Romanian Reports in Physics, Vol. 63, No. 3, P. 624–640, 2011 FROM GIANT OCEAN SOLITONS TO CELLULAR IONIC NANO-SOLITONS MILJKO V. SATARIĆ1, MILE S. DRAGIĆ2, DALIBOR L. SEKULIĆ1 1 Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia, E-mail: bomisat@neobee.net, Email: dalsek@yahoo.com 2 Production “Mile Dragić”, Makedonska 2, 23000 Zrenjanin, Serbia, E-mail: mdragic@armyequipment.com Received January 14, 2011 Abstract. It is well known that soliton bearing nonlinear differential equations show up in many areas of physics when waves can propagate in a weakly nonlinear and dispersive media. Here we will show the surprising analogy between the single hump solitary waves appearing in long wave regimes in relatively shallow water and mathematically quite similar description of nano-solitons of ionic waves propagating along microtubules in living cells. Key words: soliton, Laplace and Euler equations, microtubule, nonlinear transmission line, ionic waves. 1. WEAKLY NONLINEAR SHALLOW WATER WAVE REGIME We first consider an inviscid, incompressible and non-rotating flow of fluid of constant depth h. We take the direction of flow as x-axis and z-axis positively upward the free surface in gravitational field. The free surface elevation above the undisturbed depth h is η(x, t), so that the wave surface at height z = h + η(x, t), while z = 0 is horizontal rigid bottom [1, 2, 3]. Let φ(x, z, t) be the scalar velocity potential of the fluid lying between the bottom (z = 0) and free space η(x, t), Fig. 1. Then we could write the Laplace and Euler equation with the boundary conditions at the surface and at the bottom, respectively, as follows: ∂2ϕ ∂2ϕ + =0 ; ∂x 2 ∂z 2 0< z<h+η ; 2 ∂ϕ 1  ∂ϕ G ∂ϕ G  +  i + k  + gη = 0 ; ∂t 2  ∂x ∂z  − ∞ < x < +∞ (1) z=h+η (2) 2 From ocean solitons to cellular ionic nano-solitons 625 ∂η ∂η ∂ϕ ∂ϕ + − =0 ∂t ∂x ∂x ∂z (3) ∂ϕ =0 ; ∂z z=0 . (4) It is useful to introduce two following fundamental dimensionless parameters: σ= η0 <1 ; h 2 h δ =   <1 , l (5) where η0 is the wave amplitude (Fig. 1), and l is the characteristic length-like wavelength. Accordingly, we also take a complete set of new suitable nondimensional variables: X= x ; l Z= z ; h τ= ct ; l Ψ= η ; η0 Φ= h ϕ, η0 lc (6) where c = gh is the shallow-water wave speed, with g being gravitational acceleration. Fig. 1 – The shape of gravitational water wave for fixed t with characteristic parameters: h is undisturbed depth and η is the wave elevation with η0 being its amplitude. reads In terms of (5) and (6) the initial system of equations (1), (2), (3) and (4) now δ ∂2 Φ ∂2 Φ + =0 ; ∂X 2 ∂Z 2 (7) 626 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 2 3 2 ∂Φ σ  ∂Φ  σ  ∂Φ  +   +   +Ψ=0 ; ∂τ 2  ∂X  2δ  ∂Z  ∂Ψ  ∂Φ ∂Ψ  1 ∂Φ + σ =0 ; − ∂τ  ∂X ∂X  δ ∂Z ∂Φ =0 ; ∂Z Z = 1 + σΨ , Z = 1 + σΨ , Z =0 . (8) (9) (10) Expanding Φ(x, τ) in terms of δ Φ = Φ 0 + δ Φ1 + δ2 Φ 2 , (11) and using the dimensionless wave particles velocity in x-direction, by the definition u = ∂Φ/∂X, then substituting of (11) into (7-9), with retaining terms up to linear order of small parameters (δ, σ) in (8), and of second order in (9), we get ∂Φ 0 δ ∂ 2 u 1 − + Ψ + σu 2 = 0 , ∂τ 2 ∂τ∂X 2 (12) ∂Ψ ∂Ψ 1 ∂u δ ∂ 3 u . + σu + (1 + σΨ ) = ∂τ ∂X δ ∂X 6 ∂X 3 (13) Making the differentiation of (12) with respect to X, and rearranging (13), we get ∂u ∂u ∂Ψ 1 ∂3u + σu + − δ =0 , ∂τ ∂X ∂X 2 ∂X 2 ∂τ (14) 1 ∂3u ∂Ψ ∂ u (1 + σΨ )  − δ 3 = 0 . + ∂τ ∂X 6 ∂X (15) Returning back to dimensional variables η(x, t) and v = dφ/dx, (14) now reads ∂v ∂v ∂η 1 2 ∂ 3 v . +v +g = h ∂t ∂x ∂x 3 ∂x 2 ∂t (16) We could define the new function V(x, t) unifying the velocity and displacement of water particles as follows: v= 1 ∂V ; h ∂t η=− ∂V , ∂x (17) implying that (16) becomes 2 ∂2 V ∂ 2 V 1 ∂  ∂V  1 2 ∂4 V − + = gh h .   ∂t 2 ∂x 2 2h ∂x  ∂t  3 ∂x 2 ∂t 2 (18) 4 From ocean solitons to cellular ionic nano-solitons 627 We seek for traveling wave solutions with moving coordinate of the form ξ = x – vt and with wave speed v, which reduces (18) into ordinary nonlinear differential equation as follows:     ( v2 − gh ) ddξV2 + 2vh ddξ  dV  dξ 2 2 2 − v2 h2 d 4 V =0 . 3 dξ 4 (19) Integrating (19) once, and setting dV =W , dξ (20) d 2W = αW 2 + βW + C1 , dξ 2 (21) we get with the following abbreviations α= 3 −3 h ; 2 β= 3 ( v 2 − gh ) ( vh )2 , (22) C1 is the constant of integration. Taking the next integration of (21) one obtains 2 W3 W2 1  dW  = α + β + C1W + C .   2  dξ  3 2 (23) When v2 = v02 = gh, β = 0 and above equation reduces to 2  dW  2 3   = αW + 2C1W + 2C2 . ξ d 3   (24) The solution of this equation can be assumed in the form W = Aθ(ξ) , (25) where θ(ξ) represents the elliptic function satisfying the differential equation 2  dθ  3   = 4θ − p θ − q , d ξ   (26) with the parameters p, q, obeying the condition p 3 − 27 q 2 > 0 . (27) Inserting (25) in (24), and equating the coefficients of equal powers of θ(ξ), one gets 628 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 2α 3 A ; 3 − A2 p = 2C1 A; 5 4 A2 = (28) − A2 q = 2C2 . Last equalities yield A= 6 ; α p=− αC1 ; 3 q=− α 2 C2 . 18 (29) Since parameters p and q satisfy inequality (27), p must be positive imposing that α and C1 should be of opposite signs, (C1 < 0). The condition (27) implies the restriction 4C13 + 9αC22 < 0. The solution of (24) now reads W (ξ) = 6 θ ( ξ + C3 ; p, q ) , α (30) with new constant of integration C3. Eventually, the bounded periodic solution is now represented by W (ξ) = 6 e3 + ( e2 − e3 ) sn 2 α ( ) e1 − e3 ξ + c  ,  (31) with arbitrary constant c and parameters ej (j = 1, 2, 3) being the real roots of the cubic equation 4e3 − pe − q = 0 ; e1 > e2 > e3 . (32) Jacobian-sine elliptic function sn follows from relation θ ( ξ ) = e3 + ( e2 − e3 ) sn 2 ( ) e1 − e3 ξ + c . (33) In order to get the solitary wave limit we take the modulus of Jacobian elliptic function defined as m= e2 − e3 , e1 − e3 (34) being equal to unity. In this limit, by using definitions (17) and (20), we get bellshaped soliton describing in this case giant solitonic water wave (Fig. 2) 6 From ocean solitons to cellular ionic nano-solitons η ( x, t ) = 4 h 3 {( e − e ) sec h 1 3 2 629 }  e1 − e3 ( x − v0 t ) + c  − e1 .   (35) Fig. 2 – The shape of water soliton wave for fixed t with pertaining parameters: speed v0, length l, depression 4h3e1 and elevation η0 = 4h3(e1–e3). Let us just make a simple illustration. If we take the typical ocean depth of the order of h = 103 m, solitonic speed is of the order of v0 = gh = 102 m km = 360 . s h The usual length of a soliton created by earth-quake in such depths is of the order of l = 100 km = 105 m. It safely satisfies the condition of (5). Consequently, (35) thus gives e1 − e3 = 10−5 m −1 , or e1 − e3 = 10−10 m −2 , yielding, from (32), with appropriate integration constants, to the following real roots e1 = e2 = −10−10 m −2 ; e3 = −2 × 10−10 m −2 . The corresponding solitonic peak is η0 = 4 × 109 × 5 × 10−10 m = 2m. It brings the realistic, very moderate amplitude which can not be easily identified in the open sea. Numerical solution of this case obtained with the aid of Matlab is shown in Fig. 3. 630 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 7 Fig. 3 – Numerical solution of solitonic water wave for ocean depth h = 103 m, and length l = 100 km. 2. NONLINEAR IONIC WAVES ALONG MICROTUBULES We now examine in more detail a new microscopic biological phenomenon leading to quite analogous equations and pertinent solution as in Section 1. Microtubules (MTs) are cytoskeletal biopolymers made up of GTP-dependent α, β-tubulin protein dimer assemblies [4]. MTs have geometry of long hollow cylinders having outer diameter of 25 nm and inner diameter of 15 nm, with lengths that typically span up to several micrometers (Fig. 4). They consist of 13 parallel protofilaments in vivo. Fig. 4 – A MT hollow cylinder of 13 parallel protofilaments with denoted characteristic dimensions: outer and inner diameters of 25 nm and 15 nm and tubulin dimer length of 8 nm. 8 From ocean solitons to cellular ionic nano-solitons 631 Recently it has become apparent that neurons might utilize MT networks in cognitive processing via MT associated proteins (MAPs) in neuronal processes such as learning and memory. MTs are also linked to the regulation of a number of ion channels, thus contributing to the electrical activity of excitable cells [5]. Very recently, it was elaborated a new model of MT as the electric nonlinear transmission line [6]. In this reference one could find the details about calculations of capacity, resistivity and inductivity of basic unit of the model. This unit is the elementary ring (ER) consisting of 13 dimers with characteristic dimensions represented in Fig. 5a. Each dimer has two outward protruding rod-like tubulin tails (TTs), Fig. 5a and Fig. 5b. One of them, disposed on β-monomer, is flexible and able to change its length depending on the amount of counter-ions condensed along its negative surface. This circumstance is the essential argument why this model includes the capacity of MT surface which is nonlinear [6]. a) b) Fig. 5 – a) The landscape of a tubulin dimer with dimensions of TTs the length of 4.5 nm and diameter of 1nm, and surface charge distribution according to Tuszynski et al [7]; b) the shape of TTs, where β-TT is being shrank, thus diminishing its capacity. The basic idea here is to farther amend the underlying model taking into account the presence and the functions of nano-pores (NPs) embedded in MT wall, as presented in Fig. 6. Fig. 6 – The sketch of a NP with approximates shape and dimensions. The conical shape follows from MT geometry. 632 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 9 These NPs act like ionic channels (ionic pumps) according to the investigations performed by the group lead by Eisenberg [8, 9]. This point will be elaborated later in more details. Let us first mention the properties of an ER. MT surface is highly negative due to pertaining amino acids which have lost some protons, according to polyelectrolyte character of MTs. Around MTs surface the ionic cloud (IC) of positive counter-ions is condensed. This IC is separated of bulk cytosol by the shell depleted of any ions (Fig. 7), having the thickness named Bjerrum-length (lB), estimated to be lB = e2 = 6.7 × 10−10 m . 4πε o εkB T (36) Here e is the charge of an electron, ε0 the permittivity of vacuum, ε = 80 the relative permittivity of cytosol, kB Boltzmann’s constant, and T = 310 K is the physiological temperature. This depleted shell plays the role of dielectric and confines ionic flow within IC. Fig. 7 – The geometry of a MT with ionic cloud and depleted layer clearly depicted, including corresponding dimensions: the thickness of IC: λ = 2 nm and Bjerrum length, lB = 0.67 nm. The top part of a) shows α and β TTs. We will here completely omit the calculations of the resistive components of the ER just taking over the values from [6] as follows: the static part of capacity of an ER is estimated to be 10 From ocean solitons to cellular ionic nano-solitons 633 C0 = 1.32 × 10−15 F, (37) L0 = 8 × 10−16 H, (38) wile, the inductance is and longitudinal ohmic resistance amounts R0 = 109 Ω. (39) But, since ER’s capacitance should change with an increasing concentration of counter-ions, it implies that the charge on this capacitor diminishes with increasing the voltage in a nonlinear way as follows: Qn = C0 (1 − bvn ) vn , (40) where, the parameter b is restricted to obey the inequalities 0 < b < 1; bvn << 1, (40.a) providing that the change of capacity is being small enough (bvn<<1), since the change of the area of a β-TT is a few percents of the total outer surface of the corresponding tubulin dimer plus the surface of α-TT. Inasmuch the peak value of vn in vivo is of the order of 0.1 Volt, the condition (40.a) is fulfilled, for example if b = 0.5. We now pay more attention on the role of NPs in the context of negative incremental resistance (conductance). Eisenberg’s group [8, 9] examined a single asymmetric NP system whose rectifying properties for ionic current are changed dramatically by addition of a millimolar concentration of calcium ions. The voltage-current function for such NPs exhibits negative incremental resistance for a particular domain of negative voltage across NP, qualitatively represented on Fig. 8 (segment CC’). Fig. 8 – The qualitative shape of currentvoltage characteristics for NPs in the context of Eisenberg’s group’s work [8, 9]. The segment CC’ shows the negative incremental resistance. 634 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 11 The value of vc, from Fig. 8, is defined by the structural features of the NP and by the calcium concentration [9]. We are aware that above experiment is far for being adequate for the description of current voltage function for NPs in MT. But some general features could probably hold even for these geometrically and chemically more complex dynamical natural structures. In order take into account dynamic character of conductive properties of NPs in MT we here use the negative conductance of NP to be voltage dependent in a dynamic way as follows: dv  Gn = −G0  1 − γ n dt   ,  (41) with γ being appropriate small parameter providing the inequality γ dvn < 1. dt (41.a) Thus (41) involves an additional nonlinearity of this system. 2.1. The Model of a MT as Nonlinear Transmission Line We could now consider a long transmission-line (or ladder networks) composed of lumped sections equal to introduced ER’s. A typical section scheme is shown in Fig. 9. The length of an ER is ∆ = 8 nm is being equal to the length of a tubulin dimmer, Fig 5a. The longitudinal current consists of the series of inductance Ln, which is so small to be safely ignored, and ohmic resistance Rn for IC in an ER as being estimated in Eq. (39). The nonlinear capacity Cn is in parallel with total conductance Gn of all 13 NPs consisted in ER. Fig. 9 – An effective circuit diagram for the n-th ER with characteristic elements for Kirchhoff’s laws. 12 From ocean solitons to cellular ionic nano-solitons 635 Applying Kirchhoff’s law to the considered ladder, envisaged as the coupled electrical circuits, we write down ∂Qn + Gn vn , ∂t in − in +1 = vn −1 − vn = Rn in . (42) (43) The important point here is the facts that Ln could be discarded. Inserting (40) and (44) in (42) one gets in − in +1 = −G0 vn + γG0 vn ∂vn ∂v ∂v + C0 n − 2bC0 vn n , ∂t ∂t ∂t vn −1 − vn = R0 in ( Rn = R0 ) . (44) (45) It is very convenient to establish the characteristic reactive impedance of an ER in a natural way: Z= 1 . C0 ω (46) It enables us to introduce the new auxiliary function, u(x, t), unifying voltage and current as follows un = Z 1/ 2 in = Z −1/ 2 vn . (47) Based on the fact that the above functions change gradually from a given ER to its neighbours it is justified to expand un in a continuum approximation using a Taylor series in term of a small spatial parameter ∆ (the length of an ER):  ∂u  ∆ 2  ∂ 2 u  ∆ 3  ∂ 3 u  ∆ 4  ∂ 4 u  un ±1 = u ± ∆   +  ±  +   ± ...  ∂x  2!  ∂x 2  3!  ∂x3  4!  ∂x 4  (48) By using (44–48), we now have ∂u ∆ ∂ 2 u ∆ 2 ∂ 3 u ∆ 3 ∂ 4 u ∆ 4 ∂ 5 u G0 Z u+ + + + + − 6 ∂x 3 24 ∂x 4 120 ∂x5 ∂x 2 ∂x 2 ∆ ZC0 ∂u  γZ 3 / 2 G0 2bZ 3 / 2 C0  ∂u + + − = 0, u ∆ ∂t  ∆ ∆  ∂t (49) ∂u ∆ ∂ 2 u ∆ 2 ∂ 3 u ∆ 3 ∂ 4 u ∆ 4 ∂ 5 u R0 Z −1 − + − + − = 0. ∂x 2 ∂x 2 ∆ 6 ∂x 3 24 ∂x 4 120 ∂x5 (50) Combining last two equations and ignoring small terms beyond the order of ∆3, we get 636 2 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 13 ZC0 ∂u ( 2bC0 − γG0 ) Z 3 / 2 ∂u ∂u ∆ 2 ∂ 3 u 1 −1 + + − + − = 0. (51) R Z G Z u u ( ) 0 0 ∂x ∆ ∂t ∆ ∂t 3 ∂x3 ∆ The important step in our model is the assumption that the ohmic resistance, R0, should be balanced by the negative incremental conductance, –G0, thus discarding the third term in (51) ( R0 Z −1 − G0 Z ) = 0. (52) Similarly, as in the case of water waves, we should go over to dimensionless variables. Let us first estimate the characteristic scales of length and time. The length of an ER (∆ = 8 nm) is such length. The characteristic time could be the period of charging (discharging) of ER capacitor through the resistance R0, τ = R0 C0 . (53) Taking the numerical values from (37) and (39) we have τ = 109 Ω × 1.32 × 10−15 F = 1.32 × 10-6 s. (54) Thus, the characteristic velocity of spreading the ionic wave along the ladder could be estimated as v0 = 8 × 10−9 m cm ∆ . = = 0.6 −6 s τ 1.32 × 10 s (55) It is now convenient to use the progressive traveling-wave form as follows: x v t  t  t x x v0  = u  − s  ; u ( x, t ) = u  − v  = u  − ∆ ∆ τ ∆ ∆  ∆ v0 s= v , v0 (56) with dimensionless speed, s. Introducing the new dimensionless space-time variable, ξ ξ= x t −s , ∆ τ (17) one has ∂u 1 du = ; ∂x ∆ dξ ∂ 3 u 1 d3 u = ; ∂x3 ∆ 3 dξ3 ∂u s du =− . ∂t τ dξ (58) It enables us to get rid of partial differential equation (51) yielding the ordinary one d3u du Z 3/ 2s  ZC0 s  du 3 2 3 − − + ( 2bC0 − γG0 ) u = 0 .   3 dξ dξ τ  τ  dξ (59) 14 From ocean solitons to cellular ionic nano-solitons 637 If we use the definition (47), u = Z1/2i, and introduce the dimensionless current W = i / i0 , (60) with i0 being the peak value, we have u = Z 1/ 2 i0W = Z 1/ 2 i. (61) Z 2 i0 s d 3W dW  ZC0  dW 3 2 3 − s − + = 0. ( 2bC0 − γG0 ) W   3 dξ dξ τ  τ  dξ (62) Replacing (61) into (62) we get Performing first integration yields:  d 2W 1  Z 2 i0 s  ZC0  − − + s W 3 2 ( 2bC0 − γG0 )  W 2 + C1 = 0. 3   2 τ dξ 2  τ   (63) Comparing (62) with (21) we can use the same abbreviations α=  1  Z 2 i0 s ( 2bC0 − γG0 )  ; 3 τ 2   ZC  β = −3  0 s − 2  .  τ  (64) We here also use the conditions β = 0, leading to the velocity of ionic current defined as s= Taking Z = 2τ . ZC0 (65) 1 τ = = 1.6 × 108 Ω, we get C0 ω C0 2π s = 4π; v = 4πv0 = 7.5 cm . s (66) This is very interesting intermediate value of biological speed. For example, the speed of ionic diffusion waves in cell is of the order of 10-3 cm/s, while the speed of action potential in nerve cell ranges 3×103 cm/s. We could finally estimate the term γG0. From (41) it is possible to average the derivative ∂vn / ∂t ≈ 0.1/ τ ≈ 105 . This implies an inequality γ < 10−6 in order to obey (41.a). Otherwise from (52) we estimate G0 = 4×10-8 S. Thus, (63) now reads d 2W = αW 2 + C1 . dξ 2 (67) 638 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 15 Making the next integration of (67) one obtains 2  dW  2 3   = αW + 2C1W + C2 . d 3 ξ   (68) This is the same equation as was (24). The remaining formal procedure is already given by set of equations (25-35), yielding i ( x, t ) = i0 = { } 2τ ( e1 − e3 ) sec h 2  e1 − e3 ( x − v0 t ) + c  − e1 ; Z ( 2bC0 − γG0 ) 2 2τ . 2 Z ( 2bC0 − γG0 ) (69) Fig. 10 – The shape of ionic current nano-soliton along a microtubule given for fixed time. We will now examine the two options of solution of (69). First is the case while the term bC0 dominates over the NP nonlinearity γG0, where we have the reductions to the form (Fig. 10) i ( x, t ) = τ Z bC0 2 {( e − e ) sec h 1 3 2 }    e1 − e3 ( x − v 0 t ) + c  − e1 , τ , i0 = 2 Z bC0 (70) where ei (i = 1, 2, 3) are the roots of cubic equation: 4e3 − C1 ( 2Zbi0 ) e + C2 ( 2Zbi0 ) = 0, 2 (71) 16 From ocean solitons to cellular ionic nano-solitons 639 and which we will call ionic nano-soliton (INS). Taking b = 0.1 V-1, Z = 1.6×108 Ω and choosing that i0 has the order of 10-9 A, for appropriate integration constants one gets the following cubic equation e3 − 3 × 10−6 e + 2 × 10−9 = 0, (72) with corresponding roots: e1 = e2 = 10−3 ; e3 = −2 × 10−3. (73) It brings about the width of INS to be of the order of 20 ERs. Numerical solution of this case obtained with the aid of Matlab is shown in Fig. 11. Fig. 11 – Numerical solution of INS along microtubules for b = 0.1 V-1 and i0 = 10-9 A. Second option is more realistic while two nonlinearities are competitive. It causes the increase of INS amplitude and brings it more localized. We could comment how the value of frequency ω may influence this competition. If voltage frequency ω changes, the static part of conductivity G0 also changes through (46, 53) G0 = R0 = R0 C02 ω2 . Z2 (74) This in order to get γG0 as being competitive with bC0, ω should be of the order of 103 s-1. 640 Miljko V. Satarić, Mile S. Dragić, Dalibor L. Sekulić 17 3. CONCLUSIONS AND DISCUSSION In this paper we have shown how two very different physical phenomena can be described by formally the same mathematical model exhibiting the common features which leads to existence of the permanent profile bell-shaped soliton solutions. It results from equilibrium between two effects, the nonlinearity that tends to localize the wave, while dispersion spreads it out. The origins of respective nonlinearities are quite different. In the case of water waves the nonlinearity originates from kinetic energy, contained in Euler’s equation (2). In the other hand, the potential energy of ionic wave within INS along MTs captures nonlinearity from biophysical properties enabling the change of ER capacity and NPs conductivity according to equations (40, 41). We here paid more attention to the INSs along MTs due to the importance of this subtle phenomenon in better understanding of basic neuronal functions, including learning and memory processes. Our present model offers the satisfactory results regarding orders of magnitude of ionic current, speed of pertaining INSs in the context of still scarce available experimental evidences. We especially mention an interesting experimental assay performed by the group led by Tuszynski [11]. They have calculated that electrical amplification of ionic currents by MTs is in some sense equivalent to their ability to act as biomolecular living transistors. 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