Gustavo Martínez-Mekler

Universality of Rank-Ordering Distributions in the Arts and Sciences

Searching for generic behaviors has been one of the driving forces leading to a deep understanding and classification of diverse phenomena. Usually a starting point is the development of a phenomenology based on observations. Such is the case for power law distributions encountered in a wealth of situations coming from physics, geophysics, biology, lexicography as well as social and financial networks. This finding is however restricted to a range of values outside of which finite size corrections are often invoked. Here we uncover a universal behavior of the way in which elements of a system are distributed according to their rank with respect to a given property, valid for the full range of values, regardless of whether or not a power law has previously been suggested. We propose a two parameter functional form for these rank-ordered distributions that gives excellent fits to an impressive amount of very diverse phenomena, coming from the arts, social and natural sciences. It is a discrete version of a generalized beta distribution, given by f(r) = A(N+1-r)b/ra, where r is the rank, N its maximum value, A the normalization constant and (a, b) two fitting exponents. Prompted by our genetic sequence observations we present a growth probabilistic model incorporating mutation-duplication features that generates data complying with this distribution. The competition between permanence and change appears to be a relevant, though not necessary feature. Additionally, our observations mainly of social phenomena suggest that a multifactorial quality resulting from the convergence of several heterogeneous underlying processes is an important feature. We also explore the significance of the distribution parameters and their classifying potential. The ubiquity of our findings suggests that there must be a fundamental underlying explanation, most probably of a statistical nature, such as an appropriate central limit theorem formulation.

Discrete Dynamics Model for the Speract-Activated Ca2+ Signaling Network Relevant to Sperm Motility

Understanding how spermatozoa approach the egg is a central biological issue. Recently a considerable amount of experimental evidence has accumulated on the relation between oscillations in intracellular calcium ion concentration ([Ca]) in the sea urchin sperm flagellum, triggered by peptides secreted from the egg, and sperm motility. Determination of the structure and dynamics of the signaling pathway leading to these oscillations is a fundamental problem. However, a biochemically based formulation for the comprehension of the molecular mechanisms operating in the axoneme as a response to external stimulus is still lacking. Based on experiments on the S. purpuratus sea urchin spermatozoa, we propose a signaling network model where nodes are discrete variables corresponding to the pathway elements and the signal transmission takes place at discrete time intervals according to logical rules. The validity of this model is corroborated by reproducing previous empirically determined signaling features. Prompted by the model predictions we performed experiments which identified novel characteristics of the signaling pathway. We uncovered the role of a high voltage-activated channel as a regulator of the delay in the onset of fluctuations after activation of the signaling cascade. This delay time has recently been shown to be an important regulatory factor for sea urchin sperm reorientation. Another finding is the participation of a voltage-dependent calcium-activated channel in the determination of the period of the fluctuations. Furthermore, by analyzing the spread of network perturbations we find that it operates in a dynamically critical regime. Our work demonstrates that a coarse-grained approach to the dynamics of the signaling pathway is capable of revealing regulatory sperm navigation elements and provides insight, in terms of criticality, on the concurrence of the high robustness and adaptability that the reproduction processes are predicted to have developed throughout evolution.

Coupled trivial maps.

The first nontrivial example of coupled map lattices that admits a rigorous analysis in the whole range of the strength of space interactions is considered. This class is generated by one-dimensional maps with a globally attracting superstable periodic trajectory that are coupled by a diffusive nearest-neighbor interaction.

Periodic orbits and long transients in coupled map lattices

Abstract A coupled map lattice model is studied which presents transient and asymptotic chaotic states depending on the value of the control parameters. A first-order approximation detects the presence of asymptotic periodic attractors. The statistics of transient times as well as their dependence on lattice size are investigated.

Primordial synthesis machines and the origin of the genetic code

We characterize the dynamics of primitive molecular synthesis machines operating in outer space on quasi-one-dimensional channels where polymers interact with fixed particles. We show that a generic property of particle/polymer electrostatic interactions is an average three monomer spacing between consecutive interaction potential minima. We exhibit that this property translates into locomotion regularities with a slowing down every three monomers. We argue that this transport property may be at the origin of the three base codon composition of the genetic code. We relate these findings to present day protein synthesis mechanisms.

On a coupled map lattice formulation of the evolution of genetic sequences

Abstract A biological phenomenology for the study of the evolution of genetic sequences is developed. A hierarchy of coupled map lattice models with reaction-diffusion characteristics is proposed for this evolution based on the above phenomenology. For the case of some viruses and bacteria, uncoupled dynamics follows a “space” dependent logistic type evolution subject to a global ecological constraint. The coupling is introduced by diffusive and gradient terms related to genetic mutations. It is argued that within the formulation presented, coupled map lattices are specially suited for the modeling of the dynamics of this problem.

Translocation properties of primitive molecular machines and their relevance to the structure of the genetic code.

We address the question, related with the origin of the genetic code, of why are there three bases per codon in the translation to protein process. As a follow-up to our previous work (Aldana et al., 1998, Martínez-Mekler et al., 1999a,b), we approach this problem by considering the translocation properties of primitive molecular machines, which capture basic features of ribosomal/messenger RNA interactions, while operating under prebiotic conditions. Our model consists of a short one-dimensional chain of charged particles (rRNA antecedent) interacting with a polymer (mRNA antecedent) via electrostatic forces. The chain is subject to external forcing that causes it to move along the polymer which is fixed in a quasi-one-dimensional geometry. Our numerical and analytic studies of statistical properties of random chain/polymer potentials suggest that, under very general conditions, a dynamics is attained in which the chain moves along the polymer in steps of three monomers. By adjusting the model in order to consider present-day genetic sequences, we show that the above property is enhanced for coding regions. Intergenic sequences display a behavior closer to the random situation. We argue that this dynamical property could be one of the underlying causes for the three-base codon structure of the genetic code

Interaction of the IP3-Ca2+ and the FGF-MAPK signaling pathways in the Xenopus laevis embryo: a qualitative approach to the mesodermal induction problem

In this work we propose that the animal-vegetal gradient spatial distribution of the IP(3) receptors observed in the Xenopus embryo can effect a uniform FGF inducting input signal, allowing for different modes of transcription of the Xbra gene, producing the differentiation of the cells of the marginal zone. We analyze this hypothesis with a model for the interaction of the calcium signaling system with the MAPK cascade during the FGF mesodermal induction process, consisting of five non-linear coupled differential equations. A numerical treatment of a one- and two-cell system shows that the calcium flux between cells enhances the Raf activity levels, leading to oscillatory behavior. This qualitative result may be of consequence for the expression of the ventralizing characteristics of the FGF inducting signal.

Niflumic acid disrupts marine spermatozoan chemotaxis without impairing the spatiotemporal detection of chemoattractant gradients

Summary In many broadcast-spawning marine organisms, oocytes release chemicals that guide conspecific spermatozoa towards them through chemotaxis. In the sea urchin Lytechinus pictus, the chemoattractant peptide speract triggers a train of fluctuations of intracellular Ca2+ concentration in the sperm flagella. Each transient Ca2+ elevation leads to a momentary increase in flagellar bending asymmetry, known as a chemotactic turn. Furthermore, chemotaxis requires a precise spatiotemporal coordination between the Ca2+-dependent turns and the form of chemoattractant gradient. Spermatozoa that perform Ca2+-dependent turns while swimming down the chemoattractant gradient, and conversely suppress turning events while swimming up the gradient, successfully approach the center of the gradient. Previous experiments in Strongylocentrotus purpuratus sea urchin spermatozoa showed that niflumic acid (NFA), an inhibitor of several ion channels, drastically altered the speract-induced Ca2+ fluctuations and swimming patterns. In this study, mathematical modeling of the speract-dependent Ca2+ signaling pathway suggests that NFA, by potentially affecting hyperpolarization-activated and cyclic nucleotide-gated channels, Ca2+-regulated Cl− channels and/or Ca2+-regulated K+ channels, may alter the temporal organization of Ca2+ fluctuations, and therefore disrupt chemotaxis. We used a novel automated method for analyzing sperm behavior and we identified that NFA does indeed disrupt chemotactic responses of L. pictus spermatozoa, although the temporal coordination between the Ca2+-dependent turns and the form of chemoattractant gradient is unaltered. Instead, NFA disrupts sperm chemotaxis by altering the arc length traveled during each chemotactic turning event. This alteration in the chemotactic turn trajectory disorientates spermatozoa at the termination of the turning event. We conclude that NFA disrupts chemotaxis without affecting how the spermatozoa decode environmental cues.

Boolean Threshold Networks: Virtues and Limitations for Biological Modeling

Boolean threshold networks have recently been proposed as useful tools to model the dynamics of genetic regulatory networks, and have been successfully applied to describe the cell cycles of S. cerevisiae and S. pombe. Threshold networks assume that gene regulation processes are additive. This, however, contrasts with the mechanism proposed by S. Kauffman in which each of the logic functions must be carefully constructed to accurately take into account the combinatorial nature of gene regulation. While Kauffman Boolean networks have been extensively studied and proved to have the necessary properties required for modeling the fundamental characteristics of genetic regulatory networks, not much is known about the essential properties of threshold networks. Here we study the dynamical properties of these networks with different connectivities, activator-repressor proportions, activator-repressor strengths and different thresholds. Special attention is paid to the way in which the threshold value affects the dynamical regime in which the network operates and the structure of the attractor landscape. We find that only for a very restricted set of parameters, these networks show dynamical properties consistent with what is observed in biological systems. The virtues of these properties and the possible problems related with the restrictions are discussed and related to earlier work that uses these kind of models.