A Subsequent Fit of Time Series and Amplitude Histogram of Patch-Clamp Records Reveals
Rate Constants up to 1 per Microsecond
der, P. Harlﬁnger, T. Huth, U.P. Hansen
Center of Biochemistry and Molecular Biology, Leibnizstr. 11, 24098, Kiel, Germany
Received: 20 September 2004/Revised: 5 January 2004
Abstract. Fast gating in time series of patch-clamp
current demands powerful tools to reveal the rate
constants of the adequate Hidden Markov model.
Here, two approaches are presented to improve the
temporal resolution of the direct ﬁt of the time series.
First, the prediction algorithm is extended to include
intermediate currents between the nominal levels as
caused by the anti-aliasing ﬁlter. This approach can
reveal rate constants that are about 4 times higher
than the corner frequency of the anti-aliasing ﬁlter.
However, this approach is restricted to time series
with very low noise. Second, the direct ﬁt of the time
series is combined with a beta ﬁt, i.e., a ﬁt of the
deviations of the amplitude histogram from the
Gaussian distribution. Since the ‘‘theoretical’’
amplitude histograms for higher-order Bessel ﬁlters
cannot be calculated by analytical tools, they are
generated from simulated time series. In a ﬁrst ap-
proach, a simultaneous ﬁt of the time series and of
the Beta ﬁt is tested. This simultaneous ﬁt, however,
inherits the drawbacks of both approaches, not the
beneﬁts. More successful is a subsequent ﬁt: The ﬁt of
the time series yields a set of rate constants. The
subsequent Beta ﬁt uses the slow rate constants of the
ﬁt of the time series as ﬁxed parameters and the
optimization algorithm is restricted to the fast ones.
The eﬃciency of this approach is illustrated by means
of time series obtained from simulation and from the
channel in Chara. This shows that
temporal resolution can reach the microsecond range.
Key words: Anti-aliasing ﬁlter — Beta distributions
— Hidden Markov models — Ion channels — Rate
constants — Maximum likelihood
Ion channels are not rigid cylinders that facilitate a
steady stream of ions across biological membranes.
Instead, they are vibrating proteins leading to spon-
taneous or agent-induced interruptions of the stream.
The resulting closures and openings are modelled by
means of Markov models (Korn & Horn, 1988; Yeo
et al., 1988; Ball & Rice, 1992; Blunck et al., 1998).
The rate constants of the transitions between the
states of the Markov model even in a single channel
span a wide range from about 1 s
to at least 1 ls
der et al., 2004). The upper limit is not known,
because the temporal resolution of classical evalua-
tion algorithms leads to a limit which is about
(Parzefall et al., 1998; Farokhi et al., 2000;
Zheng et al., 2001; Hansen et al., 2003).
The diﬀerent approaches commonly employed for
the evaluation of patch-clamp time series oﬀer diﬀerent
potencies for the analysis of fast gating. Widely applied
is dwell-time analysis (Blunck et al. 1998). In order to
account for fast gating, missed-events corrections have
been suggested (Ball et al., 1993; Draber & Schultze,
1994). However, Farokhi et al. (2000) have found a
horizontal dependence of the evaluated rate-constants
on the ‘‘true’’ rate constants, which is not the optimum
condition for correction algorithms. Dwell time anal-
ysis can become powerful if 2-dimensional histograms
are evaluated (Magleby & Weiss, 1990).
The Beta ﬁt is based on the generation of am-
plitude histograms and ﬁtting their deviations from
Gaussians by beta distributions (FitzHugh, 1983;
Yellen, 1984; Klieber & Gradmann, 1993; Riessner,
1998). However, this approach is restricted to 2-state
Abbreviations: BF1, BF4, Beta ﬁt with a Bessel ﬁlter of ﬁrst or
fourth order, respectively; EP ﬁt, direct ﬁt of the time series with
extended prediction (including the ﬁlter response); C, G, Z closed
states; HMM, Hidden Markov model; A, O open state, SNR,
signal-to-noise ratio; SP ﬁt, direct ﬁt of the time series with simple
prediction (single-step prediction); SQ ﬁt, subsequent SP/BF4 Fit.
Correspondence to: U.P. Hansen; email: email@example.com-
J. Membrane Biol. 203, 83–99 (2005)