On Computing Structural and Behavioral Complexities of Threshold Boolean Networks : Application to Biological Networks.

Various threshold Boolean networks (TBNs), a formalism used to model different types of biological networks (genes notably), can produce similar dynamics, i.e. share same behaviors. Among them, some are complex (according to Kolmogorov complexity), others not. By computing both structural and behavioral complexities, we show that most TBNs are structurally complex, even those having simple behaviors. For this purpose, we developed a new method to compute the structural complexity of a TBN based on estimates of the sizes of equivalence classes of the threshold Boolean functions composing the TBN.

Formal methods for Hopfield-like networks.

Building a meaningful model of biological regulatory network is usually done by specifying the components (e.g. the genes) and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method to Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge (about structure and behavior). The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors.

On the nature and shape of tubulin trails: implications on microtubule self-organization.

Microtubules, major elements of the cell skeleton are, most of the time, well organized in vivo, but they can also show self-organizing behaviors in time and/or space in purified solutions in vitro. Theoretical studies and models based on the concepts of collective dynamics in complex systems, reaction-diffusion processes and emergent phenomena were proposed to explain some of these behaviors. In the particular case of microtubule spatial self-organization, it has been advanced that microtubules could behave like ants, self-organizing by 'talking to each other' by way of hypothetic (because never observed) concentrated chemical trails of tubulin that are expected to be released by their disassembling ends. Deterministic models based on this idea yielded indeed like-looking spatio-temporal self-organizing behaviors. Nevertheless the question remains of whether microscopic tubulin trails produced by individual or bundles of several microtubules are intense enough to allow microtubule self-organization at a macroscopic level. In the present work, by simulating the diffusion of tubulin in microtubule solutions at the microscopic scale, we measure the shape and intensity of tubulin trails and discuss about the assumption of microtubule self-organization due to the production of chemical trails by disassembling microtubules. We show that the tubulin trails produced by individual microtubules or small microtubule arrays are very weak and not elongated even at very high reactive rates. Although the variations of concentration due to such trails are not significant compared to natural fluctuations of the concentration of tubuline in the chemical environment, the study shows that heterogeneities of biochemical composition can form due to microtubule disassembly. They could become significant when produced by numerous microtubule ends located in the same place. Their possible formation could play a role in certain conditions of reaction. In particular, it gives a mesoscopic basis to explain the collective dynamics observed in excitable microtubule solutions showing the propagation of concentration waves of microtubules at the millimeter scale, although we doubt that individual microtubules or bundles can behave like molecular ants.

Influence of sparkle and saccades on tongue electro-stimulation-based vision substitution of 2D vectors.

Vision substitution by electro-stimulation has been studied since the 60s beginning with P. Bach-y-Rita. Camera pictures or movies encoded in gray levels are displayed using an electro-stimulation display device on the surface of a body part, such as the skin or the tongue. Medical-technical devices have been developed on this principle to compensate for sensory-motor disabilities such as blindness or loss of balance, or to guide specific actions, such as surgery. However, the electrical signals of stationary or moving slowly moving objects, displayed on a Tongue display unit (TDU), are quickly lost due to saturation of receptors undergoing electrostimulation. We propose to add random saccades or sparkle to the displayed visual scene to increase the quality of pattern recognition by the subjects. In the present experimental trimodal study (normal vision, TDU vision substitution, or both), we show that the presence of a moderate sparkle level enhances the perception of the direction of lines drawn on a TDU and reduces the response time.

The isochronal fibration: characterization and implication in biology.

Limit cycles, because they are constituted of a periodic succession of states (discrete or continuous) constitute a good manner to store information. From any points of the state space reached after a perturbation or stimulation of the cognitive system storing this information, one can aim to join through a more or less long return trajectory a precise neighbourhood of the asymptotic trajectory at a specific moment (or a specific place) on the limit cycle, i.e. where the information of interest stands. We propose that the isochronal fibration, initially imagined and described by A. T. Winfree may be an excellent way to connect directly those two locations. Each isochron is indeed the set of points in temporal phase with one single point of the attractor. The characterisation of the isochronal fibration of various dynamical systems is not easy and until now has principally only been done numerically but not analytically. By integrating the homogeneous solutions of the dynamical system we can solve this fibration in the case of the well known anharmonic pendulum. Other isochronal fibration on classical examples such as the van der Pol system and the non-symmetrical PFK limit cycle are obtained numerically and we also provide the first numerical study on 3-dimentional systems like the anharmonic pendulum with a linear relaxation on its third variable and the Lorenz attractor. The empirical approach seems us useful for dealing with the isochronal fibration which could constitute a powerful tool for understanding and controlling the dynamics of biological or biological-inspired systems.

RNA relics and origin of life.

A number of small RNA sequences, located in different non-coding sequences and highly preserved across the tree of life, have been suggested to be molecular fossils, of ancient (and possibly primordial) origin. On the other hand, recent years have revealed the existence of ubiquitous roles for small RNA sequences in modern organisms, in functions ranging from cell regulation to antiviral activity. We propose that a single thread can be followed from the beginning of life in RNA structures selected only for stability reasons through the RNA relics and up to the current coevolution of RNA sequences; such an understanding would shed light both on the history and on the present development of the RNA machinery and interactions. After presenting the evidence (by comparing their sequences) that points toward a common thread, we discuss a scenario of genome coevolution (with emphasis on viral infectious processes) and finally propose a plan for the reevaluation of the stereochemical theory of the genetic code; we claim that it may still be relevant, and not only for understanding the origin of life, but also for a comprehensive picture of regulation in present-day cells.

An open issue: the inner mitochondrial membrane (IMM) as a free boundary problem.

The inner mitochondrial membrane (IMM) is structured in cristae, which contributes to the best functioning of ions and adenylates exchange between the matrix and the intermembrane space. The central hypothesis of this paper is that the cristae structure favours a minimal mean free path of adenylates between translocation sites (translocase/ANT sites) and metabolic sites (ATPase sites). We propose a mathematical model and then give simulations. Based on simple hypotheses about cristae growth, they show that we can account for the major features of the IMM organization and functioning by minimizing the mean interdistance between ADP/ATP translocation and transformation sites.

Influence of glass and polymer coatings on CHO cell morphology and adhesion.

Successful development of cell-on-chip microsystems where living cells are deposited and grown in microfabricated structures is highly dependent on the control of cell/substrate interactions. In this study, several materials of interest were tested for CHO cell growth and morphology: (i) glass, fibronectin-, poly-L-lysine- and 3-aminopropyltriethoxysilane (APTES)--treated glass and UV/O(3)-modified PDMS coating on glass as well as (ii) silicon, poly-L-lysine-, APTES-, O(2) plasma-treated and oxide-coated silicon. In addition, we quantitatively characterized cell adhesion to these substrates using a radial flow detachment assay. Lack of correlation between cell adhesion and cell morphology was systematically observed for all substrates. In particular, we show that PDMS coatings on glass can be finely tuned by UV/O(3) treatment to enhance cell adhesion and induce elongated morphology. Moreover, we observed a low shear stress cell detachment mechanism on silicon oxide coatings on silicon wafers. It is therefore possible with these coatings to selectively influence either cell adhesion or morphology.

Liénard systems and potential-Hamiltonian decomposition: applications in biology.

In separated notes, we described the mathematical aspects of the potential-Hamiltonian (PH) decomposition, in particular, for n-switches and Liénard systems [J. Demongeot, N. Glade, L. Forest, Liénard systems and potential-Hamiltonian decomposition - I. Methodology, II. Algorithm and III. Applications, C. R. Acad. Sci., Paris, Ser. I, in press]. In the present note, we give some examples of biological regulatory systems susceptible to be decomposed. We show that they can be modelled in terms of 2D ordinary differential equations belonging to n-switches and Liénard system families [O. Cinquin, J. Demongeot, High-dimensional switches and the modeling of cellular differentiation, J. Theor. Biol. 233 (2005) 391-411]. Although simplified, these models can be decomposed into a set of equations combining a potential and a Hamiltonian part. We discuss about the advantage of such a PH-decomposition for understanding the mechanisms involved in their regulatory abilities. We suggest a generalized algorithm to deal with differential systems having a second part of rational-fraction type (frequently used in metabolic systems). Finally, we comment what can be interpreted as a precise signification in biological systems from the dynamical behaviours of both the potential and Hamiltonian parts.

Effect of weightlessness on colloidal particle transport and segregation in self-organising microtubule preparations.

Weightlessness is known to effect cellular functions by as yet undetermined processes. Many experiments indicate a role of the cytoskeleton and microtubules. Under appropriate conditions in vitro microtubule preparations behave as a complex system that self-organises by a combination of reaction and diffusion. This process also results in the collective transport and organisation of any colloidal particles present. In large centimetre-sized samples, self-organisation does not occur when samples are exposed to a brief early period of weightlessness. Here, we report both space-flight and ground-based (clinorotation) experiments on the effect of weightlessness on the transport and segregation of colloidal particles and chromosomes. In centimetre-sized containers, both methods show that a brief initial period of weightlessness strongly inhibits particle transport. In miniature cell-sized containers under normal gravity conditions, the particle transport that self-organisation causes results in their accumulation into segregated regions of high and low particle density. The gravity dependence of this behaviour is strongly shape dependent. In square wells, neither self-organisation nor particle transport and segregation occur under conditions of weightlessness. On the contrary, in rectangular canals, both phenomena are largely unaffected by weightlessness. These observations suggest, depending on factors such as cell and embryo shape, that major biological functions associated with microtubule driven particle transport and organisation might be strongly perturbed by weightlessness.

Diversity and survival of artificial lifeforms under sedimentation and random motion.

Cellular automata are often used to explore the numerous possible scenarios of what could have occurred at the origins of life and before, during the prebiotic ages, when very simple molecules started to assemble and organise into larger catalytic or informative structures, or to simulate ecosystems. Artificial self-maintained spatial structures emerge in cellular automata and are often used to represent molecules or living organisms. They converge generally towards homogeneous stationary soups of still-life creatures. It is hard for an observer to believe they are similar to living systems, in particular because nothing is moving anymore within such simulated environments after few computation steps, because they present isotropic spatial organisation, because the diversity of self-maintained morphologies is poor, and because when stationary states are reached the creatures are immortal. Natural living systems, on the contrary, are composed of a high diversity of creatures in interaction having limited lifetimes and generally present a certain anisotropy of their spatial organisation, in particular frontiers and interfaces. In the present work, we propose that the presence of directional weak fields such as gravity may counter-balance the excess of mixing and disorder caused by Brownian motion and favour the appearance of specific regions, i.e. different strata or environmental layers, in which physical-chemical conditions favour the emergence and the survival of self-maintained spatial structures including living systems. We test this hypothesis by way of numerical simulations of a very simplified ecosystem model. We use the well-known Game of Life to which we add rules simulating both sedimentation forces and thermal agitation. We show that this leads to more active (vitality and biodiversity) and robust (survival) dynamics. This effectively suggests that coupling such physical processes to reactive systems allows the separation of environments into different milieux and could constitute a simple mechanism to form ecosystem frontiers or elementary interfaces that would protect and favour the development of fragile auto-poietic systems.

Ground-based methods reproduce space-flight experiments and show that weak vibrations trigger microtubule self-organisation.

The effect of weightlessness on physical and biological systems is frequently studied by experiments in space. However, on the ground, gravity effects may also be strongly attenuated using methods such as magnetic levitation and clinorotation. Under suitable conditions, in vitro preparations of microtubules, a major element of the cytoskeleton, self-organise by a process of reaction-diffusion: self-organisation is triggered by gravity and samples prepared in space do not self-organise. Here, we report experiments carried out with ground-based methods of clinorotation and magnetic levitation. The behaviour observed closely resembles that of the space-flight experiment and suggests that many space experiments could be carried out equally well on the ground. Using clinorotation, we find that weak vibrations also trigger microtubule self-organisation and have an effect similar to gravity. Thus, in some in vitro biological systems, vibrations are a countermeasure to weightlessness.

Microtubule self-organisation by reaction-diffusion processes in miniature cell-sized containers and phospholipid vesicles.

Under appropriate conditions, in vitro microtubule preparations self-organise over macroscopic distances by a process of reaction and diffusion. To investigate whether such self-organisation can also occur in objects as small as a cell or an embryo we carried out experiments in miniature containers of cellular dimension. When assembled under self-organising conditions in wells of 120-500 microm, microtubules developed organised structures. Self-organisation is strongly affected by shape, being highly favoured by elongated forms. In wells of more complex shape, geometrical factors may either oppose or strengthen one another and so inhibit or reinforce self-organisation. Microtubules were also assembled within phospholipid vesicles of 2-5 microm diameter. Under self-organising conditions, we observed large shape changes from spheroids to long tubes (50-100 microm) and intertwined coils. We conclude that self-organisation of microtubules by reaction-diffusion processes can occur in containers of cellular dimensions and is capable of strongly deforming the cellular membrane.

Brief exposure to high magnetic fields determines microtubule self-organisation by reaction-diffusion processes.

A frequent feature of microtubule organisation in living systems is that it can be triggered by a variety of biochemical or physical factors. Under appropriate conditions, in vitro microtubule preparations self-organise by a reaction-diffusion process in which self-organisation depends upon, and can be triggered by, weak external physical factors such as gravity. Here, we show that self-organisation is also strongly dependent upon the presence of a high magnetic field, for a brief critical period early in the process, and before any self-organised pattern is visible. These results provide evidence that external physical factors trigger self-organisation by way of an orientational bias that breaks the symmetry of the reaction-diffusion process. As microtubule organisation is central to many cell functions, this behaviour provides a mechanism by which strong magnetic fields can intervene in biological processes.

Microtubule self-organisation by reaction-diffusion processes causes collective transport and organisation of cellular particles.

BACKGROUND: The transport of intra-cellular particles by microtubules is a major biological function. Under appropriate in vitro conditions, microtubule preparations behave as a 'complex' system and show 'emergent' phenomena. In particular, they form dissipative structures that self-organise over macroscopic distances by a combination of reaction and diffusion. RESULTS: Here, we show that self-organisation also gives rise to a collective transport of colloidal particles along a specific direction. Particles, such as polystyrene beads, chromosomes, nuclei, and vesicles are carried at speeds of several microns per minute. The process also results in the macroscopic self-organisation of these particles. After self-organisation is completed, they show the same pattern of organisation as the microtubules. Numerical simulations of a population of growing and shrinking microtubules, incorporating experimentally realistic reaction dynamics, predict self-organisation. They forecast that during self-organisation, macroscopic parallel arrays of oriented microtubules form which cross the reaction space in successive waves. Such travelling waves are capable of transporting colloidal particles. The fact that in the simulations, the aligned arrays move along the same direction and at the same speed as the particles move, suggest that this process forms the underlying mechanism for the observed transport properties. CONCLUSIONS: This process constitutes a novel physical chemical mechanism by which chemical energy is converted into collective transport of colloidal particles along a given direction. Self-organisation of this type provides a new mechanism by which intra cellular particles such as chromosomes and vesicles can be displaced and simultaneously organised by microtubules. It is plausible that processes of this type occur in vivo.

Comparison of reaction-diffusion simulations with experiment in self-organised microtubule solutions.

This article deals with the physical chemical processes underlying biological self-organization by which an initially homogenous solution of reacting chemicals spontaneously self-organizes so as to give rise to a preparation of macroscopic order and form. Theoreticians have predicted that self-organization can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence or absence of an external field, such as gravity, at a critical moment early in the self-organizing process may determine the morphology that subsequently develops. We have found that the formation in vitro of microtubules, a major element of the cellular skeleton, show this type of behaviour. The microtubule preparations spontaneously self-organise by way of reaction and diffusion, and the morphology of the state that forms depends on the presence of gravity at a critical moment early in the process. We have developed a numerical reaction-diffusion scheme, based on the chemical dynamics of a population of microtubules, which simulates the experimental self-organisation. In this article we outline the main features of these simulations and discuss the manner by which a permanent dialogue with experiment has helped develop a microscopic understanding of the collective behaviour.

Computing with bacterial constituents, cells and populations: from bioputing to bactoputing.

The relevance of biological materials and processes to computing-alias bioputing-has been explored for decades. These materials include DNA, RNA and proteins, while the processes include transcription, translation, signal transduction and regulation. Recently, the use of bacteria themselves as living computers has been explored but this use generally falls within the classical paradigm of computing. Computer scientists, however, have a variety of problems to which they seek solutions, while microbiologists are having new insights into the problems bacteria are solving and how they are solving them. Here, we envisage that bacteria might be used for new sorts of computing. These could be based on the capacity of bacteria to grow, move and adapt to a myriad different fickle environments both as individuals and as populations of bacteria plus bacteriophage. New principles might be based on the way that bacteria explore phenotype space via hyperstructure dynamics and the fundamental nature of the cell cycle. This computing might even extend to developing a high level language appropriate to using populations of bacteria and bacteriophage. Here, we offer a speculative tour of what we term bactoputing, namely the use of the natural behaviour of bacteria for calculating.