Larry S. Liebovitch

A fast algorithm to determine fractal dimensions by box counting

Abstract A new algorithm is used to determine fractal dimensions by box counting for dynamic and iterated function systems. This method is fast, accurate, and less dependent on data specific curve fitting criteria than the correlation dimension.

Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes

Abstract Ion channels have been modeled as consisting of a small number of discrete conformational states such as closed ⇌ closed ⇌ open, and the transitions between the states treated as a Markov process. We derive an alternative model based on a fractal scaling of the kinetic rate constants k, namely that k = At1-D, where t is time, A the kinetic setpoint, and D the fractal dimension. By measuring the effective kinetic rate constants at different time scales, we show how to determine if single channel records are best represented by models with discrete Markov states or continuum fractal kinetics. The fractal model suggests that the multiple closed states postulated by Markov models to fit the histograms of closed times may not exist but are an artifact of trying to fit the sum of exponentials to histograms that are not the sum of exponentials. The fractal model is also more consistent with the conformational dynamics of proteins. Analysis of patch clamp recordings from an epithelium, the corneal endothelium, shows that its channels have fractal rather than Markov kinetics.

Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and the kinetic rate constants connecting these states are constant. In the alternative fractal model the spontaneous fluctuations of the channel protein at many different time scales are represented by a kinetic rate constant k = At1-D, where A is the kinetic setpoint and D the fractal dimension. Single-channel currents were recorded at 146 mM external K+ from an inwardly rectifying, 120 pS, K+ selective, voltage-sensitive channel in cultured mouse hippocampal neurons. The kinetics of these channels were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimensions were approximately 2 for the closed times and approximately 1 for the open times and did not depend on voltage. For both the open and closed times the logarithm of the kinetic setpoint was found to be proportional to the applied voltage, which indicates that the gating of this channel involves the net inward movement of approximately one negative charge when this channel opens. Thus, the open and closed times and the voltage dependence of the gating of this channel are well described by the fractal model.

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.

The mechanism of fluid and electrolyte transport across corneal endothelium: critical revision and update of a model.

A model for endothelial transport is updated to include recent evidence. We discuss electrolyte movements based on a Na+-K+ ATPase, a Na+-H+ exchanger, a Na+-HCO3 coupler, a Cl- -HCO-3 exchanger, a K+-Cl-coupler, and K+ and anion channels. We discuss near-isotonic transport of fluid on the basis of recent findings of high endothelial osmotic permeability.

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.

Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies.

Ion channels in the cell membranes of the corneal endothelium, hippocampal neurons, and fibroblasts, and gramicidin channels in lipid bilayers have open and closed times that can be fit, in whole or part, by power law distributions. That is, the gating is self-similar when viewed at different time scales. Hence, kinetic processes at slow and fast time scales are not independent but rather are interrelated. To study how such a relationship can arise we analyze a closed in equilibrium open channel with the fractal dimension for leaving the closed state DCO approximately 2 and the fractal dimension for leaving the open state DOC approximately 1. This special case can be analyzed because it can be represented by equivalent Markov processes. We show that it is equivalent to Markov chains with forward and backward kinetic rate constants approximately equal at each stage, and forming an approximate geometric progression along the different stages. These kinetic rates determine the energy levels and activation energy barriers separating those levels. We find that there are many conformational states (substates) separated by high activation energy barriers. This is similar to the energy structure found for globular proteins such as myoglobin. However, the novel feature reported here is that the activation energy barriers are not independent but are interrelated and form an arithmetic progression. Because of this relationship the fast processes across the low activation energy barriers are linked to slow processes across the high activation energy barriers.

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 exponentH=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).

Fractal Activity in Cell Membrane Ion Channelsa

A fractal is an object that appears similar when viewed under different magnifications. We have used fractal concepts to develop new methods to analyze the sequence of open and closed times of cell membrane ion channels recorded by the patch clamp technique. The results of these analyses suggest two important properties of ionchannel proteins. First, we found that ion-channel proteins have a very large number of conformational states and many pathways connecting those states. This is consistent with recent experiments and simulations of globular proteins. Second, these many states are not independent. They are linked by a physical mechanism that results in the observed fractal properties. We describe the models that have been proposed to understand the nature of that mechanism.

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.