S) sfast (S) sw (S) sfast (S) equimolar fix b = K D fix a = K D c/k D 0 00 c/k D 0 population fast/slow (fringes) c/k D 0 00 c/k D 0 00 c/k D population fast/slow (fringes) population fast/slow (fringes)

35 So far, examples have shown clearly separate boundaries. how does it work for a small species with high D? real test for deconvolution of diffusion in c(s)

36 A (25 kda, 2.5 S) + B (40kDa,.5 S) AB (5 S), k off = 0.0/sec c(s) or ls-g*(s) (/S) 2 0 c (K D ) A B s-value (S) absorbance [OD] 0.05 small species

37 c(s) or ls-g*(s) (/S) 2 0 c (K D ) A B s-value (S) 0.25 absorbance [OD]

38 c(s) or ls-g*(s) (/S) 2 0 c (K D ) A B s-value (S) absorbance [OD]

39 2 2 0 c (K D ) A B s-value (S) absorbance [OD] c(s) or ls-g*(s) (/S)

40 c (K D ) 0. A B s-value (S) absorbance [OD] c(s) or ls-g*(s) (/S)

41 concentration (µm) concentration (µm) s w and s w,fast (S) population fast/slow (fringes)

42 K d =. µm, c = 0.,,, 0, 0 µm slow k off = /sec Kd = 0 µm, c =,, 0, 0 µm fast k off = 0.0/sec c(s) (/S) c(s) (/S) ls-g*(s) (/S) 0.2 G(s) ls-g*(s) (/S) G(s) sedimentation coefficient (S) sedimentation coefficient (S)

43 Summary of c(s) analysis of reacting systems: c(s) approximately deconvolutes diffusion from reaction boundaries c(s) distributions are approximations of the asymptotic boundary from Gilbert- Jenkins theory Gilbert-Jenkins modeling of c(s) can be quantitatively used to model isotherms of reaction boundary, in addition to s w (c) higher confidence in K D and s complex characteristic information on stoichiometry takes advantage of bimodal boundary structure, without need for detailed Lamm eq. modeling (can of course be used, afterwards, too, using parameter estimates) J. Dam, P. Schuck (2005) Sedimentation velocity analysis of heterogeneous protein-protein interactions: Sedimentation coefficient distributions c(s) and asymptotic boundary profiles from Gilbert-Jenkins theory. Biophysical Journal 89:65-666

44 Summary of c(s) based analysis methods for reacting systems second-moment method: mass balance a) integrate all c(s) peaks at once b) model weight-average (signal-average) s w (c) with isotherm model for fast reactions: bi-modal boundary according to Gilbert-Jenkins theory a) integrate fast and slow c(s) peaks b) model signal-average s(c) of fast boundary with isotherm model c) model amplitudes of fast and slow boundary with isotherm model for slow reactions: baseline-separated peaks reveal species populations a) integrate c(s) peaks individually b) species population isotherm for heterogeneous interactions: global multi-signal analysis for multi-component c(s) a) for determining stoichiometry of extended associations b) requires either k off < 0-4 /sec or c > fold K D no detailed boundary shape info used can deal with impurities outside range of interacting system useful to build model (stoichiometry, K D,s complex ) concentration series global modeling of isotherms

45 isotherm analysis weight-average s-value tot sw( c ) = å c s c ( ) free free free free A A B B A B AB l tot tot e ca s + e cb s + e + e KcA cb s sw( ca, cb ) = tot tot e c + e c tot A tot i i free free free A A B c = c + Kc c tot free free free B = B + A B c c Kc c i A A B B equimolar concentrations titration configuration only mass balance consideration, no boundary shape slow and fast s w (c) for self-association/hetero-association stoichiometry for hetero-association K D, limiting s-value s complex hydrodynamic shape?

46 isotherm analysis population isotherm free A tot tot free tot tot tot tot A B B A B AB A B c ( c, c ), c ( c, c ), c ( c, c ) tot A free free free A A B c = c + Kc c tot free free free B = B + A B c c Kc c for k off < 0-4 /sec exploit boundary shape to distinguish species for self-association/hetero-association K D

47 isotherm analysis Gilbert-Jenkins theory-based isotherms tot tot tot tot tot tot undisturbed A B reaction A B w, reaction A B c ( c, c ), c ( c, c ), s ( c, c ) model based on a) constant bath theory diffusion of reaction boundary approximately like non-interacting species b) c(s) can approximately deconvolute diffusion from reaction boundary c) prediction from Gilbert-Jenkins theory on asymptotic boundary (D = 0) d) c(s) asymptotic boundary for k off > 0 - /sec exploit bimodal nature of boundary for hetero-association K D s complex (hydrodynamic shape)