Fractal methods to analyze ion channel kinetics.

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.

Hurst analysis applied to the study of single calcium-activated potassium channel kinetics.

The gating of ion channels has been modeled by assuming that the transitions between open and closed states is a memoryless process. Nevertheless, analysis of records of unitary current events suggests that the kinetic process presents short-term memory, i.e. the open- and closed-dwell times are short-term correlated. Here the rescaled range analysis (R/S Hurst analysis) is used as a method to test long-term correlation, in single calcium-activated potassium channels present in Leydig cells. The Hurst coefficients, calculated for four different voltages (V) are: 0.634+/-0.022 (n=3) for V=+20 mV; 0.635+/-0.012 (n=4) for V=+40 mV; 0.606+/-0.020 (n=4) for V=+60 mV and 0.608+/-0.026 (n=4) for V=+80 mV. This indicates that open- and closed-dwell times are long-term correlated and do not change with the voltage applied to the patch at a 5% significance level (F=2.2402;p=0.140715). Randomly shuffling the experimental data removes the correlation in all voltages. When the Hurst method was applied to the results from a simulated three-state Markovian model, it could not account for the long-term correlation found in the experimental data. In this case, H has the following values: 0. 5498+/-0.018 (n=100) for V=+20 mV; 0.5557+/-0.0202 (n=100) for V=+40 mV; 0.5565+/-0.0246 (n=100) for V=60 mV and 0.5595+/-0.0247 (n=100) for V=+80 mV. Even a four-state Markovian model was not adequate to correctly simulate the long-term memory found experimentally, with H values significantly different from those found for the experimental data, in the same voltage range (F=15.0355;p=0.00001). In conclusion, this paper shows that: (1) the open- and closed-dwell times of the single calcium-activated potassium channel of Leydig cells are long-term correlated; (2) three- and four-state Markovian models, which describe very well the dwell time distributions, are not adequate to describe the long-term correlation found between the open and closed states of this ion channel.

Fractal ion-channel behavior generates fractal firing patterns in neuronal models.

Fractal behavior has been observed in both ion-channel gating and neuronal spiking patterns, but a causal relationship between the two has not yet been established. Here, we examine the effects of fractal ion-channel activity in modifications of two classical neuronal models: Fitzhugh-Nagumo (FHN) and Hodgkin-Huxley (HH). For the modified FHN model, the recovery variable was represented as a population of ion channels with either fractal or Markov gating characteristics. Fractal gating characteristics changed the form of the interspike interval histogram (ISIH) and also induced fractal behavior in the firing rate. For the HH model, the K+ conductance was represented as a collection of ion channels with either quasifractal or Markov gating properties. Fractal gating induced fractal-rate behavior without changing the ISIH. We conclude that fractal ion-channel gating activity is sufficient to account for fractal-rate firing behavior.

Assessing genetic markers of tumour progression in the context of intratumour heterogeneity.

This is a report from the Kananaskis working group on quantitative methods in tumour heterogeneity. Tumour progression is currently believed to result from genetic instability and consequent acquisition of new genetic properties in some of the tumour cells. Cross-sectional assessment of genetic markers for human tumours requires quantifiable measures of intratumour heterogeneity for each parameter or characteristic observed; the relevance of heterogeneity to tumour progression can best be ascertained by repeated assessment along a tumour progressional time line. This paper outlines experimental and analytic considerations that, with repeated use, should lead to a better understanding of tumour heterogeneity, and hence, to improvements in patient diagnosis and therapy. Four general principles were agreed upon at the Symposium: (1) the concept of heterogeneity requires a quantifiable definition so that it can be assessed repeatably; (2) the quantification of heterogeneity is necessary so that testable hypotheses may be formulated and checked to determine the degree of support from observed data; (3) it is necessary to consider (a) what is being measured, (b) what is currently measurable, and (c) what should be measured; and (4) the proposal of working models is a useful step that will assist our understanding of the origins and significance of heterogeneity in tumours. The properties of these models should then be studied so that hypotheses may be refined and validated.

Practical problems of determining the dimensions of heart rate data.

The practical problems are explored of determining the dimension of the phase space set generated from real, experimental and simulated data of the times between consecutive heartbeats in normal and diseased rabbits. It is determined how different measures of dimension have depended on the procedures used to construct the phase space set and on such properties of the data as the amount of data, noise, long-term trends and stationarity. Reproducible estimates of the dimensions of different physiological states are found to require considerable amounts of data recorded under stationary conditions.

Is there an error correcting code in the base sequence in DNA?

Modern methods of encoding information into digital form include error check digits that are functions of the other information digits. When digital information is transmitted, the values of the error check digits can be computed from the information digits to determine whether the information has been received accurately. These error correcting codes make it possible to detect and correct common errors in transmission. The sequence of bases in DNA is also a digital code consisting of four symbols: A, C, G, and T. Does DNA also contain an error correcting code? Such a code would allow repair enzymes to protect the fidelity of nonreplicating DNA and increase the accuracy of replication. If a linear block error correcting code is present in DNA then some bases would be a linear function of the other bases in each set of bases. We developed an efficient procedure to determine whether such an error correcting code is present in the base sequence. We illustrate the use of this procedure by using it to analyze the lac operon and the gene for cytochrome c. These genes do not appear to contain such a simple error correcting code.

What causes ion channel proteins to fluctuate open and closed?

Ion channels in the cell membrane spontaneously switch from states that are closed to the flow of ions such as sodium, potassium, and chloride to states that are open to the flow of these ions. The durations of times that an individual ion channel protein spends in the closed and open states can be measured by the patch clamp technique. We explore two basic issues about the molecular properties of ion channels: 1) If the switching between the closed and open state is an inherently random event, what does the patch clamp data tell us about the structure or motions in the ion channel protein? 2) Is this switching random?

Membrane potential fluctuations of human T-lymphocytes have fractal characteristics of fractional Brownian motion.

The voltage across the cell membrane of human T-lymphocyte cell lines was recorded by the whole cell patch clamp technique. We studied how this voltage fluctuated in time and found that these fluctuations have fractal characteristics. We used the Hurst rescaled range analysis and the power spectrum of the increments of the voltage (sampled at 0.01-sec intervals) to characterize the time correlations in these voltage fluctuations. Although there was great variability in the shape of these fluctuations from different cells, they all could be represented by the same fractal form. This form displayed two different regimes. At short lags, the Hurst exponent H = 0.76 +/- 0.05 (SD) and, at long lags, H = 0.26 +/- 0.04 (SD). This finding indicated that, over short time intervals, the correlations were persistent (H > 0.5), that is, increases in the membrane voltage were more likely to be followed by additional increases. However, over long time intervals, the correlations were antipersistent (H < 0.5), that is, increases in the membrane voltage were more likely to be followed by voltage decreases. Within each time regime, the increments in the fluctuations had characteristics that were consistent with those of fractional Gaussian noise (fGn), and the membrane voltage as a function of time had characteristics that were consistent with those of fractional Brownian motion (fBm).

Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins.

Three examples are given of how concepts from fractals and nonlinear dynamics have been used to analyze the voltages and currents recorded through ion channels in an attempt to determine the physical properties of ion channel proteins. (1) Early models had assumed that the switching of the ion channel protein from one conformational state to another can be represented by a Markov process that has no long-term correlations. However, one support for the existence of long-term correlations in channel function is that the currents recorded through individual ion channels have self-similar properties. These fractal properties can be characterized by a scaling function determined from the distribution of open and closed time intervals, which provides information on the distribution of activation energy barriers between the open and closed conformational substates of the ion channel protein and/or on how those energy barriers change in time. (2) Another support for such long-term correlations is that the whole-cell membrane voltage recorded across many channels at once may also have a fractal form. The Hurst rescaled range analysis of these fluctuations provides information on the type and degree of correlation in time of the functioning of ion channels. (3) The early models had also assumed that the switching from one state to another is an inherently random process driven by the energy from thermal fluctuations. More recently developed models have shown that deterministic dynamics may also produce the same distributions of open and closed times as those previously attributed to random events. This raises the possibility that the deterministic atomic and electrostatic forces play a role in switching the channel protein from one conformational shape to another. Debate exists about whether random, fractal, or deterministic models best represent the functioning of ion channels. However, fractal and deterministic dynamics provide a new approach to the study of ion channels that should be seriously considered by neuroscientists.

Hurst analysis in the study of ion channel kinetics.

Ion channels are protein molecules which can assume distinct open and closed conformational states. The transitions between these states can be controlled by the electrical field, ions and/or drugs. Records of unitary current events show that short open-time intervals are frequently adjacent to much longer closed-time intervals, and vice-versa, suggesting that the kinetic process has memory, i.e., the intervals are correlated in time. Here the rescaled range analysis (R/S Hurst analysis) is proposed as a method to test for correlation. Simulations were performed with a two-state Markovian model, which has no memory. The calculated Hurst coefficients (H) presented a mean +/- SD value of 0.493 +/- 0.025 (N = 100). For the Ca(2+)-activated K+ channels of Leydig cells, H was equal to 0.75, statistically different (1% level) from that calculated for the memoryless process. Randomly shuffling the experimental data resulted in an H = 0.55, not significantly different (1% level) from that found for the two-state Markovian model. For a linear three-state Markovian model, H was equal to 0.548 +/- 0.017 (N = 15), again not significantly different (1% level) from that of the memoryless process. Although the three-state Markovian model adequately describes the open- and closed-time distributions, it does not account for the correlation found in this Ca(2+)-activated K+ channel. Our results illustrate the efficacy of the R/S analysis in determining whether successive opening and closing events are correlated in time and can be of help in deciding which model should be used to describe the kinetics of ion channels.

Hurst analysis in the study of ion channel kinetics

Ion channels are protein molecules which can assume distinct open and closed conformational states. The transitions between these states can be controlled by the electrical field, ions and/or drugs. Records of unitary current events show that short open-time intervals are frequently adjacent to much longer closed-time intervals, and vice-versa, suggesting that the kinetic process has memory, i.e., the intervals are correlated in time. here the rescaled range analysis (R/S Hurst analysis) is proposed as a method to test for correlation. Simulations were performed with a two-state Markovian model, which has no memory. The calculated Hurst coefficients (H) presented a mean + or - SD value of 0.493 + or - 0.025 (N = 100). For the Ca2+ -activated K+ channels of Leydig cells, H wass equal to 0.75, statistically different (1 percent level) from that calculated for the memoryless proces. Randomly shuffling the experimental data resulted in an H = 0.55, not significantly different (1 percent level) from that found for the two-state Markovian model. For a linear three-state Markovian model, H was equal to 0.548 + or - 0.017 (N = 15), agin not significantly different (1 percent level) from that of the memoryless proces. Although the tree-state Markovian model adequately describes the open-and closed-time distributions, it does not account for the correlation found in this Ca2+ -activatedK+ channel. Our results ilustrate the efficacy of the R/S analysis in determining whether successive opening and closing events are correlated in time and can be of help in deciding which odel should be used to describe the kinetics of ion channels

Fractal analysis of pulmonary arteries: the fractal dimension is lower in pulmonary hypertension.

We analyzed the spatial structure of contact radiographs of barium-filled pulmonary arteries of rats raised in room air and in two environments that induce pulmonary arterial hypertension (PAH)--hypoxia and hyperoxia. We found that the spatial structure of the pulmonary arteries was fractal in both the control and the hypertensive lungs. The fractal dimension of the pulmonary arteries of the control lungs was 1.62 +/- 0.01 (mean +/- SEM), which is greater than that of both the hypoxic lungs 1.50 +/- 0.03 (p < 0.01) and the hyperoxic lungs 1.44 +/- 0.01 (p < 0.01). There was no significant difference between the hypoxic and hyperoxic lungs. The fractal dimension may be a useful clinical index to quantify pathologic changes in the pulmonary arterial tree.

Statistical properties predicted by the ball and chain model of channel inactivation.

It has been proposed that part of a voltage gated channel is a tethered ball and that inactivation occurs when this wandering ball binds to a site in the channel. In order to be able to quantitatively test this model by comparison to experiments we developed analytical solutions and numerical simulations of the distribution of times it takes the ball to reach the binding site when the motion of the ball is random and when it is also influenced by a directed force. If the motion of the ball is one-dimensional, at long times this distribution is a single exponential with a rate constant that is inversely proportional to the square of the length of the chain and does not depend on the starting position of the ball. This dependence on the chain length is not significantly altered if there are short range electrical forces between the ball and its binding site. These predictions suggest that to confirm the validity of this model additional experiments should be done to more precisely determine the form of this distribution and its dependence on the length of the chain.

A model of ion channel kinetics based on deterministic, chaotic motion in a potential with two local minima.

Models of the gating of ion channels have usually assumed that the switching between the open and closed states is a random process without a mechanistic basis. We explored the properties of a deterministic model of channel gating based on a chaotic dynamic system. The channel is modeled as a nonlinear oscillator, that has a potential function with two minima, which correspond to the stable open and closed states, and is driven by a periodic driving force. The properties of the model are like some properties of single channel data and unlike other properties. The model is like the data in that: the current switches between two well-defined states, this switching is nonperiodic, and there are subconductance states. These subconductance states are subharmonic resonances, due to the nonlinearities in the equation of the model, rather than stable conformational states due to local minima in the potential energy. The model is not like the data in that the current fluctuates too much within in each state and there are sometimes periodic fluctuations within a state. At the present time, the selection of the most appropriate channel model (Markov, chaotic, or other) is not possible, and in addition to chaotic models, other nonlinear models may be suitable.

A model of ion channel kinetics using deterministic chaotic rather than stochastic processes.

Models of ion channel kinetics have previously assumed that the switching between the open and closed states is an intrinsically random process. Here, we present an alternative model based on a deterministic process. This model is a piecewise linear iterated map. We calculate the dwell time distributions, autocorrelation function, and power spectrum of this map. We also explore non-linear generalizations of this map. The chaotic nature of our model implies that its long-term behavior mimics the stochastic properties of a random process. In particular, the linear map produces an exponential probability distribution of dwell times in the open and closed states, the same as that produced by the two-state, closed in equilibrium open, Markov model. We show how deterministic and random models can be distinguished by their different phase space portraits. A test of some experimental data seems to favor the deterministic model, but further experimental evidence is needed for an unequivocal decision.

Using fractals to understand the opening and closing of ion channels.

Looking at an old problem from a new perspective can sometimes lead to new ways of analyzing experimental data which may help in understanding the mechanisms that underlie the phenomena. We show how the application of fractals to analyze the patch clamp recordings of the sequence of open and closed times of cell membrane ion channels has led to a new description of ion channel kinetics. This new information has led to new models that imply: (a) ion channel proteins have many conformational states of nearly equal energy minima and many pathways connecting one conformational state to another, and (b) that these many states are not independent but are linked by physical mechanisms that result in the observed fractal scaling. The first result is consistent with many experiments, simulations, and theories of globular proteins developed over the last decade. The second result has stimulated the suggestion of several different physical mechanisms that could cause the fractal scalings observed.

The akaike information criterion (AIC) is not a sufficient condition to determine the number of ion channel states from single channel recordings.

The Akaike information criterion (AIC) and other criteria that trade off the goodness-of-fit against the number of parameters have been used to determine the number of ion channel states from single channel recordings. We show that, in general, such criteria are not valid. This is illustrated by an elementary example in which the AIC yields an incorrect result. The implications of these findings for the discrimination of different kinetic models of ion channels are discussed.