Improving sigmoidal fitting of real-timePCR

data: Linear-quadratic-sigmoidal hybrid

models, error analysis and weighting

Andrej-Nikolai Spiess

University Hospital Hamburg-Eppendorf

Department of Andrology

R: the lingua franca of statistical data analysis

• developed by John Chambers and Ross Ihaka at

the beginning of the 80´s as S

• was commercialised to S-Plus but also made

freely available (open-source) as R.

• 15 core people maintain R, ~ 2 million users

worldwide

• R‘s own repertoire of ~ 3000 functions can be

extended by ‚packages‘ (i.e. for spatial or

environmental statistics, genomics, no limit…)

=> ~ 4500 packages available

qpcR: An R package for real-time PCR analysis

Ratio

Calculation

(MC simulation,

Permutation (REST),

Error Propagation)

A typical workflow using qpcR

‚pcrimport‘

‚pcrfit‘ ‚modlist‘ ‚replist‘

Data import

Model

creation

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selectionL3, L4, L5,

L6, L7

AIC, AICc, BIC,

PRESS, F-test, R2Goodness

of fit

‚efficiency‘, ‚sliwin‘, ‚expfit‘, ‚calib‘

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2.356

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Effic

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cpD2: 15.68cpD1: 17.59

Eff: 1.795

resVar: 0.00603899 AICc: -102.58

Model: l5

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Eff/Cq value

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Eff: 1.76

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thre

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AIC: -4.55 Rsq: 0.99967

Eff: 2.04

‚ratiocalc‘‚ratiobatch‘Monte-Carlo

0.37 1 2.72 7.39 20.09 54.6

Permutation

0.37 1 2.72 7.39 20.09 54.6

Error propagation

0.37 1 2.72 7.39 20.09 54.6

5e-04

1e-03

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r1c1:g1c1:r1s1:g1s1

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Monte-Carlo

Simulation

Permutation

ErrorPropagation

r1c1:g1c1:r1s1:g1s1 r2c1:g1c1:r2s1:g1sr1c1:g1c1:r1s2:g1s2r2c1:g1c1:r2s2:g1s2

Mean.Sim 0.419531879 6.533495786 0.223828241 5.770492727

Std.dev..Sim 0.147043285 1.353377023 0.017764239 1.047243322

Median.Sim 0.396300098 6.393701355 0.223282684 5.669149665

MAD.Sim 0.133055518 1.304387664 0.017778066 1.016963311

Conf.lower.S 0.204116613 4.284885281 0.191219778 4.00151269

Conf.upper.S 0.767489852 9.538449133 0.260947425 8.068843646

Mean.Perm 0.410562957 6.513010694 0.223663646 5.749083527

Std.dev..Per 0.114406471 1.081009488 0.015364639 0.712937423

Median.Perm 0.330239654 6.74781282 0.236810989 6.37620751

MAD.Perm 0.079692836 1.010968373 0.006290143 0.169364005

Conf.lower.P 0.280518979 5.152642278 0.206810681 4.973967308

Conf.upper.P 0.560169465 7.961269931 0.240735435 6.481874382

perm > init.P 0.4995 0 0.535 0

perm == init 0.5005 0.4925 0.465 0.5125

perm < init.P 0 0.5075 0 0.4875

Mean.Prop 0.395972171 6.39779022 0.223126803 5.678021311

Std.dev..Pro 0.134606979 1.310527148 0.017678287 1.020096686

Conf.lower.P 0.13214734 3.829204209 0.188477998 3.678668546

Conf.upper.P 0.659797003 8.96637623 0.257775607 7.677374075

Data

export

What can qpcR do?Sigmoidal fitting:

3-, 4-, 5-, 6-, 7-parameter sigmoidal models. Model selection for the best model based on

F-test. Goodness of fit measures: AIC, AICc, BIC, R2, PRESS, Chi2. Extract Cq, efficiency

from any part of the curve, FDM, SDM

Other fitting methods:

Window-of-linearity, exponential fitting, LRE, Cy0

Mechanistic models: mak2, mak3, cm3, Also: maxRatio, bilinear model, linexp

Calibration curves:

Classical calibration (Cq versus Conc), confidence intervals, prediction of unknown

samples, bootstrap replicates for better efficiency estimates.

Outlier run detection:

KOD, SOD, multivariate outlier detection (FDM, SDM, slope, Fmax)

Batch analysis:

Fit 10000 (Fluidigm) curves and calculate Eff/Cq in 5 min!

Ratio calculation:

Calculate expression ratios for single or multiple setups by MC simulation/Permutation

(REST)/Error Propagation. Statistical dsplay of results. Conduct reference gene

averaging, if desired.

Visualization:

Single curves, batch curves, 3D display, heatmap display. Failed runs are automatically

marked.

Melting curve analysis:

Automatic detection of melting peaks (Tm) and automatic quality control.

qPCR data import:

Advanced import function that can import ALL data through a series of 8 steps.

Testing methods on replicate qPCR data

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Batsch et al. (2008)

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Boggy et al. (2010)

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Guescini et al. (2008)

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Lievens et al. (2012)

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All datasets available at www.dr-spiess.de

Why sigmoidal models ?

Fitting complete curve…

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Sigmoidal

Cq (SDM)

E(Cq) = F(Cq)/F(Cq – 1)

F0 = Fq/E(Cq)Cq

Fq

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Mak2 (Boggy et al. 2010)

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R^2 : 0.99949

Eff: 1.74

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Exponential (Tichopad et al., 2003) Sliding window (Ramakers et al., 2003)

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R^2 : 0.99646

Eff: 1.934 Cycles 15:19

Raw fluorescence

Effic

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LRE (Rutledge et al. 2008)

as opposed to… (curve subsets)

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AIC

8.3E18 (!)

6E5 (!)

2E6 (!)

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expGrowth 17.84

l7 23.70

l6 28.99

l5 50.77

l4 114.78

l4 l5 l6 l7

More parameters: Better fits...

How often do we need k1·x / k2·x2 ?

Estimate Pr(>|t|)

b -1.273839e+01 < 2.22e-16 ***

c -1.418255e-01 9.8655e-05 ***

d 1.082728e+01 < 2.22e-16 ***

e 1.781830e+01 < 2.22e-16 ***

f 9.283720e-01 < 2.22e-16 ***

k1 3.442433e-02 6.1591e-08 ***

k2 -5.524618e-04 4.3619e-09 ***

k2 k2

Rutledge: 117/120 83/120

Guescini: 80/84 82/84

Lievens: 90/90 81/90

Reps380: 360/380 353/380

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Cq and efficiency can differ between models

Cq Eff

Small error in efficiency and small error

in Cq => LARGE error in Ecq (Quantities) !

Efficiency

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µ = 1.8, σ = 0.036, c.v. = 2 %

Quantity

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µ = 136830, σ = 56474, c.v. = 41.3 %

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Quantity

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ECq

µ = 131025, σ = 31596, c.v. = 24.1 %

ECq

µ = 142558, σ = 70051, c.v. = 49.1 % !!!!

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Similar to Shain et al. (2008)

„maxRatio“ method

Cubic spline fitting

Splines to calculate variance in Cq/Eff

of replicate datasetsAll 380 runs

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Cq Efficiency

Fq = 1000

Creating qPCR curves with defined

error in Cq/Efficiency or both Create 1000 Cq values with [20, 21] and Eff = 1.8 at Fq = 1. Calculate F0

for each curve by F0 = Fq/(ECq). Create a sigmoidal efficiency curve and

from this, create a qPCR curve F(n + 1) = F(n) * E(n).

Cq [20, 21]

Eff [1.7, 1.8]

qPCR curves with defined

Cq or efficiency…

slope [1.2, 1.8]

Asym (‚f‘) [0.8, 1.2]

qPCR curves with defined

asymmetry or slope…

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Sim L4 L5 L6 L7 SPL Sim L4 L5 L6 L7 SPL

Performance of the different models

on the synthetic data

Cq Eff

How many replicates do we need

to estimate the error in Eff/Cq reliably ?

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st.dev

n=10: 95% n=20: 98%n=200: 99.9%

Create n (2 – 1000) samples from Normal distribution =>

calculate std. dev

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Rutledge, 20 replicates

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Lievens, 12 replicates

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VAR

Typical variance structures in

qPCR replicates

Variance Variance

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Better fit in exponential region

by using weights

unweighted Weighted by 1/Var

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log(Dilution)

Cq R2: 0.9993

AIC: -106.4

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Cq R2: 0.9995

AIC: -130.2

unweighted

Weighted by 1/Var

Increased linearity in calibration curve

analysis by using weights

Questions welcome !