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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

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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

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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)

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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

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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)

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“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

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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= = − −∑

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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.

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EVILFIT: public domain GUI for distribution analysis of optical biosensor data in MATLAB

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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

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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

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replicate experiments, same surface density of sites

Gorshkova et al., Langmuir 24 (2008) 11577

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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

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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

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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

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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

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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)

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random walk with bias, two non-interacting components

Sedimentation Velocity

(σ1 = 0.01; δ1 = 0.00005; σ2 = 0.025; δ2 = 0.0005)

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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

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( ) ( )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

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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

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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)

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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

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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⎪⎩

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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

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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

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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!

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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

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∫∫ 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)

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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?