Tuning the traffic of cellular cargos by intrinsic electric fields

PROC. 26th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2008), NIŠ, SERBIA, 11-14 MAY, 2008 Tuning the Traffic of Cellular Cargos by Intrinsic Electric Fields Miljko V. Satarić Abstract— This paper presents a plausible physical mechanism for very important role playing by microtubules and associated intrinsic cell’s electric fields in the process of regulation of very busy traffic of motor proteins within the cell. Key words: microtubule, kinesin, dynein, kink, cargo I. Introduction Cells are organized with different compartments that act as factories. Each factory creates a unique set of products which are distributed to ”consumers” along the tracks constituting the cytoskeleton. The cytoskeleton consists of three mayor types of thin rodlike filaments that span the cytoplasm; actin based filaments, tubulin based filaments called microtubules (MT) and intermediate filaments. The majority of active transport in the cell is driven along MTs via motor proteins (MP) kinesin and dynein, that move toward the plusend and the minus-end of MTs, respectively. The findings from a number of systems suggest that many cargos move bidirectionally along the MTs, reversing the course every few seconds. Mitochondria (MCH) are typically observed to move in both directions along MTs and appear to move to the cellular locations where ATP production is necessary [1], see Fig. 1. Kinesin is a force generating MP which converts the free energy of ATP into the mechanical work used to power the transport of cargos. When moving in vitro a single kinesin tipically takes about 100 steps with a fixed step siza of 8 nm, (Fig. 2). The dynein is almost ten times bigger than kinesin and moves toward minus-end of a MT by the use of multiple ATP molecules. It has been shown that both kinesin and dynein MPs are found on the same cargo at the same time, see Fig. 1, ready for action. Experimental evidencies from a number of systems show that these opposite polarity MPs are somehow coordinated so that they usually do not interfere with each others function; when dynein is active, kinesin is turned off, and vice versa. The mechanism of such coordination is largely unknown. How can the cell control different cargos in a different way in the cytoplasmic background; endosomes and Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia, E-mail: bomisat@neobee.net 978-1-4244-1882-4/08/$25.00 © 2008 IEEE Fig. 1. Cell With Microtubules and Some Cargos Carried by Motor Proteins MCH must move to opposite locations and can do so in the same cell using the same MTs? Apparently, in addition to global signaling, there must also be some sort of efficient local signaling to specifically control different cargos. In that respect we here emphasize the possible role of intrinsic cell’s electric fields associated with the active support of MTs through their excitable C-termini. II. Ferroelectric Features of MT MTs are holow cylinders with a 25 nm outer and 15 nm inner diameter, Fig.3a, constituted of 13 parallel protofilaments assembled of tubulin dimers. The length of each dimer is 8 nm, the width 5 nm and the thickness also 5 nm. Every tubulin dimer terminates with outward pointing two polypeptide chains called C-termini (carboxy-terminal end) abbreviated here as CT, Fig.3b a) b) Fig. 2. Kinesin MP stepping along a MP protofilament Located on the outside of the MT, CTs interact strongly with other proteins, primarly kinesin and dynein. The transport of MPs appears to critically involve an interaction with CTs. Removal of CTs profoundly affects the rate of the transport [2]. Being predominantly constituted of α-helices, which posses permanent dipoles, tubulin dimers exibit net dipolar character, bringing about the ferroelectric features of MT itself. Ferroelectric concept of MTs was widely elaborated by Satarić et al in several stages. For details it could be reffered to series of papers [3,4,5,6], and references therein. Every CT has a net negative charge of up to 11 electrons and interact electrostatically with a) the surface of the tubulin dimer, b) neighboring CTs and c) adjacent MPs. In our model we consider CTs as a thin rigid rods of the length ℓ = 4, 5nm and molecular mass of approximately m = 2kD = 3, 4 · 10−24 kg and an electric dipole which under ionic screening in physiological condition has effective value p = 90 Debye. The collection of CTs of a MT could be conceived as a thin cylindric liquid crystal-like layer where the dominant degree of freedom is the tilt angle θ with respect to the radial axis r = y, Fig. 4. The energy of this layer is the combination of elastic energy, dipolar energy and the dipole-electric field interaction. After straightforward procedure given elsewhere [7] the coubic nonlinear equation of motion in x-direction is obtained as follows γ ∂2θ ∂θ = ρ 2 − aθ3 + bθ + cE ∂t ∂x (1) where γ is the constant of viscous friction, ρ contains elasticity constant and geometry of cyliner, a, b, c depend on elastic a, d polarization and geometric parameters, of MT. Eventually, E is the intrinsic electric field oriented along the cylinder axis-x. Fig. 3. a) The shape of a MT with characteristic dimensions b) Tubulin dimer with extended C-termini The solution of Eq. (1) is slightly distorted kink of the form � 3b �1/2 θ = sin(θ0 + π/3)· �� �a � (2) · 1 − tanh X−XW0s−vs τ + � b �1/2 + a cos(θ0 + 2π/3) where the kink speed has the form: � b �1/2 cos(θ0 + π/3); vs = 3 2a � √ � 1 −1 3 3 a1/2 c E θ0 = 3 cos 2 b and its width is: �−1 � �1/2 �� b �1/2 sin(θ0 + π/3) Ws = 32 a X−X0 −vs τ Ws = ξ. (3) (4) X and τ are the dimensionless coordinate along cylinder axis and the time, and X0 stands for the dimensionless position of the center of a moving kink. Fig. 4. C-termini as the collection of coupled rods The greater nonlinearity b, the faster and the narower kink is. For the condition a1/2 c E << 1 b being satisfied, the kink’s terminal speed is expressed by √ � √ � 3 3 c a ρE = µE (5) vs = √ γb 2 2 where µ stands for kink’s mobility. Interestingly, this linear response law holds even for fields of the order of 105 V /m. The point is that we expect kink should be elicited (launched) by the action of some molecule from the region of the cell where the need for transport of certain cargo exists. III. The Role of The Intrinsic Electric Fields In what follows we pay attention on tuning the kink’s propagation along MTs by the active role of the intrinsic electric fields. The most interesting case is presented by MTs in long neuron cells. The longest neuron in the human body has a single thread like projection (axon) that spans from the base of the spine to the foot, a distance of up to one meter. The most cellular components, including large organells such as MCHs, are synthesized within the cell bodu that lies in the spinal cord. They must be delivered down the axon to the synapse, where the neuron forms an electrically active connection with the muscles that flex the toe. Estimated times that would take for a MCH to diffuse that distance range up to 100 years. In fact this yourney is achieved in just a few days. The axon potential propagating along the neuron carries an electric field of the order of 104 V /m. This field even partially screened along MT causes estimated, Eq. (5), terminal velocitites of kinks involved of the order of mm/s. We farther assume that certain intrinsic alternating electric fields (IAEF) could be elieited in a cell itself. It was suggested that oriented (vicinal) water molecules form the water electric dipole field (WEDF) occuring on either side of the cell membrane. Del Guidice et al [7] have proposed that electromagnetic fields arising from the coherent oscillations of WEDF represent signals comparable in size with the dimensions of cell’s MTs. These fields would be expected to exert an impact on MT dynamics throught the control of CT kink’s motion and consequently regulate the intensive traffic of MPs along MTs. Let us consider the motion of a kink obeying Eq. (1) driven by a harmonic electric force generated by WEDF. The Eq. (1) can be reduced to elimensionless form of nonlinear ordinary differential equation ψξζ + βψξ − ψ 3 + ψ + ε = 0 ψ=θ (6) � a �1/2 . 2b Subscript ξ stands for the derivative with respect to dimensionless argument ξ, Eq. (4). It is reasonable that the dimensionless driving force has a wavelength greater than the average MT length and it harmonically depends on the dimensionless time τ f (τ ) = f0 cos(Ωτ + ϕ0 ) (7) where f0 is the amplitude, Ω is the frequency and ϕ0 is the initial phase of IAEF. Adding Eq. (7) to Eq. (6) we could examine numerically the kink’s dynamics in the new regime. First we examined the case where a strong field E carried by the action potential in a nerve cell is switched on in parallel with the harmonic IAEF, Eq. (7). An examination based on our model provides the order of magnitude for damping and forcing parameters, respectively, as β = 0, 001 and ε = 0, 01. Choosing f0 = 0, 01 and Ω = 0, 01 with the initial conditions ϕ0 = 0 and ∂ψ/∂τ (0) = 0 we find that unidirectional motion on entering in the terminal regime attains the dimensionless speed v = vs /v0 = 0, 00004, see Fig. 5a The actual kink speed is of the order of vs = v · v0 ≈ m 1cm s , where v0 ≈ 220 s is the speed of acoustic CT wave. If the initial phase of IAEF is ϕ0 = π/2 under the same other conditions as above, unidirectional kink motion is faster (v = 0, 0001) and accompanied by oscillations, Fig. 5.b. When the initial phase is reversed ϕ = −π/2, the direction of kink’s motion also reverses and proceeds with velocity vs = −1, 5cm/s, Fig. 6.a. By increasing the amplitude f0 = 0, 1 the velocity of unidirectional motion increases, Fig. 6.b IV. Conclusion Within the framework of our model proposed here MTs are not only passive tracks for intensive transport of cargos in the cell, but also signal relays for electrical, mechanical and biochemical stimuli that may be transduced over distances comparable to the cell size. Here a) b) Fig. 5. a) Trajectory of a kink in the strong constant electric field in parallel with IAEF field (f◦ = 0, 01, Ω = 0, 01) b) Trajectory of a kink under the same conditions as 5a) except ϕ = π/2 a) b) Fig. 6. a) The change of kink’s direction caused by change of initial angle ϕ = −π/2 b) The same conditions as in 5b), but incrised amplitude of IAEF Acknowledgement we made the further refinement of our nonlinear model [5] of dipolar kinks associated with the collective tilts of CTs and excited in regions of a cell where the delivery of specific cargon is needed. The second assumption of our model is that such excited kink could be tuned with respect to the velocity (sign and intensity), the amplitude and frequency of its moving center. The central role in such tunning is attributed to so called WEDF or other coherent fields sustained by the involved cell compartments. The essential point is that an appropriately tuned kink could turn on a proper MP already existing on the cargo enabling it to establish processivity in the right direction along MT. This work was supported by Serbian Ministry of Science with the framework of project No 141018. References [1] [2] [3] [4] [5] [6] [7] [8] R.L. Morris and P.J. Hollenbeck, J. Cell. Sci 104, 917 (1993) Z. Wang and M.P. Sheetz, Biophys. J. 78, 1955 (2000) M.V. Satarić, J.A. Tuszyński and R.B. Žakula, Phys. Rev. E48, 589 (1993) M.V. Satarić, S. Zeković, J.A. Tuszyński and J. Pokorni, Phys. Rev. E58, 6333 (1998) M.V. Satarić and J.A. Tuszynski, Phys. Rev. E67, 011901 (2003) M.V. Satarić, L. 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