EMPLOYMENT OF THE METHOD OF FACTORIZATION FOR SOLVING PROBLEMS IN NONLINEAR DYNAMICS OF MICROTUBULES

In the present paper we rely on the so called u-mod el f the dynamics of microtubules. We demonstrate how a crucial differen tial equation can be solved using the method of factorization. It was shown that the solu ti ns match the ones obtained using the standard procedure.


INTRODUCTION
Cell cytoskeleton of eukaryotes has a dynamic nature and is composed of three types of filaments: actin filaments, intermediate filaments and microtubules [1]. Microtubules are the most stiff of the three types of filaments. They are composed of heterodimers consisting of α and β tubulins. The heterodimers have the same orientation all along microtubules and their α and β ends are called minus (-) and plus (+) end, respectively.
A row of αβ heterodimers represents a protofilament, while 11-16 protofilaments form a hollow tube (cylinder) which is a microtubule. If the number of protofilaments in microtubule is 13 they all lie parallel to the microtubule axis. Otherwise, they wind slowly helicoidally around the microtubule axis. The length of the αβ heterodimer is nm 8 = l [2,3] and its dipole moment is Debye 337 = p [2,3]. α tubulin is positively and β tubulin is negatively charged [2]. The minus (-) end of microtubule usually starts at an organizing center of microtubules in the cell represented with the so called centrosome, usually with a pair of centrioles [4]. In the interphase cells the centrosome is approximately at the center of the cell. The other, (+) end of microtubule, radiates to the periphery of the cell. An exception to that rule is a neuron, where about 40% of microtubules in their dendrites are oriented with their (-) ends towards the neuron's periphery [4].
Outer and inner diameters of a microtubule are nm 25 and 5nm 1 , respectively. Both α and β monomers bind one GTP molecule. Soon after polymerization into a microtubule the β monomer hydrolyses its GTP molecule to GDP and phosphate P i [4]. This hydrolysis makes the microtubules unstable and prone to disintegration (which happens always from the ends, never from the walls of the microtubule). Microtubules with slowly hydrolysable analogs of GTP are much more stable [4]. Microtubules can be as long as µm 20 in vivo and mm 3 in vitro [4].
In this paper we consider the so called u-model of the microtubule dynamics. A crucial partial differential equation, which describes this dynamics, can be transformed into an ordinary differential equation. It is demonstrated that the latter one can be solved using the method of factorization. Finally, we compare solutions obtained by the method of factorization with the solutions obtained by the standard procedure. Of course, the same procedure can be applied for solving similar equation, i.e. to different models explaining dynamics of microtubules.

THE U-MODEL OF DYNAMICS OF MICROTUBULES
The u-model [2] has the same form as the z-model [3] and we get the same crucial ordinary differential equation, but the corresponding coordinates n u and n z of an n-th αβ heterodimer have different meaning. In the u-model αβ heterodimers rotate but n u is the projection of the tip of the n-th dimer on the protofilament direction [2]. Hence, it is common to call the model as the longitudinal one. In the z-model αβ heterodimers have a longitudinal degree of freedom and n z is the longitudinal displacement of the αβ heterodimer along protofilament direction [3].
Interaction between αβ heterodimers in one protofilament is much stronger than interaction between αβ heterodimers belonging to neighboring protofilaments [5,6]. Influence of neighboring protofilaments is included through a potential ) ( n u V : where A , B and C are positive model parameters and is the intrinsic electric field and 0 > q is the surplus of charge of the αβ heterodimer. We assume that αβ heterodimer has only one degree of freedom [2,3,5]. From now on we choose the label n u . The Hamiltonian for one protofilament of the microtubule reads [2,3,5] (the Hamiltonian in the z-model is the same except for n z playing the role of n u ): where dot means the first derivative with respect to time, m is a mass of the αβ heterodimer, k is an interdimer stiffness constant and n counts αβ heterodimers in the protofilament. Taking the Hamiltonian becomes where NP n Q ) is a nonpotential generalized force. Our nonpotential generalized force is viscous force where γ is a viscosity coefficient.
Combining (4), (5) and (6) we get: Now we apply the continuum approximation ) , where l is the length of the αβ heterodimer. All this brings about a following partial differential equation [2]: We are interested in the solutions of the above equation which have the form of the traveling and κ and ω are constants. We straightforwardly obtain an ordinary differential equation where ξ d du u = ′ . Introducing a dimensionless function ψ defined as we finally get where A kl m and ξ ψ ψ Equation (13) is the crucial ordinary differential equation that we want to solve using the method of factorization.
The potential which describes the influence of the neighboring protofilaments is which can be obtained according to equations (4) and (12). The values of ψ for which the potential ) (ψ V has extreme values are the solutions of the following equation Equation (16) The function ) (ψ f is shown in Fig. 1 for two values of the parameter σ . One can see that 1 ψ , 2 ψ and 3 ψ describe a right minimum, a maximum and a left minimum, respectively.  (16) is [2]: (21)

SOLVING EQUATION (13) BY THE METHOD OF FACTORIZATION
Our ordinary differential equation that should be solved in this Section is equation (13).
Because of the free term σ − , the method of factorization cannot be immediately applied [7]. To get rid of this free term we introduce the following substitution [7]: where δ is a constant to be determined.
Inserting equation (22) into (13) we obtain: which indicates We should notice that equation (24) is the same as (16). Hence, because of (24) equation (23) becomes We seek (25) in the form [7]: where 1 f and 2 f are functions of ϕ to be determined. This yields: Comparing equations (25) and (27)   Equation (29) indicates that we should look for the functions 1 f and 2 f in the form: where the parameters a and α should be determined, while k is the "working" parameter and, of course, the final result should not depend on it. From (28) and (31) we easily obtain . which gives the following system of equations: Equation (33) can be written in a more convenient form It was pointed out that k was a "working" parameter in a sense that the final result will not depend on it. Hence, we can safely assume an integer. According to (35) We can easily check that and ) ( 2 The functions 1 f and 2 f could have been sought in the form: which was studied recently [10]. Of course, both procedures should give the same results, as was explained above. A next step is to obtain the function ) (ξ ϕ . Equations (26) and (40) indicate [7]: Its direct integration brings about ( )    This obviously represents a transition from the right minimum to the maximum (See Fig. 1). If we had chosen the minus sign in (45) we would have obtained 2 ) ( ψ ψ = −∞ and 1 ) ( ψ ψ = +∞ , representing the same transition but in opposite direction. Following the same procedure we obtain the remaining two pairs of the asymptotic values. All this is shown in the upper half of Table 2. It is obvious that (31) is not a unique choice for 1 f and 2 f . Alternative possibility is Following the procedure explained above we obtain the remaining two rows in Table 2 as well as (36) and (37). Also, equation (39) becomes The absolute value exists because the sign of 1 2 2ϕ ϕ − depends on the choice of δ as shown in Table 1. Table 2 shows that there are three pairs of the asymptotic values. However, a careful inspection of the solutions ) (ξ ψ shows that that there are actually three pairs of the solutions. Namely

CONCLUSIONS AND DISCUSSIONS
In the present paper we showed how the method of factorization can be used to solve problems in nonlinear dynamics of MTs. The procedure was demonstrated using u-model of MTs. It was shown that nonlinear MT dynamics can be represented by existence of kink and antikink solitons in MT chain. The same procedure can be applied to somewhat simpler ϕmodel [8].
Of course, equations like (13) can be solved using different methods. Examples could be modified extended tanh-function method [2] and the procedure based on Jacobian elliptic functions [6]. These two methods require trial functions. In this sense, the procedure used here is more general as no trial function is required.
Finally, we show that the function ) (ξ ψ , derived in this paper, is the same as the one obtained more than 20 years ago using different procedure [5,9,10], which we can call as standard method. To obtain the function u u ≡ ) (ξ we should use equations (12), (22) and (45). As an example we pick up the case 3 ψ δ = , for which one can easily show that Hence, we easily obtain To compare (50) with, for example, equation (22) in Ref. [9], we should comment the expression for α . Namely, following [9] we can write ( ) ( ) where 0 c and v are sound and solitonic velocities, respectively. It was assumed that v c >> 0 as well as vt x s − = instead of (10) above. Hence, under these circumstances, equation (50) which are equations (21) and (22) in Ref. [9]. Notice that 3 u in this paper corresponds to 2 u in Gordon's notation.
It is important to point out that our approach is more general as nothing was assumed for α . It was shown here and in Ref. [2] that α is negative. However, different model [8] may bring about positive α . This means that ) (ξ u represents a supersonic soliton, which is not uncommon in physics [11]. Finally, one very important point deserves to be discussed. To derive equation (13) the continuum approximation was used. A key question in nonlinear biophysics is either an essentially discrete system can be considered as a continuum one. As for the MTs this question was answered recently [12]. Namely, the initial discrete equation (7) was solved numerically and the result excellently matches the one shown here, based on the continuum approximation. This proves that the continuum approximation is applicable in MT research.