To further investigate the effects of normalization using house-keeping or recombinant proteins, we perform an in silico study on an artificially generated signal and normalizer signal.
We take a look at the signal intensity in terms of amplification factors on the noisy mea- surements of the artificial signal itself, and of the normalized signal, processed by using the artificial normalizer measurements.
3.5.5.1 Generation of the artificial signal
We apply a piecewise linear time transformation to a modified sine function to generate an artificial signal that is remarkably similar to the experimentally observed kinetics in the IL6+ setting (compare section 3.6 on page 78 to the artificial signal in figure 3.15b).
The underlying sine function that we use as signal generator is given as
f (x) = 24.5· 1 + sin(x + 1.5π)+ 1. (3.13)
Further, the continuous piecewise linear time transformation,
T (t) = π 25t t∈ [0, 30] π 75t + 45π t∈ [30, 60] π 150t + 6 5π t∈ [60, 120] (3.14)
(a) raw sinusoidal signal... (b) ... transformed to artifical signal Figure 3.15: Raw and time-transformed artificial signal.
The sinusoidal signal in (a) is generated by evaluating eq. (3.13) at time points ˜ti = i/5· π (i = 0..10), equidistantly sampling the interval [0, 2π] for a whole period.
The artificial signal (b) is formed by applying the time-transformation T (eq. (3.14)) to the measurement time points {0, 5, 10, 15, 20, 25, 30, 45, 60, 90, 120}.
To generate the artificial signal, for each time point ti∈ {0, 5, 10, 15, 20, 25, 30, 45, 60, 90, 120},
the signal generator function f is evaluated at f (T (ti)) and scaled(25)by a fixed random num-
ber k ∼ U 500000, 1000000, independent of i. Figure 3.15 shows both the artificial signal generated by the procedure above (b) and the underlying sinusoidal base signal (a).
The artificial normalizer signal is generated by choosing a number ˜k∼ U 500000, 1000000 and assigning it to all sample time points.
Both, the artificial and the normalizer signals are subsequently distorted by a multiplicative normally distributed noise. It is obvious, that this distortion certainly changes the observed maximum amplification factor. Indeed, it might also influence the time point where it is observed, as, by chance, the (truly) highest signal might be distorted in a way that it is lower than the signal observed at other time points.
3.5.5.2 Simulation results
As figure 3.16 shows, both expected types of errors occur. Ideally, i.e. without any error in the measurements, a single bar at 50-fold containing 100% of the simulations would be observed, and the time point histogram in the right column of figure 3.16 would consist of a single bar at time point 25min.
Obviously, applying the normalization to the sinusoidal signal before calculating the ampli- fication factors results in a much broader distribution of the observed maximum amplification factor.
For a noise level of 10%, both mean maximum amplification factors of the signal itself
(52.1-fold) and of the (normalized signal (54.3-fold) are close to the true value of 50-fold;
however, their standard deviations differ considerably (6.7 vs. 10.0). For higher noise levels, the mean value of the normalized signal deviates much faster from the true value than the mean value of the unprocessed signal (noise level of 20%: 57.8-fold vs. 65.6-fold, noise level of 30%: 65-foldvs.78.1-fold). For the standard deviation, these effects are even worse (noise level of 20%: 15.5 vs.25.3, noise level of 30%: 26vs. 39.2).
The impact of measurement errors on the time point at which the maximum signal intensity is detected is demonstrated in the rightmost pictures in figure 3.16. If solely the signal itself is used to calculate the amplification factor (depicted as green boxes ), with increasing noise level, the time point at which the maximum signal is observed spreads: For a noise level of 0.1, the correct time point is identified in more than two thirds of all cases (figure 3.16a); this declines to still more 40% if a noise level of 0.3 (figure 3.16c) is simulated.
Again, the situation is worse if the signal is at first normalized (red boxes ): Even for small disturbances (CV = 0.1), the time point of the maximum signal intensity is correctly detected in only about half of all cases. For larger noise levels, this already declines to less than 40% (CV = 0.2) or roughly 30% (CV = 0.3). In the case of the highest simulated noise, two out of five measurements will show the maximum level more than 10 minutes away from the true time point – in contrast to only 5% if the signal is not normalized.
3.7 Conclusion
Normalization with a house-keeping or manually added calibrator protein may massively disturb the intensity and time point of the peak signal already at moderate noise levels. Under same noise intensities, using solely the raw signal – as proposed in the amplifi- cation factors method – delivers much better estimates on peak amplification and peak time point (see figure 3.16).
We finish this small study with the remark that the underlying sampling is also influencing the observation distribution. For example, a thinner sampling around the time point of the maximum signal level (25min) would result in a narrower distribution of the detected time points. Taking more samples generally results in a broadened histogram. For the simulation study, we used the same sampling as in the real experiments (section 3.3).
(a) Distribution of maximum amplification factors and their time points for a CV of 10%.
(b) Distribution of maximum amplification factors and their time points for a CV of 20%.
(c) Distribution of maximum amplification factors and their time points for a CV of 30%. Figure 3.16: Normalization effects on distribution of maximum amplification factors and their
time point for different levels of noise: (a) CV = 0.1, (b) CV = 0.2, CV = 0.3.
In this simulation study, 1000000 artificial sinusoidal signals with a peak amplification factor of 50 reached at 25min (see figure 3.15b) as well as the same amount of artificial normalizer signals were created; both distorted by identically distributed noise to simulate measurement errors.
Shown are histograms of the observed maximum amplification factors with ( ) and without ( ) prior
normalization. Without any disturbance, a single bar at the 50-fold mark would be observed. From both histograms, an estimated probability density was derived by nonparametric kernel smoothing using normal
kernel functions (thin green—and red—lines), to which normal distributions were fitted by maximiz-
ing the overlap (integral) of the respective estimated probability density function and the normal density (thick green---- and red----lines). The values ofmean,sd(in green colour),mean,sd (in red colour) de- note the empirical mean and standard deviations for each series. Amplification factors exceeding the inter- val(1, 200) were excluded from all calculations.