Motor protein transport along inhomogeneous microtubules
NNoname manuscript No. (will be inserted by the editor)
Motor Protein Transport Along InhomogeneousMicrotubules
S. D. Ryan · Z. McCarthy · M. Potomkin* the date of receipt and acceptance should be inserted later
Abstract
Many cellular processes rely on the cell’s ability to transport materialto and from the nucleus. Networks consisting of many microtubules and actin fila-ments are key to this transport. Recently, the inhibition of intracellular transporthas been implicated in neurodegenerative diseases such as Alzheimer’s disease andAmyotrophic Lateral Sclerosis (ALS). Furthermore, microtubules may contain so-called defective regions where motor protein velocity is reduced due to accumulationof other motors and microtubule associated proteins. In this work, we propose anew mathematical model describing the motion of motor proteins on microtubuleswhich incorporate a defective region. We take a mean-field approach derived from afirst principle lattice model to study motor protein dynamics and density profiles.In particular, given a set of model parameters we obtain a closed-form expres-sion for the equilibrium density profile along a given microtubule. We then verifythe analytic results using mathematical analysis on the discrete model and MonteCarlo simulations. This work will contribute to the fundamental understanding ofinhomogeneous microtubules providing insight into microscopic interactions thatmay result in the onset of neurodegenerative diseases. Our results for inhomo-geneous microtubules are consistent with prior work studying the homogeneouscase.
Keywords
Mathematical Biology, Motor Proteins, Microtubules, Phase Transi-tions, Defective Transport
S. D. RyanDepartment of Mathematics and Statistics, Cleveland State University, Cleveland, OH 44115,USACenter for Applied Data Analysis and Modeling, Cleveland State University, Cleveland, OH44115, USAE-mail: [email protected]. McCarthyDepartment of Mathematics and Statistics, York University, Toronto, ON, CanadaLaboratory for Industrial and Applied Mathematics, Toronto, ON, CanadaCentre for Disease Modelling, York University, Toronto, ON, CanadaFields-CQAM Mathematics for Public Health Laboratory, Toronto, ON, CanadaM. PotomkinDepartment of Mathematics, University of California, Riverside, CA, 92521, USA*Corresponding Author E-mail: [email protected] a r X i v : . [ q - b i o . S C ] M a y S. D. Ryan et al.
Mathematics Subject Classification (2010)
Cells strongly rely on the ability to efficiently transport cargo via motor pro-teins. Fast active transport (FAT) is required to deliver materials such as proteins,mRNA, mitochrondria, vescicles, and organelles for use in a variety of cellularprocesses [7,8]. One of the three major means of intracellular transport is withinnetworks of microtubules (MTs) [8]. To accommodate FAT, motor proteins “walk”along MTs and actin filaments (AFs) with their cargo forming a “superhighway”for cellular transportation [15,20,21]. Motor-protein families kinesin and dyneinbind to the cellular material or cargo they carry as they move up and down themicrotubule [4,8]. For a more general overview of molecular motor protein motionsee review [18].Defects in microtubules are known to exist, but the current literature has yetto clarify their impact on molecular motor-based transport [17]. Defects in activetransport, particularly axonal, have been implicated in the progression of variousdiseases [8]. For instance, the defining characteristics of many neurodegenerativediseases such as Alzheimer’s disease and Amyotrophic Lateral Sclerosis (ALS) maybe related to deficiencies in active transport within neurons [8]. One such area ofneed for immediate study is the scenario where the microtubule paths used by mo-tor proteins become congested, obstructed, or defective. Hallmarks and early indi-cators of neurodegenerative diseases are an accumulation of organelles and proteinsin the cell body or axon, which inhibits active transport [8]. Hence, understandingthe nature of motor protein dynamics will provide insight in understanding theonset of these diseases and developing control strategies.Advances in biophysical tools and imaging technology have allowed for manyrecent insightful in vitro experiments of motor protein behavior on microtubules [7,8,17]. Motor-proteins may change directions, stop or pause briefly, increase theirvelocity, and also attach/detach from the microtubule [7]. Experimentally it isobserved that this behavior may be attributed to the presence of MT-associatedproteins [7]. Therefore, the cell’s ability to regulate active transport may be stud-ied through tau MAP regions where motion of motor proteins is inhibited. Werefer to these patches of high tau MAP concentration as defective regions due totheir effects on the reduction of motor motility. We focus our study to modelingcollective motor protein motility on MTs, first homogeneous and then defective.Modeling motor protein motility on MTs has received recent theoretical at-tention. A one-dimensional discrete lattice model was studied in detail using amean-field approach, including a full phase diagram for the stationary states [1,3,5,13,14]. The model was capable of predicting the emergence of interior layers bysplitting the equilibrium density of motor proteins along the MT into two phases:low and high density. Such a co-existence corresponds to a traffic jam consisting ofmotor proteins translating along the MT in one direction, from the region with low density to the one with high density. In addition, a generalization of this latticegas model has also been studied to account for local interactions between motors[12]. In particular, the effects of adjacent motors enhancing the detachment rateas well as motor crowding enhancing the frequency which motors become inactiveor paused [12]. In a similar light, models featuring multiple “lanes” on a MT and otor Protein Transport Along Inhomogeneous Microtubules 3 motor proteins may switch lanes have also been studied [9] or between a filamentand a tube in [19]. The common feature among these lattice gas models is theyfollow the totally asymmetric simple exclusion process (TASEP) framework [1].These models also feature attachment and detachment processes for motors whosestationary states are described by the theory of Langmuir dynamics [1].In this paper, we seek to advance the current understanding of subcellulartransport along microtubules with a defective region. In Section 2, we introducea new discrete model for motor protein motility on a microtubule with a one-dimensional lattice following the TASEP paradigm and attachment/detachmentdynamics. The distinctive feature of this model is that the MT consists of a de-fective region with decreased motor protein motility rate. From a discrete latticeformulation, we take the limit to recover the mean-field form of these equations.Next, in Section 2.3, we verify that our mean-field approximations produce resultsconsistent with existing studies (e.g., [1,3,5]) before moving to the defective regioncase where motility will be hindered. The main analytical results are then presentedin Section 2.4, where we consider examples with the domain split into two regions,fast and slow, whereas the local and boundary attachment/detachment rates arevaried as parameters. From these studies we find a closed-form expression of thesolution given a set of the model parameters for attachment, detachment, andboundary conditions. We then verify that the analytical solution of the mean-fieldmodel is consistent with corresponding Monte Carlo simulations of the originaldiscrete lattice model in Section 3. We note that if one needs to consider a widerange of problem parameters, for example, in model calibration or a control prob-lem (e.g., finding the location and width of a defective region for a desired motordistribution), Monte Carlo simulations are prohibitively time consuming as com-pared to a closed-form expression when available. In the Appendix, we provide anew analytical approach to the solution of the homogeneous problem, based on theanalysis of the phase portrait of the corresponding system of ODEs; then we giveexamples of applications of the approach for specific problem parameters. Overall,this work provides a critical result for inhomogeneous MTs consisting of multipleparts with different motility properties; namely, it can be modeled as segments ofhomeogeneous MTs linked by a matching flux condition. This greatly expands theutility of past studies that developed the theory of homogeneous MTs. (i) each binding site maybe occupied by a maximum of one motor; (ii) motors move unidirectionally on thelattice and (iii) motors enter the lattice on the left side and exit the lattice on the right side [1]. Here we also account for the attachment and detachment of motorproteins on the MT interior as in works similar in scope focused on modeling [1,3,5] and experiment [16]. Another recent work has focused on stochastic model-ing with the goal of revealing how motor protein and filament properties affecttransport [22].
S. D. Ryan et al.
Specifically, consider a one-dimensional lattice { x < x < ... < x M = (cid:96) } ,representing sites which a motor may occupy on a microtubule of length (cid:96) . In-troduce ρ ni which is the probability of finding a motor at site x i at time step n .Probabilities at each lattice site 0 ≤ ρ ni ≤ ≤ i < M change during one timestep via ρ n +1 i − ρ ni = v i − ρ ni − (1 − ρ ni ) − v i + ρ ni (1 − ρ ni +1 ) + ω A (1 − ρ ni ) − ω D ρ ni . (1)The first term on the right-hand side of (1) says that the probability of finding amotor at site x i increases due to a possible jump of a motor from site x i − to x i provided that the following jump condition is satisfied: there is a motor at site x i − and site x i is vacant. As it is done in previous works on one-dimensional transportof active motors along a microtubule [1,3,5], consider the problem in the mean-field approximation, that is, correlations are negligibly small and the probabilityof the jump condition is simply ρ ni − (1 − ρ ni ). Additionally, the coefficient v i − accounts for inhomogeneity of the microtubule: if the jump condition is satisfiedon the interval [ x i − , x i ], then the jump occurs with probability (motility rate) v i − , and these coefficient may change from site to site. The second term on theright-hand side of (1) is similar to the first term, but accounts for the decreasein probability ρ ni due to a possible jump from site x i to x i +1 . The third term onthe right-hand side of (1) describes the interaction of the microtubule with theexterior environment: a motor from outside can attach to the microtubule at site x i as well as a motor already occupying site x i can detach from the microtubule.Parameters ω A and ω D are the corresponding attachment and detachment rates.Stationary states of (1) solve the following system:0 = v i − ρ i − (1 − ρ i ) − v i + ρ i (1 − ρ i +1 ) + ω A (1 − ρ i ) − ω D ρ i . (2)This system is supplemented with boundary conditions corresponding to the at-tachment rate α at the left end and detachment rate β at the right end of themicrotubule: ρ = α and ρ M = 1 − β. (3)We note here that it is assumed that boundary attachment rates have a corre-sponding relationship to those inside the microtubule, that is, α = ω A ω A + ω D and β = ω D ω A + ω D , (4)and v i ≡ v , then the solution of (2)-(3) is simply a constant: ρ ≡ ω A / ( ω A + ω D ).However, in practice, rates α and β are different from (4) which leads to non-trivial stationary solutions possessing interior jumps, even in the homogeneouscase v i ≡ const [1,3].We end this subsection with another form of (2) which is helpful for under-standing the continuous limit presented in Section 2.2. Introduce the followingnotation for the flux between sites x i − and x i : J i − = v i − ρ i − (1 − ρ i ) . (5)Note that if one interpolates ρ i by a smooth function ρ ( x ) such that ρ ( x i ) = ρ i ,then performing Taylor expansions one can verify that ∆x ρ (cid:48) i − = − v − i − J i − + ρ i − (1 − ρ i − ) + o ( ∆x ) , (6) otor Protein Transport Along Inhomogeneous Microtubules 5 where ρ i − = ρ ( x i − ), ∆x = x i − x i − , and x i − = x i − ∆x .Going back to definition (5), we point out that, in terms of J i ± , equation (2)has the following form: J i + − J i − = ω A (1 − ρ i ) − ω D ρ i (7)2.2 Limiting continuous problemWe focus on the asymptotic behavior of solutions of (2)-(3) as M → ∞ in the frame-work of the mean-field limit. Specifically, we introduce parameter ε := (cid:96)M − (cid:28) (cid:96) represents the total length of microtubule) and lattice { x i = iε, i = 0 , .., M } with the distance between lattice points ε . For small ε , we approximate the solu-tion with the continuous function ρ ε ( x ) defined on 0 < x < (cid:96) and derived from adiscrete set of unknowns ρ i associated with the lattice points x i , i.e., ρ ε ( x i ) = ρ i .Then for ε (cid:28)
1, the system of algebraic equations (2) for the unknown ρ i ’s becomesa second order ordinary differential equation for unknown function ρ ε ( x ): ∂ x (cid:16) v ( x ) (cid:16) − ε ∂ x ρ ε + ρ ε (1 − ρ ε ) (cid:17)(cid:17) = Ω A − ( Ω A + Ω D ) ρ ε , (8)where v ( x ) is the velocity the motor proteins move with at location x , and Ω A/D = Mω A/D are properly rescaled attachment/detachment rates. The equalities in (3)become boundary conditions for ρ ε ( x ): ρ ε (0) = α and ρ ε ( (cid:96) ) = 1 − β. (9) Remark 1
In order to obtain (8) from (2) we take the discrete-to-continuous limit ε →
0. Specifically, we write both (6) and (7), which considered together areequivalent to (2), with ∆x = ε (cid:28) (cid:40) ε ρ (cid:48) i − = − v − i − J i − + ρ i − (1 − ρ i − ) + o ( ε ) ,J (cid:48) i = Ω A (1 − ρ i ) − Ω D ρ i + o ( ε ) . (10)We then disregard the o ( ε ) terms and write resulting equations for all x ∈ (0 , (cid:96) ): (cid:40) ε ρ (cid:48) = − v − J + ρ (1 − ρ ) ,J (cid:48) = Ω A (1 − ρ ) − Ω D ρ (= Ω A − ( Ω A + Ω D ) ρ ) . (11)Finally, we find J from the first equation of the system above and substitute itinto the second equation to derive (8) for ρ = ρ ε . Here both notations ρ (cid:48) and ∂ x ρ denote the derivative in x . It turns out that the phase portrait of system (11) of twocoupled first order differential equations is the key to constructing the solutions of(8)-(9), see Appendix A. Remark 2
Alternatively, in the case of the continuous velocity v ( x ), this equationcan be formally derived from (2) by using a Taylor expansion of a smooth function ρ ε ( x ) which is again obtained by interpolation of the values ρ i on the mesh points x i . If v ( x ) is piecewise continuous with a finite number of jumps, then one needsto supplement equation (8), which holds inside the intervals of continuity of v ( x ),with a continuity condition for the flux J ε ( x ) := v ( x ) (cid:16) − ε ρ (cid:48) ε + ρ ε (1 − ρ ε ) (cid:17) . S. D. Ryan et al.
One can also consider (8) in the distributional sense. Then using the defini-tion of J ε ( x ), the integration of (8) in x , and the absolute continuity of integralstogether imply the continuity of flux J ε ( x ): J ε ( x ) = J ε (0) + x (cid:90) [ Ω A − ( Ω A + Ω D ) ρ ε ( s )] d s = J ε (0) + Ω A x − ( Ω A + Ω D ) x (cid:90) ρ ε ( s ) d s. (12)In what follows below, we assume that for all ε >
0, there exists unique smoothsolution 0 ≤ ρ ε ( x ) < ρ ( x ), referred to as the “outersolution”, such that lim ε → ρ ε ( x ) = ρ ( x ) , for all 0 ≤ x ≤ (cid:96). (13)By calling ρ ( x ) the outer solution we stick to the standard terminology ofsingularly perturbed ordinary differential equations [6]. While the outer solu-tion ρ ( x ) is the pointwise limit of ρ ε ( x ) for 0 ≤ x ≤ (cid:96) , it does not approxi-mate the exact solution ρ ε ( x ) uniformly on 0 ≤ x ≤ (cid:96) . Specifically, the outersolution ρ ( x ) approximates ρ ε ( x ) poorly in the vicinity of the jumps { x J } . Tomake the approximation uniform, one takes into account boundary layer termsof the form ρ corrector ( x ) = Y (( x − x J ) /ε ) whose distinguishing feature is thatits slope, derivative in x , is of the order of ε − . Observe also that, even though ρ (0) = α and ρ ( (cid:96) ) = 1 − β , it is possible that the outer solution ρ ( x ) does notsatisfy the boundary condition in the following sense: either lim x → + ρ ( x ) (cid:54) = α orlim x → (cid:96) − ρ ( x ) (cid:54) = 1 − β . Remark 3
We note that (1) possesses a unique constant solution ρ i ≡ Ω A / ( Ω A + Ω D ). In what follows, we are interested in regimes where jamming can occur and,thus, for the sake of simplicity we restrict ourselves to the case when the attach-ment rate exceeds the detachment rate, that is, Ω A > Ω D . This implies that theconstant solution of (1) is greater than 1 / v ( x ), representing a homogeneousmicrotubule, that is, v i ≡ v . The solution in the homogeneous case has beenstudied previously (e.g., [1,2,3]). In this case, (8) reduces to v ∂ x (cid:16) − ε ∂ x ρ ε + ρ ε (1 − ρ ε ) (cid:17) = Ω A − ( Ω A + Ω D ) ρ ε , < x < (cid:96), (14) ρ ε (0) = α, ρ ε ( (cid:96) ) = 1 − β. (15)Recall that (cid:96) is the length of a microtubule.Equation (14) is a second order nonlinear ODE. If one formally passes to thelimit ε → ρ ε vanishes and thisequation becomes first order where the solution cannot, in general, satisfy both otor Protein Transport Along Inhomogeneous Microtubules 7 boundary conditions in (15) as the boundary value problem is overdetermined.To describe the limiting solution, lim ε → ρ ε ( x ), we introduce an auxiliary function g ( x ; s, a ), which is the solution of the initial value problem of the first order ob-tained from the formal limit as ε → (cid:26) v (1 − g ) ∂ x g = Ω A − ( Ω A + Ω D ) g,g ( x = s ; s, a ) = a. (16)In other words, function ρ ( x ) := g ( x ; s, a ) is the smooth outer solution subjectto a single boundary condition (or, equivalently, initial condition): ρ | x = s = a .Equation (16) can be solved in an explicit form in terms of special functions(see [3]).Note that initial value problem (16) is not well-posed for a = 0 .
5. If a = 0 . x > s , and for x < s we define g as the function solvingthe differential equation in (16) subject to the following condition:lim x → s g ( x ; s, .
5) = 0 . g ( x ; s, . > . x < s. (17)In other words, for the initial condition with a = 0 .
5, equation (16) admits twosolutions for x < s : one solution is less than 0 .
5, another one is above 0 .
5, andto describe the outer solution ρ ( x ) we will need restrict our consideration to theupper one (the upper solution is stable in a certain sense, see Appendix A).Function g ( x ; s, a ) is not necessarily defined globally, for all x . For example,consider x ≥ s , then the solution exists on the interval ( s, s + x a ) for some x a > x = s + x a the slope of g becomes unbounded. For example, if a < .
5, thenthe value of x a can be found from the condition g ( s + x a ; s, a ) = 0 . x a = (cid:90) . a d g v ( g ) = (cid:90) . a v (1 − g ) d gΩ A − ( Ω A + Ω D ) g , (18)where function v ( g ) is introduced in such a way that the differential equation from(16) is equivalent to ∂ x g = v ( g ). One can compute the integral on the right-handside of (18) to obtain an analytic formula for x a : x a = v Ω A − Ω D ( Ω A + Ω D ) log Ω A − Ω D Ω A − a ( Ω A + Ω D )) + v − aΩ A + Ω D . (19)The following theorem gives an explicit formula for the limiting solution of the(14)-(15) as ε → Theorem 1
Define ρ ( x ) := lim ε → ρ ε ( x ) for ≤ x ≤ (cid:96) and ρ ε solving (14) - (15) . Then ρ ( x ) = α, x = 0 ,g ( x ; 0 , α ) , < x < max { , x J } ,g ( x ; (cid:96), max { . , − β } ) , max { , x J } < x < (cid:96), − β, x = (cid:96). (20) If α ≥ / , then x J = 0 . If α < / , then x J is determined by x J := min { x ≥ | g ( x ; 0 , α ) + g ( x ; (cid:96), max { . , − β } ) ≤ } . (21) S. D. Ryan et al.
The result of this theorem is consistent with previous works where the system(14)-(15) was studied (e.g., see [5,9,12]). We relegate the proof of this theorem andexamples of the application of the representation formula (20) to Appendix A.
Remark 4
The point x J , if 0 < x J < (cid:96) , is the location of the interior jump, that is,the outer solution ρ ( x ) is a smooth solution of the differential equation from (16)on intervals (0 , x J ) and ( x J , (cid:96) ) and it has one jump inside (0 , (cid:96) ) at x = x J . Thevalue of x J can also be found via numerical simulations of the following equation g ( x J ; 0 , α ) + g ( x J ; (cid:96), max { . , − β } ) = 1 . (22)This equation is equivalent to the continuity of fluxes J = ρ (1 − ρ ) at the pointof the jump of the outer solution ρ = ρ ( x ): v ρ α (1 − ρ α ) | x → x J − = v ρ β (1 − ρ β ) | x → x J + , (23)where ρ α ( x ) = g ( x ; 0 , α ) and ρ β ( x ) = g ( x ; (cid:96), max { . , − β } ). Remark 5
If one varies boundary conditions (15), then the outer solution ρ ( x ) maystay unchanged (except values at x = 0 and x = (cid:96) ) for wide range of parameters α and β . For example, denote ρ β ( x ) := g ( x, (cid:96), max { . , − β } ). Theorem 1 impliesthat outer solution ρ ( x ) coincides with ρ β ( x ) on the interval 0 < x < α : 1 − ρ β (0) ≤ α ≤ . (24)Once α becomes smaller than the lower limit, 1 − ρ β (0), an interior jump appearsin the outer solution ρ ( x ).The following corollary is important for the study of inhomogeneous micro-tubules in Section 2.4. Corollary 1
Assume α ≥ / or . (cid:90) v (1 − ρ ) Ω A − ( Ω A + Ω D ) ρ d ρ ≤ (cid:96). (25) Then(i) lim x → (cid:96) − ρ ( x ) = max { . , − β } .(ii) If β < / , then ρ ( x ) is continuous at x = (cid:96) , that is, lim x → (cid:96) − ρ ( x ) = ρ (0) = 1 − β .(iii) If α ≥ / , then there is no interior jump, that is, x J ≤ .(iv) If < x J < (cid:96) , then α < / and lim x → ρ ( x ) = α (that is, ρ ( x ) is continuous at x = 0 ). Condition (25) excludes the case of low density solutions, that is, we excludeouter solutions of the form: ρ ( x ) < . x ∈ [0 , (cid:96) ). This regime is notconsistent with jamming, which is the focus of this work. In other words, condition(25) imposes that the “left” part of solution, g ( x ; 0 , α ), cannot be extended toentire interval [0 , (cid:96) ). The reason we would like to impose this condition below in the inhomogeneous case is because we then focus on cases when regions with highdensities emerge, and thus traffic jams in motor transport are possible. By directintegration, condition (25) can be written as v Ω A + Ω D (cid:20) − Ω A − Ω D Ω A + Ω D log (cid:18) Ω A − Ω D (cid:19)(cid:21) ≤ (cid:96). (26) otor Protein Transport Along Inhomogeneous Microtubules 9 v ( x ). Biologically, it cor-responds to a inhomogeneous microtubule with different motor protein mobilitiesin different regions of the microtubule. Without loss of generality, we take (cid:96) = 1.We restrict ourselves to the case v ( x ) = (cid:26) v L , ≤ x ≤ x ,v R , x < x ≤ . (27)This case may be considered as two coupled homogeneous microtubules meetingat interface x = x with coupling through the continuity of densities and fluxes.Specifically, we have the following system of equations: v L ∂ x (cid:104)(cid:16) − ε ρ (cid:48) ε + ρ ε (1 − ρ ε ) (cid:17)(cid:105) = Ω A − ( Ω A + Ω D ) ρ ε , < x < x , (28) v R ∂ x (cid:104)(cid:16) − ε ρ (cid:48) ε + ρ ε (1 − ρ ε ) (cid:17)(cid:105) = Ω A − ( Ω A + Ω D ) ρ ε , x < x < , (29)and two coupling conditions:(1) continuity of ρ ε ( x ) at x : ρ ε ( x − ) = ρ ε ( x +0 ) =: ρ ε . (30)(2) continuity of flux J ε ( x ) at x : v L (cid:16) − ε ρ (cid:48) ε ( x − ) + ρ ε (1 − ρ ε ) (cid:17) = v R (cid:16) − ε ρ (cid:48) ε ( x +0 ) + ρ ε (1 − ρ ε ) (cid:17) . (31)The outer solution ρ ( x ) = lim ε → ρ ε ( x ) is not necessarily continuous, neverthelessdue to (12) it satisfies the flux continuity condition: v L ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) x → x − = v R ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) x → x +0 . (32) Remark 6
By looking at the system (28)-(29) one may think that the case of in-homogeneous v ( x ) is equivalent to the case of inhomogeneous attachment/detach-ment rates but with constant motility rates v ( x ) ≡ ∂ x (cid:16) − ε ρ (cid:48) ε + ρ ε (1 − ρ ε ) (cid:17) = Ω A ( x ) − ( Ω A ( x ) + Ω D ( x )) ρ ε (33)with attachment/detachment rates Ω A ( x ) and Ω D ( x ) are Ω A /v L and Ω D /v L in-side the left half of the microtubules, x < x , and Ω A /v R and Ω D /v R inside theright half, x > x . For differential equation (33), from the continuity of ρ ε ( x ) and J ε ( x ), one concludes that ρ ε is necessarily continuously differentiable, whereas thesolution of (28)-(29) possesses a discontinuous derivative at x for v L (cid:54) = v R whichfollows from (31) and is written as ρ (cid:48) ε ( x +0 ) − ρ (cid:48) ε ( x − ) = 2 εv R v L ( v R − v L ) J ε ( x ) . (34)Therefore, problems for inhomogeneous v ( x ) and inhomogeneous Ω A,D ( x ) are notequivalent. By analogy with function g from (16) in the case of homogeneous microtubules,we introduce g L ( x ; s L , a L ) and g R ( x ; s R , a R ) as solutions of the following initialvalue problems: (cid:26) v L (1 − g L ) ∂ x g L = Ω A − ( Ω A + Ω D ) g L , g L ( s L ; s L , a L ) = a L ,v R (1 − g R ) ∂ x g R = Ω A − ( Ω A + Ω D ) g R , g R ( s R ; s R , a R ) = a R . (35)In what follows we consider separately two cases:(i) fast - slow microtubule : v L > v R ,(ii) slow - fast microtubule : v R > v L .Before we formulate our main result for these two cases, we note that thedifficulty in the determination of the outer solution ρ ( x ) is finding the value of ρ at the interface, A := ρ ( x ). Once A is found, one can use Theorem 1 to restore ρ ( x ) in both intervals [0 , x ] and [ x , Theorem 2
Consider v L > v R and assume that condition (25) holds with v = v L and (cid:96) = x . Let ρ ( x ) be the outer solution of system (28) - (29) equipped with couplingconditions (30) - (31) . Then function ρ ( x ) has a jump at x = x and is given by ρ ( x ) = α, x = 0 ,g L ( x ; 0 , α ) , < x < max { , x J } ,g L ( x ; x , A ) , max { , x J } < x ≤ x ,g R ( x ; 1 , max { . , − β } ) , x < x < , − β, x = 1 , (36) where x J is determined from the continuity of fluxes: g L ( x J ; 0 , α ) + g L ( x J ; x , A ) = 1 , (37) and A = (cid:16) v L + (cid:113) v L − J R (cid:17) / (2 v L ) where J R = v R g R (1 − g R ) with g R := g R ( x ; 1 , max { . , − β } ) . Proof
First, we show that the outer solution ρ ( x ) has a jump at x = x . Indeed,from continuity of fluxes (31) with ρ ε ( x ) = A + o (1) we get: v L ρ (cid:48) ε ( x − ) = v R ρ (cid:48) ε ( x +0 ) + v L − v R ε A (1 − A ) + o (cid:18) ε (cid:19) . (38)Since A is strictly between 0 and 1, we find that one of the derivatives (either theleft or right one) is of the order ε − corresponding to a jump.Next, denote the limits of the outer solution from the left and right at x by A L := ρ | x → x − and A R := ρ | x → x +0 . Then the continuity of fluxes (32) is written as v L A L (1 − A L ) = v R A R (1 − A R ),and since v L > v R we have A L (1 − A L ) < /
4. By applying Corollary 1 (ii) wefind A = A L > . x . Then, followingCorollary 1 (iii), there is no interior jump, and ρ ( x ) in interval [ x ,
1] is determinedby Corollary 1 (i), which is a boundary condition for ρ ( x ) at x = 1. otor Protein Transport Along Inhomogeneous Microtubules 11 Fig. 1
Examples 1 & 2 for inhomogeneous microtubulus. The upper row depicts plots of outersolutions ρ ( x ). The lower row depicts solutions as trajectories in the ( ρ, J )-plane; note thatthese trajectories are continuous in J and have discontinuities (jumps) in ρ . Red (lower) andblue (upper) dashed lines are arcs J = v R ρ (1 − ρ ) and J = v L ρ (1 − ρ ), respectively. Specifically, by Theorem 1 we have ρ ( x ) = g R ( x ; 1 , max { . , − β } ) for x 1. Then A R = g R := g R ( x ; 1 , max { . , − β } ) and A is the solution of thequadratic equation v L A (1 − A ) = v R A R (1 − A R ), which is strictly greater than0 . 5. Thus, we found A , and the expression for ρ ( x ) in [0 , x ] is found by using therepresentation formula (20) from Theorem 1. (cid:117)(cid:116) Next, we illustrate formula (36) of Theorem 2 by the following two exampleswith x = 0 . v L = 1 . v R = 0 . Ω A = 0 . Ω D = 0 . Example 1. α = 0 . β = 0 . ρ ( x ) = . , x = 0 ,g L ( x ; 0 . , . , < x ≤ . , ( A = 0 . ± √ − J R ) ≈ . g R ( x ; 1 . , . , . < x ≤ . . Example 2. α = 0 . 05 and β = 0 . ρ ( x ) = g L ( x ; 0 . , . , ≤ x < x J , ( x J ≈ . g L ( x ; 0 . , . , x J < x ≤ . ,g R ( x ; 1 . , . , . < x < . , ( A ≈ . . , x = 1 . . Solutions from Examples 1 and 2 are depicted in Fig. 1. The main differencebetween Examples 1 and 2 is that in Example 2 there is an interior jump in theleft sub-interval (0 , . , . 5) and right (0 . , 1) (or, equivalently, x J = 0 in (36) for Example 1). Theorem 3 Consider v R > v L and assume condition (25) holds with both v = v L , (cid:96) = x and v = v R , (cid:96) = 1 − x . Let ρ ( x ) be the outer solution of system (28) - (29) equipped with coupling conditions (30) - (31) . Denote also ˆ A R := g R ( x ; 1 , max { . , − β } ) . (39) Then A = − (cid:114) − v L v R , v R ˆ A R (1 − ˆ A R ) > . v L , 12 + (cid:114) − v R v L ˆ A R (1 − ˆ A R ) , v R ˆ A R (1 − ˆ A R ) ≤ . v L , (40) and function ρ ( x ) is given by ρ ( x ) = α, x = 0 ,g L ( x ; 0 , α ) , < x < max { , x ( L ) J } ,g L ( x ; x , max { A, . } ) , max { , x ( L ) J } ≤ x < x ,g R ( x ; x , min { A, . } ) , x ≤ x ≤ x ( R ) J ,g R ( x ; 1 , max { . , − β } ) , x ( R ) J < x < , − β, x = 1 . (41) Here x ( L ) J and x ( R ) J are determined from the continuity of fluxes: g L ( x ( L ) J ; 0 , α ) + g L ( x ( L ) J ; x , max { A, . } ) = 1 ,g R ( x ( R ) J ; x , min { A, . } ) + g R ( x ( R ) J ; 1 , max { . , − β } ) = 1 . Proof As in the proof of Theorem 2, introduce A L := ρ | x → x − and A R := ρ | x → x +0 .Then continuity of fluxes at x reads v L A L (1 − A L ) = v R A R (1 − A R ) . (42)By Corollary 1 (i) we get A L ≥ . 5. Next, we consider two cases. If v R v L ˆ A R (1 − ˆ A R ) ≤ / , then A R = ˆ A R and A L ≥ . A = A L = 1 + (cid:114) − v R v L ˆ A R (1 − ˆ A R ) . If v R v L ˆ A R (1 − ˆ A R ) > / , then ρ ( x ) does not coincide with g R ( x ; 1 , max { . , − β } )on the entire interval ( x , x < x ( R ) J < A R < / A = A R . Then A L = 1 / ρ ( x ) onintervals (0 , x ) and ( x , 1) is found by (20). (cid:117)(cid:116) otor Protein Transport Along Inhomogeneous Microtubules 13 Fig. 2 Examples 3 & 4 for a inhomogeneous microtubule. The upper row depicts plots of outersolutions ρ ( x ). The lower row depicts solutions as trajectories in ( ρ, J ) plane, red (lower) andblue (upper) dashed lines are arcs J = v L ρ (1 − ρ ) and J = v R ρ (1 − ρ ), respectively. Next we illustrate formula (41) by two examples with v R > v L . Namely, set v L = 0 . v R = 1 . x = 0 . Ω A = 0 . Ω D = 0 . Example 3. α = 0 . β = 0 . ρ ( x ) = g L ( x ; 0 . , . , ≤ x < x ( L ) J , ( x ( L ) J ≈ . g L ( x ; 0 . , . , x ( L ) J < x < . ,g R ( x ; 0 . , A ) , . ≤ x < x ( R ) J , ( A = (1 − √ ) , x ( R ) J ≈ . g R ( x ; 1 . , . , x ( R ) J < x < . , . , x = 1 . . Example 4. α = 0 . β = 0 . ρ ( x ) = g L ( x ; 0 . , . , ≤ x < x ( L ) J , ( x ( L ) J ≈ . g L ( x ; 0 . , A ) , x ( L ) J ≤ x ≤ . , ( A ≈ . g R ( x ; 1 . , . , . < x ≤ . Solutions from Examples 3 and 4 are depicted in Fig. 2. These two examplesillustrate two possibilities for slow-fast microtubules: when ρ ( x ) is continuousfrom the right at x (Example 3) and when it is continuous from the left at x (Example 4). Fig. 3 Examples 3 & 4 are simulated with R = 10 realizations of the lattice model, describedin Section 3. Simulations use the same parameters as Examples 3 & 4 for inhomogeneousmicrotubules above (analytical solution is solid black line; see also Figure 2). We see a strongquantitative agreement between the Monte Carlo (red) and analytical (black) solutions. We now show that the results from Section 2.4 involving the mean-field continuousmodel in the inhomogeneous case are consistent with those from Monte Carlosimulations. To this end, we return to the discrete problem (2)-(3) with M = 500lattice sites. We perform R = 10 realizations and compare the resulting discretedensities with the continuum equation (8) as ε → 0, computed by representationformulas from Theorem 2 and 3.Specifically, we adapt a similar algorithm to the one in [1]. In each realization r = 1 , ..., R we consider tuple { ν ( n ) ( i ) } where n stands for the iteration step numberand ν ( n ) ( i ) = 1 if the i th site is occupied at the n th iteration step and ν ( n ) ( i ) = 0,if otherwise. Initially, microtubule is empty, i.e., ν (0) ( i ) = 0 for i = 1 , ..., M . Foreach time step, n = 1 , ..., N discrete dynamics of (cid:110) ν ( n ) ( i ) (cid:111) Mi =1 is described by thefollowing procedure: Choose randomly site i (all sites are equiprobable). If i = 1 and ν ( n ) (1) = 0, then ν ( n +1) (1) = 1 with probability α . If i = M and ν ( n ) ( M ) = 1, then ν ( n +1) ( M ) = 0 with probability β . If 1 < i < M , then – if ν ( n ) ( i ) = 0, then ν ( n +1) ( i ) = 1 and ν ( n +1) ( i − 1) = 0 with probability v i − provided that ν ( n ) ( i − 1) = 1; – if ν ( n ) ( i ) = 1, then ν ( n +1) ( i ) = 0 and ν ( n +1) ( i + 1) = 1 with probability v i + provided that ν ( n ) ( i + 1) = 0. If 1 < i < M , then – if ν ( n +1) ( i ) = 1 after step 4, then let ν ( n +1) ( i ) = 0 with probability ω d ; – if ν ( n +1) ( i ) = 0 after step 4, then let ν ( n +1) ( i ) = 1 with probability ω a .Finally, after running N steps and R realizations, we assign ρ MCS i := (cid:104) ν ( N ) i (cid:105) r .Monte Carlo simulations are in very good agreement with the outer solutionsderived in Section 2.4. Specifically, results of Monte Carlo simulations correspond-ing to Example 3 and 4 from Section 2.4 are depicted in Fig. 3; one can seeagreement between histograms obtained from Monte Carlo simulations (red) and otor Protein Transport Along Inhomogeneous Microtubules 15 analytical solutions (black). Observe that number of realizations R and numberof iteration steps N needed to reach equilibrium are critical for recovering thesharp transitions observed near the interface of the inhomogeneous microtubule.For Monte Carlo histograms in Fig. 3, we were to take N = 2 . · time stepsin order to guarantee that a transient solution has reached equilibrium for each of R = 10 realizations. The large numbers for both realizations and iteration stepslead to the observation that the reproduction of equilibrium profiles (solutions)in Examples 1-4 by Monte Carlo simulations to be very time consuming, whereasTheorems 2 and 3 give explicit formulas which require only numerical integrationof at most four ODEs for the auxiliary functions g ( x ; s, a ). In this work, we present a mathematical model to describe dynamics of motorproteins on microtubules. Using methods from asymptotic analysis, we provideclosed-form expressions for motor protein density solutions. We also provide veri-fication of the results of mathematical analysis by Monte Carlo simulation with thediscrete MT model. The mathematical model may serve as a convenient frameworkfor studying experimental data. Even more, the modeling and analysis may assistin inferring in vivo dynamics where biophysical imaging is limited in the crowdedcellular environments. It is also important to note that the model presented hereinis consistent with prior theoretical results for the homogeneous case (e.g., [1,3,5]).The model approach developed herein provides additional advantages over theprior approach of Frey et al. [12] and others [13,14] while remaining faithful inthe homogeneous case. Most notably, the model is developed to study inhomoge-neous regimes where large density profiles can result in the emergence of internalboundary layers. Beyond the obvious application to motor protein dynamics alonga microtubule, this also provides insight into traffic flow problems. The PDE gov-erning the density of cars has a similar form to the equation governing the densityof motor proteins here. This work also provides an additional example of the powerof analyzing discrete ODE model systems by passing to the limit and obtaining amean-field PDE.What made this work challenging is that a priori initial data cannot predictregions of low or high density. Even within the Monte Carlo simulations we ob-serve that they must be run for a significant length of time to capture all thefeature of the solutions (e.g, interior boundary layers, sharp transitions etc.). Anadditional challenge lies in experimental verification given the current state oftechnology. Once imaging technology improves combined with advancements inbiophysical knowledge, the theory developed in this manuscript can be rigorouslytested experimentally both in vitro and in vivo . This will be crucial in verifyingmodel parameter regimes corresponding to biologically realistic results.This work lays the foundation for future work in understanding inhibited trans-port along microtubules. The model we present may be augmented to account for more biological realism in describing motor protein dynamics and intracellulartransport. Realistically there are several “lanes” on these MTs which motor pro-teins move laterally and they may switch lanes. Similarly, motor proteins may alsochange directions when encountering patches [7]. Furthermore, transport takesplace on highly complex 3-dimensional networks of many MTs and AFs. Hence modeling the intersections between MTs would be of interest as well as analyzingthe composite density profiles using the analysis presented in this work. In addi-tion, the cargo transported to and from the cell nucleus and cell wall is carried bymotor proteins [8,4] and the given model may be augmented to account for thiscargo. We also note that motor proteins transfer from MT to MT within the cell,and the model as well as analysis developed here may serve as a foundation forthis study.Overall, the model for an inhomogeneous microtubule presented here can in-form motor protein dynamics in rough regimes where transport properties are notconsistent along given trajectories. This will ultimately lay the groundwork forfundamental understanding of the onset of neurodegenerative diseases. The inho-mogeneous microtubule model may be used to investigate how one can controltransport properties of motor proteins in high density regimes along microtubules.Given the structure of a microtubule, can one devise conditions so that the equi-librium solution contains no high density regimes (jams) by understanding orimposing defects along its surface? Also, given a distribution of inhomogeneousregions ( N > 2) on a microtubule can we predict the equilibrium solution? Theanswers to these questions may be the source of further investigation in a futurework. Acknowledgements The work of SR was supported by the Cleveland State University Officeof Research through a Faculty Research Development Grant. References 1. Parmeggiani, A., Franosch, T. and Frey, E., 2003. Phase coexistence in driven one-dimensional transport. Physical review letters, , p.086601.2. Popkov, V., R´akos, A., Willmann, R.D., Kolomeisky, A.B. and Sch¨utz, G.M., 2003. Local-ization of shocks in driven diffusive systems without particle number conservation. PhysicalReview E, , p.066117.3. Parmeggiani, A., Franosch, T. and Frey, E., 2004. Totally asymmetric simple exclusionprocess with Langmuir kinetics. Physical Review E, , p.046101.4. Gross, S. P., Vershinin, M. and Shubeita G. T., 2007. Cargo transport: Two motors aresometimes better than one. Current Biology, , R478-R486.5. Reese, L., Melbinger, A. and Frey, E., 2011. Crowding of molecular motors determinesmicrotubule depolymerization. Biophysical Journal, p. 2190-2200.6. M. H. Holmes, Introduction to Perturbation Methods , 2nd edition. Texts in Applied Math-ematics, 20. Springer, New York, 2013. xviii+436 pp.7. Ross, J. L., Ali, M. Y. and Warshaw, D. M., 2008. Cargo transport: molecular motorsnavigate a complex cytoskeleton Current Opinion in Cell Biology, p. 41-47.8. Lakadamyali, M., 2014. Navigating the cell: how motors overcome roadblocks and trafficjams to efficiently transport cargo Phys. Chem. Chem. Phys., p. 5907.9. Reichenbach, T., Frey, E. and Franosch, T., 2007. Traffic jams induced by rare switchingevents in two-lane transport New Journal of Physics, p. 159.10. Tan, R., Lam, A.J., Tan, T., Han, J., Nowakowski, D.W., Vershinin, M., Simo, S., Ori-McKenney, K.M. and McKenney, R.J., 2019. Microtubules gate tau condensation to spatiallyregulate microtubule functions. Nature cell biology, (9), pp.1078-1085.11. Oelz, D. and Mogilner, A., 2016. A drift-diffusion model for molecular motor transportin anisotropic filament bundles. Discrete and Continuous Dynamical Systems, :8 p. 4553-4567.12. Rank, M. and Frey, E., 2018. Crowding and pausing strongly affect dynamics of kinesin-1motors along microtubules. Biophysical journal, (6), 1068-1081.otor Protein Transport Along Inhomogeneous Microtubules 17 Fig. 4 Left: Phase portrait for (43) with ε = 0 . Ω A = 0 . Ω D = 0 . 3; Right: Sketch ofthe phase portrait for (43) with ε (cid:28) 1, the black circle represents the stationary point.13. Klumpp, S., Chai, Y., and Lipowsky, R., 2008. Effects of the chemomechanical steppingcycle on the traffic of molecular motors. Physical Review E, , 041909.14. Klumpp, S., Nieuwenhuizen, T. M., and Lipowsky, R., 2005. Self-organized density patternsof molecular motors in arrays of cytoskeletal filaments. Biophysical Journal, , pg. 3118-3132.15. Klumpp, S., and Lipowsky, R., 2005. Cooperative cargo transport by several molecularmotors. PNAS, (48), pg. 17284-17289.16. Varga, V., Leduc, C., Bormuth, V., Diez, S., and Howard, J., 2009. Kinesin-8 motorsact cooperatively to mediate length-dependent microtubule depolymerization. Cell, , pg.1174-1183.17. Liang, W. H., Li, Q., Rifat Faysal, K. M., King, S. J., Gopinathan, A., and Xu, J., 2016. Microtubule defects influence kinesin-based transport in vitro. Biophysical Journal, , pg.2229-2240.18. Mallik, R., and Gross, S. P., 2004. Molecular motors: Strategies to get along. CurrentBiology, , pg. R971-R982.19. M¨uller, M. J. I., Klumpp, S., and Lipowsky, R., 2005. Molecular motor traffic in a half-open tube. Journal of Physics: Condensed Matter, , pg. S3839-S3850.20. Rossi, L. W., Radtke, P. K., and Goldman, C., 2014. Long-range cargo transport oncrowded microtubules: The motor jamming mechanism. Physica A, , pg. 319-329.21. Striebel, M., Graf, I. R., and Frey, E., 2020. A mechanistic view of collective filamentmotion in active nematic networks. Biophysical Journal, (2), pg. 313-324.22. Dallon, J. C., Leduc, C., Etienne-Manneville, S., Portet, S., 2019. Stochastic modelingreveals how motor protein and filament properties affect intermediate filament transport. Journal of Theoretical Biology, , pg. 132-148. A Homogeneous microtubules: Proof of Theorem 1 and Corollary 1 Equation (14) may be rewritten in the form of a system of two first order ODEs for density ρ ε and flux J ε (see also (11)): (cid:40) ε ρ (cid:48) ε = − v − J ε + ρ ε (1 − ρ ε ) ,J (cid:48) ε = Ω A − ( Ω A + Ω D ) ρ ε . (43)Next, we discuss the phase portrait for this system with ε (cid:28) 1, depicted in Fig. 4. Awayfrom curve γ defined by γ := (cid:26) ( ρ, J ) (cid:12)(cid:12)(cid:12)(cid:12) J = v ρ (1 − ρ ) and 0 ≤ ρ ≤ , ≤ J ≤ v / (cid:27) , (44)8 S. D. Ryan et al.the trajectories of (43), parametrized by 0 ≤ x ≤ (cid:96) , have almost horizontal slope in ( ρ, J )plane. This is because the slope of ρ ε is of the order ε − , that is ρ (cid:48) ε ( x ) ∼ ε − , whenever thepoint ( ρ ε ( x ) , J ε ( x )) is away from γ (it follows from the first equation in (43)). It would benatural to expect that as ε vanishes, trajectory { ( ρ ε ( x ) , J ε ( x )) , ≤ x ≤ (cid:96) } approaches the arch γ and this trajectory is contained in a given thin neighborhood of γ for sufficiently small ε . Inthis subsection, it will be shown that the behavior of the solution is more complicated thansimply evolving near γ .To describe how the solution ρ ε ( x ) behaves for ε (cid:28) 1, we introduce the following notationfor parts of curve γ . Namely, γ l := γ ∩ { ≤ ρ < . } ,γ r, + := γ ∩ { . ≤ ρ ≤ ρ eq } ,γ r, − := γ ∩ { ρ eq ≤ ρ ≤ } . Here ρ eq := Ω A / ( Ω A + Ω D ). Let us also introduce the following horizontal segment Γ := { ( ρ, J ) : J = v / , ≤ ρ ≤ . } , and the solution g ( x ; s, a ) to (16), i.e., the initial value problem of the first order obtained bythe formal limit as ε → v (1 − g ) ∂ x g = Ω A − ( Ω A + Ω D ) g, g ( s ; s, a ) = a. (45)First, note that γ l , which is the left part of the curve γ , is unstable, that is all trajectories,excluding γ l , are directed away from γ l in the vicinity of γ l . The right part of the curve γ ,consisting of curve segments γ r, + and γ r, − , is stable, attracting all trajectories in its vicinity,except those that follow Γ . We note that this exception, when γ r, + loses its stability, occursat the interface point ( ρ = 1 / , J = v / 4) where γ r, + meets γ l . All trajectories reaching thispoint near (not necessarily intersecting) the curves γ r, + and γ l continue along Γ .Given specific values of α, β ∈ (0 , 1) in boundary conditions (15), the statement of The-orem 1 as well as representation formula (20) can be simply verified by careful inspection ofthe phase portrait depicted in Fig. 4. Specifically, for all 0 < α, β < 1, one can draw a path { ( ρ ( x ) , J ( x )) : 0 ≤ x ≤ (cid:96) } along arrows in Fig. 4 (right), which starts at vertical line ρ = α andends at vertical line ρ = 1 − β , and such a path will be unique for given α and β (see also leftcolumn of Fig. 5 for specific examples). Instead of checking each couple ( α, β ), one would splitranges of ( α, β ) into sub-domains within which the outer solution has constant or smoothlyvarying shape, as it is done in proof below. Proof of Theorem 1. Consider the following functions: ρ α ( x ) = g ( x ; 0 , α ) and ρ β ( x ) = g ( x ; (cid:96), max { . , − β } ) . These functions can be thought of as one-sided solutions (i.e., satisfying one of the boundaryconditions, either ρ (0) = α or ρ ( (cid:96) ) = max { . , − β } ) of Equation (14) for ε = 0. Thereason we choose ρ ( (cid:96) ) = max { . , − β } instead of ρ β ( (cid:96) ) = 1 − β is because there is nosolution continuous at x = (cid:96) with ρ ( (cid:96) ) < . γ is unstable in region { ≤ ρ < . } ).Introduce also the corresponding fluxes: J α ( x ) = v ρ α ( x )(1 − ρ α ( x )) and J β ( x ) = v ρ β ( x )(1 − ρ β ( x )) . From the definition of function g it follows that J α ( x ) and J β ( x ) are both monotonic functions,and function J β ( x ) is defined for all 0 ≤ x < (cid:96) . Moreover, J β ( x ) can be extended onto ( −∞ , (cid:96) ]and lim x →−∞ J β ( x ) = J eq , where J eq := v Ω A Ω D ( Ω A + Ω D ) . Consider case α ≥ . 5. From Fig. 4, it follows that a trajectory emanating for initial point( α, J ) for any 0 < J < v / γ r and stays on γ r ∪ Γ for 0 < x ≤ (cid:96) .Thus, at x = 0 trajectory { ( ρ ( x ) , J ( x )) : 0 ≤ x ≤ (cid:96) } , describing the outer solution, jumpsfrom ( α, J (0)) at t = 0 to γ r : ρ ( x ) = (cid:26) α, x = 0 ,ρ β ( x ) , < x ≤ (cid:96). (46)otor Protein Transport Along Inhomogeneous Microtubules 19In the case where α < . 5, denote by 0 ≤ x J ≤ (cid:96) location at which fluxes J α ( x ) and J β ( x )intersect, that is, J α ( x J ) = J β ( x J ) . (47)Equality (47) implies that either ρ α ( x J ) = 1 − ρ β ( x J ) or ρ α ( x J ) = ρ β ( x J ). If ρ α ( x J ) = ρ β ( x J ),then since ρ α and ρ β are solutions of the same first order ordinary differential equation, thesetwo functions coincide ρ α ( x ) ≡ ρ β ( x ).We show now that eitherthere exists at most one x J ≤ ρ α ( x ) ≡ ρ β ( x ). (48)Indeed, since α < . 5, trajectory ( ρ α ( x ) , J α ( x )) evolves on γ l for all 0 ≤ x ≤ (cid:96) where solution ρ α ( x ) exists, and J α ( x ) monotonically increases. Trajectory ( ρ β ( x ) , J β ( x )) evolves also for all0 ≤ x ≤ (cid:96) within either γ r, + or γ r, − . If ( ρ β ( x ) , J β ( x )) evolves within γ r, − , then J β ( x ) ismonotonically decreasing in x whereas J α ( x ) is monotonically increasing x , and thus equation J α ( x ) = J β ( x ) can have at most one root in this case. If ( ρ β ( x ) , J β ( x )) evolves within γ r, + , thenboth J α ( x ) and J β ( x ) increase with x . Assume that there are at least two distinct numbers x (1) J , x (2) J such that x (1) J < x (2) J and J α ( x ( i ) J ) = J β ( x ( i ) J ), i = 1 , 2. Assume also that x (1) J and x (2) J are neighbor roots of equation J α ( x ) = J β ( x ), i.e., for all x ∈ ( x (1) J , x (2) J ) we have J α ( x ) (cid:54) = J β ( x ). Then due to ∂ x J = Ω A − ( Ω A + Ω D ) g, where J ( x ) = v g ( x )(1 − g ( x ))and ρ α ( x ( i ) J ) < . ρ α ( x ( i ) J ) > . i = 1 , 2, we have that ∂ x J α ( x ( i ) J ) > ∂ x J β ( x ( i ) J ), i = 1 , x J is at most one and (48) is shown.If J α ( x ) (cid:54) = J β ( x ) for all 0 ≤ x ≤ 1, then define x J as follows: x J = (cid:26) , J β ( x ) < J α ( x ) for all 0 < x < (cid:96), , J α ( x ) < J β ( x ) for all 0 < x < (cid:96). We note that point x = x J is where the outer solution jumps from ρ α ( x ) to ρ β ( x ), thus ρ ( x ) = (cid:26) ρ α ( x ) , ≤ x < x J ,ρ β ( x ) , x J < x < (cid:96). (49)and ρ ( (cid:96) ) = 1 − β .Formulas (46), (49), and (15) complete the proof of Theorem 1. (cid:3) B Examples of solutions given by (20) To illustrate the result of Theorem 1 we continue with the following examples. We take v = 1, (cid:96) = 1, Ω A = 0 . Ω D = 0 . 2, and we vary the boundary rates α and β . The outer solutionfor each example, as both a trajectory in ( ρ, J ) plane and the plot of ρ ( x ), is depicted inFig. 5. Example 1. α = 0 . β = 0 . ρ ( x ) = (cid:26) . , x = 0 g ( x ; 1 , . , < x ≤ . Example 2. α = 0 . β = 0 . ρ ( x ) = (cid:26) g ( x ; 0 , . , ≤ x ≤ x J , x J ≈ . g ( x ; 1 , . , x J < x ≤ . Example 3. α = 0 . β = 0 . Fig. 5 Left: The thick line represents the trajectories from Examples 1-4; it starts at ρ = α and ends at ρ = 1 − β , the black circle at (0.8,0.16) represents the stationary solution. Right:The thick line represents the outer solution ρ ( x ) for Examples 1-4. In Examples 2 and 3,branches g ( x ; 0 , α ) and g ( x ; 1 , max { . , − β } ) extend slightly beyond the intervals where theyare a part of the outer solution ρ ( x ) (thin curves).otor Protein Transport Along Inhomogeneous Microtubules 21 Fig. 6 The thick line represents the trajectories from Example 5; it starts at ρ = α and endsat ρ = 1 − β , the black circle at (0.8,0.16) represents the stationary solution. Right: The thickline represents the outer solution ρ ( x ) for Example 5. ρ ( x ) = g ( x ; 0 , . , ≤ x ≤ x J , x J ≈ . ,g ( x ; 1 , / , x J < x < , . , x = 1 . Example 4. α = 0 . β = 0 . ρ ( x ) = . , x = 0 ,g ( x ; 1 , / , < x < , . , x = 1 . The case x J > ρ ( x ) < . x ∈ (0 , Example 5. α = 0 . β = 0 . Ω A = 0 . 16 and Ω D = 0 . ρ ( x ) = (cid:26) g ( x, , α ) , ≤ x < , − β, x = 1 ..