fredholm integral equations in the analysis of optical ... documents... · national institute for...
TRANSCRIPT
1
Fredholm Integral Equations in the Analysis of Optical Biosensor and AnalyticalOptical Biosensor and Analytical
Ultracentrifugation Data
Peter SchuckDynamics of Macromolecular AssemblyLaboratory of Bioengineering & Physical ScienceNational Institute for Biomedical Imaging and BioengineeringNational Institutes of HealthBethesda, Maryland
2
Acknowledgments
H B k iHacene BoukariLIMB, NICHD, National Institutes of Health, USA
Zhaochun Chen, Robert PurcellLID, NIAID, National Institutes of Health, USA
Jon Houtman, Lawrence SamelsonLCMB, NCI, National Institutes of Health, USA
Jim Arthos, Claudia Cicala, Anthony FauciLI NIAID National Institutes of Health USALI, NIAID, National Institutes of Health, USA
Thomas Vorup-Jensen, Timothy SpringerHarvard Medical School
Shlomo TrachtenbergShlomo TrachtenbergThe Hebrew University of Jerusalem
Richard LeapmanLBPS, NIBIB, National Institutes of Health, USA
Patrick Brown, Inna Gorshkova, Andrea Balbo, Juraj Svitel, Huaying ZhaoDMA, LBPS, NIBIB, National Institutes of Health, USA
3
Background
• data often report only indirectly on the quantity of interest ( )a τ ( )c R
max
min
( ) ( ) ( , )R
Ra c R g R dRτ = τ∫
2 12(1)
1( , ) a q Rg R a e−− ττ =
∑ πλ−=i
rrEtI )2sin()( 00
PM
4
Background
• data often report only indirectly on the quantity of interest ( )a τ ( )c R
max
min
( ) ( ) ( , )R
Ra c R g R dRτ = τ∫
2 12(1)
1( , ) a q Rg R a e−− ττ =
• Stephen Provencher (1982): fit integral equation to experimental DLS data with CONTIN
• use Tikhonov regularization (or maximum entropy), scale with F-statistics or
{ }2 2
α 2 1Δχ =
{ }2 2
0icMin a c c≥
− +αG B
0.15
10 20 300.00
0.05
0.10
c(R
)( )a τ
10 20 30
Stokes radius R (nm)
5
Optical Biosensing
Physical Principle of the Experiment
immobilization of protein to serve as binding sites(usually amine coupling to CM dextran)dextran)
detection with, e.g., surface plasmon resonance
• highly reproduciblehighly reproducible• great signal/noise ratio• flow-injection of analyte
( ) 0(1 / )( )1( , , , ) 1 off Dk c K t ta D off Ds K k c t K c e− + −−= +0 0b t t⎧ <
⎪⎪
For a single class of sites with equilibrium dissociation constant KD and off-rate constant koff (and on-rate constant kon = koff/KD), we expect piece-wise single-exponential traces:
( )ff
( )( , , , ) ( , , , ) off ck t td D off a D off cs K k c t s K k c t e− −=
0 0
0 0
( , , , ) ( , , , )
( , , , )D off a off D c c
d off D c
s K k c t s k K c t b t t t t
s k K c t b t t t
=⎪⎪ + < < +⎨⎪ + > +⎪⎩
with
Ann. Rev. Biophys. Biomol. Struct. 26(1997):541
6
Data: family of traces at different concentrations (obtained from the same surface)Fit with single discrete site model
experimental data are highly reproducible, but usually not single exponentials(unless truncated, or collected at extremely low signal/noise)
7
“God made the solid state, he left the surface to the devil” (Enrico Fermi)
min
ar fl
ow
f 0
lam
L in
n ch
ains
ilized
rece
ptor
und
ligan
d
ensemble of surface sites may exhibit a continuum of free energies of binding and rate constants
dext
ra+
imm
ob+
bou
8
kK
The Model Optical Biosensing
,max,max
,min ,min
( , )offD
D off
kK
off DK
tot D offk
P P K k dk dK= ∫ ∫Let P(KD,koff)dKDdkoff be the population of molecules with affinity between KD and KD+dKD, and with chemical off-rate constants between koff and koff+d koff
kK ,max,max
,min ,min
( , ) ( , , , ) ( , )offD
D off
kK
ooffob ff DK k
s off D Ds c t s k K c P k Kt dk dK= ∫ ∫…………….…………….…………….…………….…………….…………….…………….…………….
log(koff)
log(Kd)
0 0
0 0( , , , ) ( , , , )D off a off D c c
b t ts K k c t s k K c t b t t t t=
⎧ <⎪⎪ + < < +⎨⎪
without transport limitationlinear, independent sites
( ) ( )s c t s c t= ∑
with a single ‘effective’ transport rate constantcompartment model for transport, large system of rate equations coupled by competition for surface concentration of analyte
0 0( , , , )ff ff
d off D cs k K c t b t t t⎪ + > +⎪⎩
( ) 0(1 / )( )1( , , , ) 1 off Dk c K t ta D off Ds K k c t K c e− + −−= +
0/ ( )s t s rdc dt k c c j= − −
( )( )idsj k P K k s c k s∑
( , ) ( , )tot ii
s c t s c t= ∑
Svitel et al. (2003) Biophys. J. 84:4062-4077Svitel et al. (2007) Biophys. J. 92:1742
( )( , , , ) ( , , , ) off ck t td D off a D off cs K k c t s K k c t e− −=
( ), , , ,( , )ir on i D i off i i s off i i
ij k P K k s c k s
dt= = − −∑
9
naïve inversion would be ill-conditioned, amplify noise
need regularization (e g CONTIN)need regularization (e.g., CONTIN)
selects from all statistically indistinguishable fits the most parsimonious one
F-statistics sets an upper limit for acceptable rmsd values
maximum entropy or Tikhonov-Phillips regularizationmaximum entropy or Tikhonov Phillips regularization
with a Bayesian approach to incorporate prior expectation
( )222 d PMin signal signal dP
⎛ ⎞= + ⎜ ⎟∑ ∫α( )exp model 2
intdata po sMin signal signal d
p rP oP
d ri= − + ⎜ ⎟
⎝ ⎠∑ ∫α
• prior = constant: all binding parameters are equally likely
• prior = δ-function: there should be a single class of sites
• specific shape of prior could be from previous experiment, mutant, etc.
11
example for the effect of different prior: data with low information content
concentrations from 5 to 80 μM
prior = δ-function: gives the distribution closest to a single δ-peakbut not sacrificing quality of fit
prior = constant: calculates the broadest possible distribution
probing the space of possible distributions that fit the datathis flexibility of interpretation will be small for data with high signal/noise ratio
12
example for the effect of different prior: data with high information content
concentrations from 20 nM to 2 μM
prior = constant: calculates the broadest possible distribution
prior = δ-function: gives the distribution closest to a single δ-peakbut not sacrificing quality of fit
probing the space of possible distributions that fit the data
14
replicate experiments, same surface density of sites
Gorshkova et al., Langmuir 24 (2008) 11577
from computer simulations: with experimental signal/noise ratio of SPR data, we need > 3-5fold affinity difference in order to resolve two different sites
15
Application: Microheterogeneity Optical Biosensing
Binding of an antigen to its monoclonal antibody immobilized on a short-chainantibody immobilized on a short chain carboxy-methyl dextran surface. Analyte concentrations 1, 10, and 100 nM
main peak cannot be regarded as a single site
16
Application: Continuum of Sites Optical Biosensing
soluble integrin αX domain binding to immobilized fibrinogen:
native proteolyzed and guanidine-treated fibrinogen 0.28 to 10.6 μMnative, proteolyzed, and guanidine treated fibrinogen
as a model for leukocytes recognition of structurally decayed protein
Vorup-Jensen et al., PNAS (2005) 102:1614
17
Summary: Functional Heterogeneity of Surface Sites in Biosensing
accounting for surface heterogeneity gives (for the first time) an excellent description ofaccounting for surface heterogeneity gives (for the first time) an excellent description of experimental data in most cases
inversion may not be unique, but choice of Bayesian prior allows to probe the space of possible distributions consistent with the data
can discriminate conformational change and mass transport processes
allows naturally to distinguish between ‘specific’ and ‘non-specific’ sites
reveals ‘number of sites’
reveals micro-heterogeneity
appropriate for ‘complex’ functionalized surfaces, optimization of surfaces
implemented in shareware software EVILFITimplemented in shareware software EVILFIT
LIMITATION: Information content of experiment, essentially exponential kernel
19
analytical ultracentrifugation: diffusion with bias due to centrifugal force
Sedimentation Velocity
diffusion with bias due to centrifugal force
#particlesper volume
(σ = 0.025; δ = 0.0005)
20
random walk with bias, two non-interacting components
Sedimentation Velocity
(σ1 = 0.01; δ1 = 0.00005; σ2 = 0.025; δ2 = 0.0005)
21
Physical Principle of the Experiment
Sedimentation Velocity
• Nobel Prize 1926
laser interferometry (refractive index ~ mass)
• Nobel Prize 1926
• sample in optically transparent compartment• inserted in rotor ~ 300,000 g• measure evolution of radial concentration distribution• multiple optical systems (absorbance, Rayleigh interferometry)• since ~ 10 years, fast finite element solutions of Lamm equation (msec) y , q ( )
2 21 rD s rt r r r
∂ ∂ ∂⎡ ⎤= −⎢ ⎥∂ ∂ ∂⎣ ⎦χ χ χ ω
1000 kDa, 30 S
100 kDa, 5 S ( )2 3(1 )s M vρ−∼
10 kDa, 1 S
( )1DM vs
RT
ρ−=
1 2 3 2( ) b
rD s as s f−∼ ∼
( )
1 kDa, 0.3 S RT
Biophys. J. 74 (1998) 466Comput. Phys. Commun. 178 (2008) 105
22
( ) ( )max max
( )s f
a r t c s f s f r t dsdfχ= ∫ ∫
The Model Sedimentation Velocity
Size-and-shape distribution ( )c s f ( ) ( )min min
1( , ) , , , ,r r rs f
a r t c s f s f r t dsdfχ= ∫ ∫Size-and-shape distribution
Sedimentation coefficient distribution ( ) ( )max
1 ,( , ) , , ,s
r wa r t c s s f r t dsχ= ∫
( ), rc s f
( )c s
mins∫
2
0 1 ,k
K
l k kl k kc l k k
Min a c c H cκ κκ
χ α≥ =
⎡ ⎤⎛ ⎞⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥− +⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦∑ ∑ ∑discretized problem of the form
typically with l = 10,000 – 100,000 data points, distribution discretized to k = 100 – 1000 species, usenormal equations and NNLS
with Tikhonov regularization for broad continuous particle distributions
maximum entropy for biological macromolecules
scale regularization iteratively with goal to achieve confidence limits from F-statistics
,k k
k
c H cκ κκ
α∑log( )k k
k
c cα∑
Bayesian prior distribution to probe space of possible (statistically indistinguishable) solutions
implemented in public domain software SEDFIT and SEDPHATBiophys. J. 78 (2000) 1606Biophys. J. 89 (2005) 651Biomacromolecules 8 (2007) 2011
24
Applications: Very Large Particles Sedimentation Velocity
for it is 0D =2 22
1( , ) ( )s t s tr t e H r me−= −ω ωχ
calculating distribution is similar to differentiation of noisy data
( ) 2 22( , ) ( )s t s ta r t c s e H r me dsω ω−= −∫2
2 3
( )tr da
c sω∗ =
( )
( )
2
2
2
2
( )
( )
da rc s r r ds
dr mr dr
c s s s dsm ds
δ
δ
∗∗
∗ ∗∗
∗
= −
⎛ ⎞⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎜⎝ ⎠
∫
∫
2( )c s
drm=
clearly highly ill-conditioned if
( )2 1
2 3
ms noise da dr
trω
−Δ < ×
resolution of species with different s limited by signal/noise and rotor speed
boundary spread is sufficiently different from diffusion effect
25
Applications: Very Large Particles Sedimentation Velocity
1.26
1.28
1.30
1 14
1.16
1.18
1.20
1.22
1.24
sign
al (f
ringe
s)
http://berkeley.edu/news/media/releases/2005/09/22_spiro.shtml
6.25 6.3 6.35 6.4 6.45 6.51.10
1.12
1.14
radius (cm)
26
Applications: Protein Complexes Sedimentation Velocity
6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
sed coefficient [S]
c(s)
approximately like step-functions, but diffusion broadened
optimize average scaling parameters in non-linear regression,
7.2%as well as meniscus position
diffusion creates additional correlation between species of similar s
highly quantitative for trace species 1 %0.5%
highly quantitative for trace species ~ 1 %
very precise s-values
time-average s-values of systems of rapidly interacting molecules, interpreted as binding isotherms
27
Applications: Regularization Sedimentation Velocity
log( )k kk
c cα∑
log( )kkk k
cc
pα∑
1 3
0k
k
for sp
else
⎧ =⎪⎪= ⎨⎪⎪⎩
1 4
0k
k
for sp
else
⎧ =⎪⎪= ⎨⎪⎪0 else⎪⎩
28
Applications: Regularization Sedimentation Velocity
BSAkc∑
log( )k kk
c cα∑
log( )kkk k
cc
pα∑
with from integration of peaks of initial distribution
kp
kc
NKR
29
Applications: Multi-Signal SV Analysis Sedimentation Velocity
with multiple signals aλ(r,t): sedimentation coefficient distributions of component k
max
min
11
,( , ) ( , , , ) 1... ,ˆ ( )s
r ws
K
k kk
r t s f r ta c s ds Kλ λ χ λε=
≅ = Λ ≤ Λ∑ ∫
aλ(r,t) sedimentation signal at wavelength λ (or RI)εkλ extinction coefficient of component k at signal λ
5000
6000
1/M
cm)
trp• different fraction of tryptophan residues will give characteristic RI / ε280
• ratio of tyrosine and tryptophan ε / ε
250 260 270 280 290 300 310
0
1000
2000
3000
4000
mol
ar e
xtin
ctio
n (1
l th ( )
4 tyr• ratio of tyrosine and tryptophan ε280 / ε250
• glycosylation will contribute to signal RI / ε280
• chromophoric labels
• different spectral decomposition in different s-rangeswavelength (nm)
PNAS 102 (2005) 81
30
mix IgG and aldolase
RI 280 nm 250 nm
multi-λ ck(s) from mixturestandard c(s) for each λ separately
0.5
S) (
uM/S
)
0.4
0.6
0.8
1.0
(S) (
uM/S
)
RI
280 nm
IgGaldo
2 3 4 5 6 7 8 9 10 11 120.0
c(
s-value (S)2 3 4 5 6 7 8 9 10 11 12
0.0
0.2
c(
s-value (S)
250 nm
Spectral resolution → hydrodynamic resolution!
31
binding of Grb2 and Sos1 to LAT1pY
Grb2 24 μMLAT1pY 12 μMLAT1pY 12 μMSos1 12 μM
Y
YYP
P
P
Y
Y
Y
YP
P
P
P
Y
YYP
P
P
Y
YP
P
LAT
P
LAT
P
LAT
YPPLC-γP
PLC-γP
P PSos1 Sos1Y
Y
YP
P
P
Nat Struct Mol Biol 2006, 13:798
PP
PGrb2 ComplexGrb2
Complex
PP
P
Y
Y
Y
P
P
PSLP-76
Gads
P
GRAPComplex
32
∫∫ d dDtDLDCt )()()( : SolEqLammL
Applications: Multi-Method Size-And-Shape Analysis Sedimentation Velocity
{ }∫∫ τ−γ≅τDs
LS dsdDDqDsMDscg,
2)1( exp),(),()(
∫∫≅Ds
SVELSVEL dsdDtrDsLDscCtra,
),,,(),(),( ..: SolEqLammL
)1(),(
ρ−=
vDsRTDsM
sdDdrMEDscCraDs
EQEQ ∫∫ ω≅ω,
),,(),(),( RTrrvM
mbekrME 2)1(
20
22
),,(−
ωρ−=ω
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
α+⎥⎦
⎤⎢⎣
⎡−
σ ∑∑∑ ∑ji
ijije n ji
enji
eije
en
en
njiceCccxDsKcCxa
e
eee
e ,
2
,
)()()()(
,,ln),,()(1Min
calculate as
33
15
16
Oligomerized D1-D2 construct
12
13
14
Rs
(nm
)
3 4 5 6 7 8 9 10 119
10
11
3 4 5 6 7 8 9 10 11molar mass (monomer units)
34
One of Many Problems: Solution With Fewest Discrete Species? Sedimentation Velocity
IgG
( ) ( )max max
1( , ) , , , ,s f
r r ra r t c s f s f r t dsdfχ= ∫ ∫
data fit as size-and-shape distribution
( ) ( )min min
1( , ) , , , ,r r rs f
f f fχ∫ ∫
no regularizationlack of resolution of diffusion dimension justifies scaling lawbut may not be strictly fulfilled
with Tikhonov regularizationbroad distribution, incompatible with single species“Bayesian prior” regularization method sometimes too roughprior assumption of few discrete species often reasonablesusceptibility to systematic errors?