Slobodan Zdravković

Modified extended tanh-function method and nonlinear dynamics of microtubules

Abstract We here present a model of nonlinear dynamics of microtubules (MT) in the context of modified extended tanh-function (METHF) method. We rely on the ferroelectric model of MTs published earlier by Sataric et al. [1] where the motion of MT subunits is reduced to a single longitudinal degree of freedom per dimer. It is shown that such nonlinear model can lead to existence of kink solitons moving along the MTs. An analytical solution of the basic equation, describing MT dynamics, was compared with the numerical one and a perfect agreement was demonstrated. It is now clearer how the values of the basic parameters of the model, proportional to viscosity and internal electric field, impact MT dynamics. Finally, we offer a possible scenario of how living cells utilize these kinks as signaling tools for regulation of cellular traffic as well as MT depolymerisation.

Nonlinear dynamics of microtubules —A longitudinal model

In the present letter we describe a model of nonlinear dynamics of microtubules (MTs) assuming a single longitudinal degree of freedom per tubulin dimer. This is a longitudinal displacement of a dimer at a certain position with respect to the neighbouring one. A nonlinear partial differential equation, describing dimers dynamics within MT, is solved both analytically and numerically. It is shown that such a nonlinear model can lead to the existence of kink solitons moving along the MTs. The internal electrical field strength is calculated using two procedures and a perfect agreement between the results is demonstrated. This enabled the estimation of the total energy, kink velocity and kink width. To simplify the calculation of the total energy we stated and proved a useful auxiliary theorem.

Localized modulated waves and longitudinal model of microtubules

We here study nonlinear dynamics of microtubule (MT). A so-called u - model is explained in detail. A single longitudinal degree of freedom per MT subunits is assumed. It is known that a continuum approximation of a basic discrete dynamical equation of motion enables existence of kink and antikink solitons along MT. In this paper we use semi-discrete approximation for this equation and show that modulated solitonic waves could propagate as well. We suggest possible biological implications of these waves. Also, a detailed parameter analysis is performed.

Transverse interaction in DNA molecule

Interaction between nucleotides at a same site belonging to different strands is studied. This interaction is modelled by a Morse potential which depends on two parameters. We study a relationship between the parameters characterizing AT and CG pairs. We show that certain circumstances, i.e. certain values of these parameters, bring about a negligible influence of inhomogeneity on the solitonic dynamics.

DNA-RNA transcription as an impact of viscosity

The impact of viscosity on DNA dynamics is studied both analytically and numerically. It is assumed that the viscosity exists at the segments where DNA molecule is surrounded by RNA polymerase. We demonstrate that the frictional forces destroy the modulation of the incoming solitonic wave. We show that viscosity, crucial for demodulation, is essential for DNA-RNA transcription.

Nonlinear dynamics of C-terminal tails in cellular microtubules.

The mechanical and electrical properties, and information processing capabilities of microtubules are the permanent subject of interest for carrying out experiments in vitro and in silico, as well as for theoretical attempts to elucidate the underlying processes. In this paper, we developed a new model of the mechano-electrical waves elicited in the rows of very flexible C-terminal tails which decorate the outer surface of each microtubule. The fact that C-terminal tails play very diverse roles in many cellular functions, such as recruitment of motor proteins and microtubule-associated proteins, motivated us to consider their collective dynamics as the source of localized waves aimed for communication between microtubule and associated proteins. Our approach is based on the ferroelectric liquid crystal model and it leads to the effective asymmetric double-well potential which brings about the conditions for the appearance of kink-waves conducted by intrinsic electric fields embedded in microtubules. These kinks can serve as the signals for control and regulation of intracellular traffic along microtubules performed by processive motions of motor proteins, primarly from kinesin and dynein families. On the other hand, they can be precursors for initiation of dynamical instability of microtubules by recruiting the proper proteins responsible for the depolymerization process.

Role of nonlinear localized Ca2+ pulses along microtubules in tuning the mechano–sensitivity of hair cells

This paper aims to provide an overview of the polyelectrolyte model and the current understanding of the creation and propagation of localized pulses of positive ions flowing along cellular microtubules. In that context, Ca(2+) ions may move freely on the surface of microtubule along the protofilament axis, thus leading to signal transport. Special emphasis in this paper is placed on the possible role of this mechanism in the function of microtubule based kinocilium, a component of vestibular hair cells of the inner ear. We discuss how localized pulses of Ca(2+) ions play a crucial role in tuning the activity of dynein motors, which are involved in mechano-sensitivity of the kinocilium. A prevailing notion holds that the concentration of Ca(2+) ions around the microtubules within the kinocilium represents the control parameter for Hopf bifurcation. Therefore, a key feature of this mechanism is that the velocities of these Ca(2+) pulses be sufficiently high to exert control at acoustic frequencies.

Morse potential in DNA molecule – An experiment proposal

We rely on the helicoidal Peyrard-Bishop model for DNA dynamics. Interaction between nucleotides at a same site belonging to different strands is modelled by a Morse potential energy. This potential depends on two parameters that are different for AT and CG pairs, which is a possible source for inhomogeneity. It was shown recently (Zdravković and Satarić 2011) that certain values of these parameters bring about a negligible influence of inhomogeneity on the solitonic dynamics. We propose an experiment that should be carried out in order to determine the values of both of these parameters.

A biophysical model of how α–tubulin carboxy–terminal tails tune kinesin–1 processivity along microtubule

It appears that so-called post-translational modifications of tubulin heterodimers are mostly focussed at positions of amino acid sequences of carboxy-terminal tails. These changes have very profound effects on microtubule functions especially in connection with cellular traffic in terms of motor proteins. In this study, we elaborated the biophysical model aimed to explain the strategy governing these subtle interplays between structural and functional properties of microtubules. We relied onto Langevin equations including fluctuation-dissipation processes. In that context we found out that small interaction between a charged motor neck domain and oppositely charged carboxy-terminal tail of the α-tubulin plays the decisive role in tuning kinesin-1 motor processivity along microtubules.

Nonlinear DNA dynamics: Nonlinearity versus dispersion

In the present paper we study the impact of dispersion and nonlinearity on DNA dynamics. We rely on the helicoidal Peyrard-Bishop model and use the fact that nonlinear DNA dynamics represents an interplay between nonlinearity and dispersion. We state that a dispersion coefficient P and a coefficient of nonlinearity Q, existing in nonlinear Schrodinger equation, are mutually dependent and show how function Q(P) and P(Q) can be obtained. Also, we show how all this can be used to find a possible interval for the parameter describing helicoidal structure of DNA.