Effective normalization of complexity measurements for epoch length and sampling frequency.

The algorithmic complexity of a symbol sequence is sensitive to the length of the message. Additionally, in those cases where the sequence is constructed by the symbolic reduction of an experimentally observed wave form, the corresponding value of algorithmic complexity is also sensitive to the sampling frequency. In this contribution, we present definitions of algorithmic redundancy that are sequence-sensitive generalizations of Shannon's original definition of information redundancy. In contrast with algorithmic complexity, we demonstrate that algorithmic redundancy is not sensitive to message length or to observation scale (sampling frequency) when stationary systems are examined.

Information content of protein sequences.

The complexity of large sets of non-redundant protein sequences is measured. This is done by estimating the Shannon entropy as well as applying compression algorithms to estimate the algorithmic complexity. The estimators are also applied to randomly generated surrogates of the protein data. Our results show that proteins are fairly close to random sequences. The entropy reduction due to correlations is only about 1%. However, precise estimations of the entropy of the source are not possible due to finite sample effects. Compression algorithms also indicate that the redundancy is in the order of 1%. These results confirm the idea that protein sequences can be regarded as slightly edited random strings. We discuss secondary structure and low-complexity regions as causes of the redundancy observed. The findings are related to numerical and biochemical experiments with random polypeptides.

Measures of complexity in neural spike-trains of the slowly adapting stretch receptor organs.

Discrete sequence analysis methods were applied to study spike-trains generated by the isolated neuron of the slowly adapting stretch receptor organ. Calculation of the algorithmic complexity and block entropies of digitized individual spike-train forms allowed us to distinguish different classes of neural behavior. While some spike-trains exhibited significant structure, others displayed diverse degrees of randomness. The sequences recorded during the stimulated portions of the intermittent and walk-through forms, differed considerably from their randomly shuffled surrogates. Informational and grammar complexity measures (in two, four and eight-letter alphabets), tell us things about the structure of spike-trains that are not obtained with conventional spike analysis. Comparison of the conditional entropies for the digitized signals showed that the method distinguishes between different stimulated conditions. Additionally, comparison of the different stimulated conditions with their corresponding surrogates showed that, both, conditional entropies and complexities were significantly different for the two groups. Although the original and the randomly shuffled sequences had the same distribution and average firing rate, their complexity values were different. The results obtained with both measures of sequence structure were quite consistent.

Protein evolution drives the evolution of the genetic code and vice versa.

A model for the developmental pathway of the genetic code, grounded on group theory and the thermodynamics of codon-anticodon interaction is presented. At variance with previous models, it takes into account not only the optimization with respect to amino acid attributes but, also physicochemical constraints and initial conditions. A 'simple-first' rule is introduced after ranking the amino acids with respect to two current measures of chemical complexity. It is shown that a primeval code of only seven amino acids is enough to build functional proteins. It is assumed that these proteins drive the further expansion of the code. The proposed primeval code is compared with surrogate codes randomly generated and with another proposal for primeval code found in the literature. The departures from the 'universal' code, observed in many organisms and cellular compartments, fit naturally in the proposed evolutionary scheme. A strong correlation is found between, on one side, the two classes of aminoacyl-tRNA synthetases, and on the other, the amino acids grouped by end-atom-type and by codon type. An inverse of Davydov's rules, to associate the amino acid end atoms (O/N and non-O/non-N) of 18 amino acids with codons containing a weak base (A/U), extended to the 20 amino acids, is derived.

The hypercube structure of the genetic code explains conservative and non-conservative aminoacid substitutions in vivo and in vitro.

A representation of the genetic code as a six-dimensional Boolean hypercube is described. This structure is the result of the hierarchical order of the interaction energies of the bases in codon-anticodon recognition. In this paper it is applied to study molecular evolution in vivo and in vitro. In the first case we compared aligned positions in homologous protein sequences and found two different behaviors: (a) There are sites in which the different amino acids may be explained by one or two 'attractor nodes' (coding for the dominating amino acid(s)) and their one-bit neighbors in the codon hypercube; and (b) There are sites in which the amino acids correspond to codons located in closed paths in the hypercube. In the second case we studied the 'Sexual PCR'1 experiment described by Stemmer [Stemmer (1994)] and found that the success of this combination of usual PCR and recombination is in part due to the Gray code structure of the genetic code.

The algorithmic complexity of neural spike trains increases during focal seizures.

The interspike interval spike trains of spontaneously active cortical neurons can display nonrandom internal structure. The degree of nonrandom structure can be quantified and was found to decrease during focal epileptic seizures. Greater statistical discrimination between the two physiological conditions (normal vs seizure) was obtained with measurements of context-free grammar complexity than by measures of the distribution of the interspike intervals such as the mean interval, its standard deviation, skewness, or kurtosis. An examination of fixed epoch data sets showed that two factors contribute to the complexity: the firing rate and the internal structure of the spike train. However, calculations with randomly shuffled surrogates of the original data sets showed that the complexity is not completely determined by the firing rate. The sequence-sensitive structure of the spike train is a significant contributor. By combining complexity measurements with statistically related surrogate data sets, it is possible to classify neurons according to the dynamical structure of their spike trains. This classification could not have been made on the basis of conventional distribution-determined measures. Computations with more sophisticated kinds of surrogate data show that the structure observed using complexity measures cannot be attributed to linearly correlated noise or to linearly correlated noise transformed by a static monotonic nonlinearity. The patterns in spike trains appear to reflect genuine nonlinear structure. The limitations of these results are also discussed. The results presented in this article do not, of themselves, establish the presence of a fine-structure encoding of neural information.

Patterns in the complementary determining regions of immunoglobulins (CDRs).

An analysis of the frequency of use of amino acids on the CDR-1 and CDR-2 of 1500 immunoglobulins showed that the frequencies of amino acids in different positions could be fitted by two types of distribution. For some positions the frequencies were fitted by an inverse power law and for other positions by an exponential distribution. In order to see whether the more frequently used amino acids for specific positions had physicochemical properties or attributes in common, they were clustered using an algorithm normally applied to artificial intelligence problems. It was found that the amino acids in those positions fitted by the inverse power law have similar hydrophobicity and volume, which are commonly attributes of amino acids in structural positions. Thus, if these positions are critical to maintaining the structural features of the CDR domains, the rest of the positions should be either properly involved in the recognition process or irrelevant. The frequencies of amino acids in these recognition positions were fitted by the exponential law, and it was found by the clustering analysis that these amino acids share properties of a more general type, such as capability of forming hydrogen bonds, polarity, etc. This suggests that at least part of the recognition mechanism requires general properties rather than specific amino acids. Amino acids sharing the required attributes for each one of these positions are then used with random frequency.

On the syntactic structure and redundancy distribution of the genetic code.

By means of an algorithm for finding rules in data, it is shown that the genetic code may be written as a codon-tree, independent of amino acid assignments. Considering this tree as a structural description, a low-complexity, context-free grammar of the code is built and its grammar complexity and grammar redundancy calculated. The relationship between the codon-tree and the hierarchy of amino acid categorizations previously introduced by the author is investigated. Interpreting the obtained code's structure as a record of its evolution, some inferences about the divergences of the code series are made.

Toward a quantitative characterization of patient-therapist communication.

The efficacy of psychotherapy, in its many and varied forms, is one of the most intensely contested issues in clinical practice. Though many theories have been advanced, quantitative evidence in their defense is limited. This contribution is directed to relatively unambitious objectives. Rather than establish yet another qualitative theory of psychotherapeutic practice, we wish to contribute to the construction of research methodologies that can quantitatively characterize the dynamic patterns of patient-therapist communication. It is hoped that a theoretical understanding of psychotherapy might eventually emerge naturally from a growing body of quantitative data.

On the skew distribution of immunoglobulins and the inverted protein-folding problem.

The question of antibody specificity is discussed in the framework of the inverted protein-folding problem (i.e. the characterization of protein sequences with a common fold). A stochastic model of the immune response, patterned after a model for the distribution of words in natural languages is proposed. It is shown that the steady-state probability distribution of immunoglobulin variable-region frequencies is the Yule distribution.