Physico-chemical model of surface hybridisation and post- hybridisation washing Conrad Burden Mathematical Sciences Institute Australian National University,

Physico-chemical model of surface hybridisation and post-

hybridisation washing

Conrad Burden

Mathematical Sciences Institute Australian National University, Canberra

Canberra

A comprehensive model of hybridisation plus post-hybridisation washing is set out in

C.J. Burden and H. Binder: “Physico-chemical modelling of target depletion during hybridization on oligonucleotide microarrays”, Phys. Biol. 6, (2010) 016004

Basic idea: it’s all equilibrium physical chemistry driven by the law of mass-action

ADSORPTION

PROBE + TARGET DUPLEX DESORPTION

Image courtesy of Affymetrix

... plus all the other chemical reactions involving nonspecific hybridisation, hybridisation in bulk solution, etc.

P

S S’ N

P.S P.NP’

N.N

Assume a set of chemical reactions r each in equilibrium, with equilibrium constant Kr:

Assume a set of chemical reactions r each with equilibrium constant Kr:

In addition to the Kr the input parameters are

• specific target concentration

xS = [S] + [S’] + [P.S] + [S.N] + [S.S]

(= spike-in conc. in spike-in experiments)

• effective non-specific target concentration

xN = [N] + [N’] + [P.N] + [S.N] + [N.N]

• effective probe concentration on surface of feature

p = [P] + [P’] + [P.S] + [P.N]

The aim (in the first instance) is to determine the coverage fraction (0 ≤ θ ≤ 1) on a particular feature of the microarray of fluorescent-dye carrying duplexes

in terms of the input parameters xS, xN, p, Kr (r = PS, PN, SN, ...).

From this the observed flourescence intensity is simply

I(xS) = a + bθ

physical background saturation intensity

Solving this system for θ(xS) is straightforward algebra:

where θsum is the solution to

with the sum taken over all features contributing to the depletion of specific target species S.

€

θ(xS ) =XN + KS (xS − pθ sum )

1+ XN + KS (xS − pθ sum )

€

θ sum =KS

f (xS − pθ sum )

1+ KSf (xS − pθ sum )features f

∑

Solving this system for θ(xS) is straightforward algebra:

Note: • Three parameters:

KS (effective equilibrium const.) depends on xN and Kr (r = PS, PN, SN, ...)

XN (non-specific binding strength) depends on xN and Kr (r = PS, PN, SN, ...)

p (probe density)

• If depletion of target from supernatant solution can be ignored, p = 0, and θ(xS) is a rectangular hyperbola;

• If xN = 0 then XN = 0 and θ(0) = 0;

• At saturation spec. target concentration xS −> ∞, θ(xS) −> 1.

€

θ(xS ) =XN + KS (xS − pθ sum )

1+ XN + KS (xS − pθ sum )

Saturation asymptote θ(∞) = 1 does not agree with spike-in experiments:

For PM/MM pair, theory gives θPM(∞) = θMM(∞) = 1

Spike-in experiments always give θPM(∞) > θMM(∞)

θ = 1

PM

MM

log

I(t)

/I(0

)

Differing asymptotes explained by post-hybridisation washing:

Decaying fluorescence intensity with washing is confirmed by experiment(Binder, H. et al. BMC Bioinformatics 11:291 (2010))

Differing asymptotes explained by post-hybridisation washing

0

1

Differing asymptotes explained by post-hybridisation washing

0

1

Saturation asymptote θ(∞) = 1 does not agree with experiment:

Spike-in experiments always give θPM(∞) > θMM(∞)

Putting it all together, the fluorescence intensity on a given feature is related to the specific target concentration by a 4-parameter ‘isotherm’:

where θsum is the solution to

with the sum taken over all features contributing to the depletion of specific target species S.

The 4 fitting parameters A, B, K and p can be fitted well to spike-in data ...

€

I(xS ) = A + BK(xS − pθ sum )

1+ K(xS − pθ sum )

€

θ sum =K f (xS − pθ sum )

1+ K f (xS − pθ sum )features f

∑

Example of fits to Affy U133a spikes:

PM

MM

The fitting parameters

A, B, K and p

are complicated functions of physical input parameters

xN, Kr (r = PS, PN, SN, ...), p, a, b, wS, wN.

Is it practical to ‘predict’ these parameters from the available data (probe sequences, distribution of intensities across microarray, ...)?

Maybe/Maybe not, but the model does give a physical understanding which helps with building algorithms (Hook curve, Inverse Langmuir , ...)

€

I(xS ) = A + BK(xS − pθ sum )

1+ K(xS − pθ sum )

1. With no non-spec. hybridisaton, target depletion or washing, ‘Langmuir isotherm’ is:

I(xS)

€

a +1

2b

0

€

KPS-1

xS

€

I(xS ) = a + bKPSxS

1+ KPSxS

€

a + b

€

a

Understanding the model bit by bit:

ADSORPTION

PROBE + SPEC. TARGET DUPLEX DESORPTION

€

I(xS ) = A + BKxS

1+ KxS

I(xS)

€

A

€

A + B

€

A +1

2B

0

€

K−1xS

€

a + b

€

a

€

KS-1

2. Switch on non-specific hybridisation and washing

Understanding the model bit by bit:

I(xS)

€

A

€

A + B

€

A +1

2B

0

€

K−1xS

€

a + b

€

a

€

KS-1

2. Switch on non-specific hybridisation and washing

Understanding the model bit by bit:

washing

washing

non-spec. hybr.

non-spec. hybr.

€

I(xS ) = A + BKxS

1+ KxS

I(xS)

€

A

€

A + B

€

A +1

2B

0

€

K−1xS

Non-spec + a

Specific

2. Switch on non-specific hybridisation and washing

Understanding the model bit by bit:

€

I(xS ) = A + BKxS

1+ KxS

I(xS)

€

A

€

A + B

€

A +1

2B

0

€

K−1xS

2. Switch on non-specific hybridisation and washing

Understanding the model bit by bit:

€

I(xS ) = A + BKxS

1+ KxS

I(xS)

€

A

€

A + B

€

A +1

2B

0

€

K−1xS

3. Switch on target depletion

Understanding the model bit by bit:

€

I(xS ) = A + BK(xS − pθ sum )

1+ K(xS − pθ sum )

I(xS)

€

A

€

A + B

0 xS

4. Switch on target depletionand consider PM/MM pair

Understanding the model bit by bit:

€

I(xS ) = A + BK(xS − pθ sum )

1+ K(xS − pθ sum )

€

A +1

2B

€

K−1

PM

MM

Inflection point

No inflection

Example of fits to Affy U133a spikes:

PM

MM

Example of fits to Affy U133a spikes:

PM

MM

Discussion topics: ‘Langmuir-based model’

1.Is the model sufficiently detailed? Are there other things happening (e.g. non-equilibrium chemistry: Hooyberghs et al. Phys. Rev. E 81, 012901 (2010))

2.Is it too detailed? (e.g. could we leave out folding? target depletion?)

3.How do we make practical use of it? There are many ‘physical’ input parameters: xN, Kr (r = PS, PN, SN, ...), p, wS, wN, a, b and 3 or 4 ‘fitting’ parameters: A, B, K, p.

Can we infer the fitting parameters from a combination of first principles and the data (e.g. Hook Curve or Inverse Lanmuir)?

Discussion topics: ‘Langmuir-based model’

1.Is the model sufficiently detailed? Are there other things happening (e.g. non-equilibrium chemistry: Hooyberghs et al. Phys. Rev. E 81, 012901 (2010))Equilibrium constant K should be governed by the Arrhenius eqn.

K = const. × exp(ΔG/RT).

Using binding energy ΔG from nearest-neighbour stacking models, fits to spike-in data typically require

K = const. × exp(λΔG/RT) with λ ≈ 0.1.

Suggests a non-equilibrium model

probe + target <−> unstable duplex <−> stable duplex

with a slow relaxation time for the second reaction.

Discussion topics: ‘Langmuir-based model’

3.How do we make practical use of it? There are many ‘physical’ input parameters: xN, Kr (r = PS, PN, SN, ...), p, wS, wN, a, b and 3 or 4 ‘fitting’ parameters: A, B, K, p.

I believe these parameters are NOT universal functions of the probe sequence alone:

different experiments different non-specific background xN

different A and K.

This is bourne out by experiment (see our ‘Chip-to-chip normalisation paper’, J.Phys.Chem B 113, 2874 (2009))