On the role of calcium diffusion and its rapid buffering in intraflagellar signaling.

We have considered the realistic mechanism of rapid Ca2+ (calcium ion) buffering within the wave of calcium ions progressing along the flagellar axoneme. This buffering is an essential part of the Ca2+ signaling pathway aimed at controlling the bending dynamics of flagella. It is primarily achieved by the mobile region of calmodulin molecules and by stationary calaxin, as well as by the part of calmodulin bound to calcium/calmodulin-dependent kinase II and kinase C. We derived and elaborated a model of Ca2+ diffusion within a signaling wave in the presence of these molecules which rapidly buffer Ca2+. This approach has led to a single nonlinear transport equation for the Ca2+ wave that contains the effects brought about by both as necessary buffers for signaling. The presence of mobile buffer calmodulin gives rise to a transport equation that is not strictly diffusive but also exhibits a sink-like effect. We solved straightforwardly the final transport equation in an analytical framework and obtained the implied function of calcium concentration. The effective diffusion coefficient depends on local Ca2+ concentration. It is plausible that these buffers' presence can impact Ca2+ wave speed and shape, which are essential for decoding Ca2+ signaling in flagella. We present the solution of the transport equation for a few specified cases with physiologically reasonable sets of parameters involved.

Calcium signal transmission by axonemal microtubules as an optimized information pathway in cilia and flagella.

Calcium plays a key role in signal transduction in eukaryotic cells. Besides controlling local functions of cells calcium ions are responsible for the generation of global signals such as waves and spikes. Pulsatile increases of calcium concentrations are generally considered to have a much higher fidelity of information transfer than simple tonic changes, since they are much less prone to noisy fluctuations. In that respect, it was clearly revealed that Ca2+ has very crucial involvement in many signaling pathways in cilia and flagella. We earlier established a model in which axonemal microtubules exhibit the features of nonlinear polyelectrolitic electric transmissions lines for efficient transport of cations, primarily Ca2+. These microtubules guide accumulated "ionic clods" which serve as the pulsatile signals aimed to regulate pertaining motor proteins, dyneins and kinesis. We here consider such Ca2+ signals in axoneme in the context of Shannon's and Fisher's information theories. It appears that the fast drift of these "ionic clouds" represents the optimized calcium signaling for control of "flagellary beats" as well as intraflagellary transport of proteins essential for the construction, elongation and maintenance of eukaryotic cilia and flagella themselves.

Calcium signaling modulates the dynamics of cilia and flagella.

To adapt to changing environments cells must signal and signaling requires messengers whose concentration varies with time in space. We here consider the messenger role of calcium ions implicated in regulation of the wave-like bending dynamics of cilia and flagella. The emphasis is on microtubules as polyelectrolytes serving as transmission lines for the flow of Ca2+ signals in the axoneme. This signaling is superimposed with a geometric clutch mechanism for the regulation of flagella bending dynamics and our modeling produces results in agreement with experimental data.

Demodulated standing solitary wave and DNA-RNA transcription.

Nonlinear dynamics of DNA molecule at segments where DNA-RNA transcription occurs is studied. Our basic idea is that the solitary wave, moving along the chain, transforms into a demodulated one at these segments. The second idea is that the wave becomes a standing one due to interaction with DNA surrounding, e.g., RNA polymerase molecules. We explain why this is biologically convenient and show that our results match the experimental ones. In addition, we suggest how to experimentally determine crucial constant describing covalent bonds within DNA.

Nonlinear ionic pulses along microtubules.

Microtubules are cylindrically shaped cytoskeletal biopolymers that are essential for cell motility, cell division and intracellular trafficking. Here, we investigate their polyelectrolyte character that plays a very important role in ionic transport throughout the intra-cellular environment. The model we propose demonstrates an essentially nonlinear behavior of ionic currents which are guided by microtubules. These features are primarily due to the dynamics of tubulin C-terminal tails which are extended out of the surface of the microtubule cylinder. We also demonstrate that the origin of nonlinearity stems from the nonlinear capacitance of each tubulin dimer. This brings about conditions required for the creation and propagation of solitonic ionic waves along the microtubule axis. We conclude that a microtubule plays the role of a biological nonlinear transmission line for ionic currents. These currents might be of particular significance in cell division and possibly also in cognitive processes taking place in nerve cells.

A nonlinear model of ionic wave propagation along microtubules.

Microtubules (MTs) are important cytoskeletal polymers engaged in a number of specific cellular activities including the traffic of organelles using motor proteins, cellular architecture and motility, cell division and a possible participation in information processing within neuronal functioning. How MTs operate and process electrical information is still largely unknown. In this paper we investigate the conditions enabling MTs to act as electrical transmission lines for ion flows along their lengths. We introduce a model in which each tubulin dimer is viewed as an electric element with a capacitive, inductive and resistive characteristics arising due to polyelectrolyte nature of MTs. Based on Kirchhoff's laws taken in the continuum limit, a nonlinear partial differential equation is derived and analyzed. We demonstrate that it can be used to describe the electrostatic potential coupled to the propagating localized ionic waves.

Modelling the role of intrinsic electric fields in microtubules as an additional control mechanism of bi-directional intracellular transport.

Active transport is essential for cellular function, while impaired transport has been linked to diseases such as neuronal degeneration. Much long distance transport in cells uses opposite polarity molecular motors of the kinesin and dynein families to move cargos along microtubules. It is clear that many types of cargo are moved by both sets of motors, and frequently in a reverse direction. The general question of how the direction of transport is regulated is still open. The mechanism of the cell's differential control of diverse cargos within the same cytoplasmic background is still unclear as is the answer to the question how endosomes and mitochondria move to different locations within the same cell. To answer these questions we postulate the existence of a local signaling mechanism used by the cell to specifically control different cargos. In particular, we propose an additional physical mechanism that works through the use of constant and alternating intrinsic (endogenous) electric fields as a means of controlling the speed and direction of microtubule-based transport. A specific model is proposed and analyzed in this paper. The model involves the rotational degrees of freedom of the C-termini of tubulin, their interactions and the coupling between elastic and dielectric degrees of freedom. Viscosity of the solution is also included and the resultant equation of motion is found as a nonlinear elliptic equation with dissipation. A particular analytical solution of this equation is obtained in the form of a kink whose properties are analyzed. It is concluded that this solution can be modulated by the presence of electric fields and hence may correspond to the observed behavior of motor protein transport along microtubules.

Gravitational symmetry breaking leads to a polar liquid crystal phase of microtubules in vitro.

Recent space-flight experiments performed by Tabony's team provided further evidence that a microgravity environment strongly affects the spatio-temporal organization of microtubule assemblies. Characteristic time and length scales were found that govern the organization of oriented bundles under Earth's gravitational field (GF). No such organization has been observed in a microgravity environment. This paper discusses physical mechanisms resulting in pattern formation under gravity and its disappearance in microgravity. The subtle interplay between chemical kinetics, diffusion, gravitational drift, thermal fluctuations, electrostatic interactions and liquid crystalline characteristics provides a plausible scenario.

Nonlinear dynamics of microtubules: biophysical implications.

A recently developed model of nonlinear dynamics for microtubules is further expanded based on the biophysical arguments involving the secondary structure of the constitutive protein tubulin and on the ferroelectric properties of microtubules. It is demonstrated that kink excitations arise due to GTP hydrolysis that causes a dynamical transition in the structure of tubulin. The presence of an intrinsic electric field associated with the structure of a microtubule leads to unidirectional propagation of the kink excitation along the microtubule axis. This mechanism offers an explanation of the dynamic instability phenomenon in terms of the electric field effects. Moreover, a possible elucidation of the unidirectional transport of cargo via motor proteins such as kinesin and dynein is proposed within the model developed in this paper.

Models of spatial and orientational self-organization of microtubules under the influence of gravitational fields.

Tabony and co-workers [C. Papaseit, N. Pochon, and J. Tabony, Proc. Natl. Acad. Sci. U.S.A. 97, 8364 (2000)] showed that the self-organization of microtubules from purified tubulin solutions is sensitive to gravitational conditions. In this paper, we propose two models of spatial and orientational self-organization of microtubules in a gravitational field. First, the spatial model is based on the dominant chemical kinetics. The pattern formation of microtubule concentration is obtained (1) in terms of a moving kink in the limit when the disassembly rate is negligible, and (2) for the case of no free tubulin and only assembled microtubules present. Second, the orientational pattern of striped microtubule domains is consistent with predictions from a phenomenological Landau-Ginzburg free energy expansion in terms of an orientational order parameter.

Relationship between the nonlinear ferroelectric and liquid crystal models for microtubules.

Microtubules (MTs), which are the main components of the cytoskeleton, are important in a variety of cellular activities, but some physical properties underlying the most important features of their behavior are still lacking satisfactory explanation. One of the essential enigmas regarding the energy balance in MTs is the hydrolysis of the exchangeable guanosine 5'-triphosphate bound to the beta monomer of the molecule. The energy released in the hydrolysis process amounts to 6.25 x 10(-20) J and has been the subject of many attempts to answer the questions of its utilization. Earlier, we put forward a hypothesis that this energy can cause a local conformational distortion of the dimer. This distortion should have nonlinear character and could lead to the formation of a traveling kink soliton. In this paper we use the formalism of the liquid crystal theory to consider the nonlinear dynamics of MTs. We demonstrate that this new model is formally equivalent to our earlier ferroelectric model which was widely exploited in an attempt to elucidate some important dynamical activities in MTs. We also study the stability of kink solitons against small perturbations and their unusual mutual interactions as well as the interactions with structural inhomogeneities of MTs. Our new approach based on liquid crystal properties of microtubules has been recently corroborated by new insights gained from the electrostatic properties of tubulin and microtubules.

Impact of regulatory proteins on the nonlinear dynamics of DNA.

In this paper we examine the nonlinear dynamics of a DNA chain whose exciton modes are affected by regulatory proteins that may become bound to the DNA chain by hydrogen bonds. The dynamics of the DNA chain is described by the Peyrard-Bishop model. Since this model gives rise to large-amplitude broad oscillations of base pairs, we consider the impact of attached regulatory proteins on the so-called breathers or bubbles. Assuming that an ideal gas of bubbles may exist in the DNA chain at physiological temperatures we adopt a statistical approach to calculate the average size of base-pair stretching under the prevailing conditions.

Nonlinear dynamics of a DNA chain affected by endogenous AC fields.

In this paper we investigate the nonlinear dynamics of a DNA chain in the presence of endogenous AC fields (EACF) generated by the living cell itself. The dynamics of the DNA chain is described in the framework of the nonlinear breather mode. The transition of breather localized modes into open states, affected by AC fields is calculated by using Kubo's formalism for the linear response of the system.

The enigma of microtubules and their self-organizing behavior in the cytoskeleton.

The cytoskeleton of eukaryotic cells contains networks of protein polymers called microtubules which structurally and functionally organize their interiors. Both in vivo and in vitro microtubules exhibit a fascinating and yet poorly understood array of important functions involving complex self-organization phenomena which are very sensitive to physiological and laboratory conditions, respectively. In this paper we discuss the main physical characteristics of microtubules focusing our attention on four particular aspects: (a) the dynamics of their assembly and disassembly processes (b) the types and the range of existence of ordered dipolar phases and (c) modes of energy transfer and (d) information processing capabilities.

The change of microtubule length caused by endogenous AC fields in cell.

In this paper we investigate the change of microtubule cylinder's length in the presence of endogenous AC fields generated by the cell itself. The dynamics of microtubule is described on the basis of classical u4 model. The average stretching is calculated by using Kubo's formalism for linear response of system.