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