


Nonlinear analysis of a Drosophila ECG time series.
Skinner, James E.^{1},
Elka D. Yankulova^{2,3}, George Yannopoulos^{2}, and Tassos
Bountis^{3}. ^{1}Delaware
Water Gap Science Institute, Bangor, PA 18013, USA; Department of Biology^{2}, Department of Mathematics^{3},
University of Patras, 26500 Patras, Greece.
Email addresses: jskinner@vicortech.com; eyankulova@yahoo.com; yannop@upatras.gr; bounties@math.upatras.gr
The objective is to introduce a new computational approach to the genetic analysis of Drosophila cardiac function, using nonlinear dynamics together with gene knockout.
Chaos theory and a unique software, the Point Correlation Dimension, PD2i, were applied for the first time to the study of the genetic nature of Drosophila cardiac dynamics. The PD2i software (Skinner et al., 1994), calculates the degrees of freedom in small subepochs of a data series, such as the digitized electrocardiogram (ECG). The PD2i algorithm has an advantage over other chaos quantifiers in that it can analyse a nonstationary data series. Originally the PD2i software was designed for the detection of human heartbeat pathologies, and it was found to have a 100% sensitivity in predicting ventricular fibrillation among highrisk patients (Skinner et al., 1993). In the present study the PD2i was applied for the first time to an abnormality in Drosophila cardiac function resulting from gene knockout of potassium channels. The result was that the Drosophila mutants eag (which encode subunits of K+ channels) show periodic lowdimensional excursions in the PD2i range of 2.2 to 3.7 degrees of freedom that neither the normal control nor the randomizedphase surrogate data show (p < 0.01). Statistical measurements (e.g., running SD’s, Power spectrum, 1/f noise) did not detect any differences in the same data set.
The
Drosophila eag mutant heartbeat data are similar to that of the ischemic
myocardium of a mammalian heart, although in the latter the excursions are
to 1.2 dimensions. Low dimensional chaos turns out to be the most serious
indicator of genetically induced cardiac disorders. Only in cardiac mutations
are observed the characteristic low dimensional excursions registered by the
PD2i.
Drosophila heartbeat: Optical recording and digitization
All measurements were made on Drosophila melanogaster. eaggene mutants (etheragogo), concerning K^{+ }channels which are related with heart pathologies, and wild type controls were provided by the Bloomington Drosophila Stock Center, U.S.A. Optical ECG records were taken at a stage P1 (white puparium) of Drosophila development (Bainbridge and Bownes, 1981) when it is both immobile and transparent and the dorsal vessel (Rizki, 1978) is easily viewed. The object was placed on a glass slide in a drop of distilled water under a microscope (magnification 350´). Fluctuation in light intensity due to movement of the dorsal vessel tissue is captured by photocells fitted to the one eyepiece of the microscope. Experts from the Onassis Cardiac Center, Athens, Greece, designed this analogue signal acquisition equipment.
The captured analogue signal was then digitized at 1 kHz sampling rate by data acquisition card and LabVIEW data capturing software supplied by "National Instruments" and stored in a
Figure 1. A low dimensional excursion in the Drosophila eag mutant ECG time series detected by the PD2i software.
Pentium III computer. 600000 data points (more than 1000 heartbeats) were taken for each sample. Optimal data gain was empirically established to be equal to 5.

Mathematical stationarity
is presumed during the collection of data in the above D2 application, a presumption
which is not tenable for biological systems. The "pointwise" scaling
dimension was suggested by Farmer, Ott and Yorke (1983) to be an estimate
of D2 that was perhaps less sensitive to nonstationarities, because the reference
vector is fixed. Since the reference vector is chosen sequentially for each
digitized point in the timeseries, dimension is estimated as a function of
time.
The "pointD2"
estimate of the correlation dimension (PD2i) was developed by Skinner and
associates (1991). Like the D2i, each PD2i reference ivector remains fixed,
while each of the jvectors run through the whole data series. But for the
PD2i, the jvectors that will contribute to the small logr values must arise
from a subepoch that manifests scaling characteristics similar to those surrounding
the ivector. Basically the PD2i reference vector seeks its own sunspecies
of stationary data with which to make the vectordifference lengths; this
occurs by a process that involves, 1) the plot length (PL) of the small logr
values in the scaling region (i.e., as observed in the loglog
plot of the cumulative histogram of the rankordered vector difference lengths
vs the range), 2) the linearity criterion (LC) for this scaling region,
and 3) the convergence criterion (CC) of the slope of this
scaling region vs the embedding dimension. Part of the success of the PD2i software revolves around the
rejection of values that do not result in linear scaling and clear convergence;
these rejections also eliminate PD2i estimates that could result from contamination
of the small logr values (i.e., by other nonstationary subepochs,
noise or artifacts in the data; contamination by cardiac arrhythmias is also
eliminated).
The
Chaos software is unique in its class, since it is a deterministic data processor
and can analyze nonstationary time series. Its basic advantages are:
· Reliability: it is a deterministic
(not a stochastic) measure whose results are valid for individual subjects
and representative at any point of the data series. Thus, it displays unique
precision and predictability compared to other, traditional quantifiers of
chaos as Lyapunov exponents, Generalized dimensions, Entropies, etc.;
· Sensitivity: deterministic
dimensional measures are inherently more sensitive to the output of the system
than classical stochastic measures as the mean, standard deviation, power
spectrum, etc.;
· Efficiency: it does not
require data stationarity and can track rapid dimensional changes within small
points of time;
It was not until recently considered that biological data are random and spurious. Chaos theory and the PD2i software for the first time allow interpreting them as deterministic. Standard tools for the analysis of biological data as: Power spectrum, Fourier transform, Mean and Standard deviation, Entropy, and so forth treat these data "averagely" (statistically), while the PD2i software is capable of identifying individual qualitative specificities within an unfolding data series. This capability of PD2i allowed us to apply it for distinguishing the individual dynamical properties of the various Drosophila mutants heart dynamics.
Acknowledgments: This research has been supported by a Marie Curie Fellowship of the European Community programme "Quality of Life and Management of Living Resources" under contract number QLK5CT200051155.
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