article: Intracellular transport is an essential function in eucaryotic cells, facilitated by motor proteins—proteins converting chemical energy into kinetic energy. It is understood that motor proteins work in teams enabling unidirectional and bidirectional transport of intracellular cargo over long distances. Disruptions of the underlying transport mechanisms, often caused by mutations that alter single motor characteristics, are known to cause neurodegenerative diseases. For example, phosphorylation of kinesin motor domain at the serine residue is implicated in Huntington’s disease, with a recent study of phosphorylated and phosphomimetic serine residues indicating lowered single motor stalling forces. In this article we report the effects of mutations of this nature on transport properties of cargo carried by multiple wild-type and mutant motors. Results indicate that mutants with altered stall forces might determine the average velocity and run-length even when they are outnumbered by wild type motors in the ensemble. It is shown that mutants gain a competitive advantage and lead to an increase in the expected run-length when the load on the cargo is in the vicinity of the mutant’s stalling force or a multiple of its stalling force. A separate contribution of this article is the development of a semi-analytic method to analyze transport of cargo by multiple motors of multiple types. The technique determines transition rates between various relative configurations of motors carrying the cargo using the transition rates between various absolute configurations. This enables a computation of biologically relevant quantities like average velocity and run-length without resorting to Monte Carlo simulations. It can also be used to introduce alterations of various single motor parameters to model a mutation and to deduce effects of such alterations on the transport of a common cargo by multiple motors. Our method is easily implementable and we provide a software package for general use.
  Motor proteins- kinesin, dynein and myosins- are nanoscale machines that are the main effectors of intracellular transport. They play a critical role in the growth and sustenance of healthy cells by enabling a transport of intracellular cargo over networks of microtubules [1]. Disruption of the functions performed by these molecular motors is linked to neurodegenerative diseases such as Huntington’s, Parkinson’s and Alzheimer’s Disease [2,3], muscular disorders such as heart disease, uterine complications and high blood pressure. The mechano-chemical behavior of a single motor moving along the microtubule substrate to transport a cargo is relatively well understood [4]. There is considerable prior work on understanding how a single-motor transports a cargo on a microtubule. In [5] the Fokker–Planck equation for a single particle moving in a one-dimensional potential is obtained and used to determine the probability distribution of the position of the particle in a single repeat length of the polymer track via discretization. The underlying approach was extended to a motor connected to a cargo by a linear spring [6,7] while travelling under the influence of a tilted periodic potential. In vivo it is known that multiple motors work in teams to transport a common cargo [16]; how multiple motor-proteins coordinate to transport a common cargo is not well understood [16–18]. Many studies employ a probabilistic description of the behavior of a single motor protein to construct models that describe how multiple motors transport a common cargo [16,19–21]. In [22], Gross and coworkers employed Monte-Carlo simulation studies built on a model of a single kinesin in to explore how multiple identical kinesin motors might interact to transport a cargo against a hindering load force. Their work indicates, counter intuitively, the existence of a form of strain-gating, where the motors of an ensemble share loads unequally enabling cargo transport over longer distances. Xu and coworkers examined the effects of ATP concentration on the transport of cargo carried by single and two motors in [23]. At decreased levels of ATP concentration, the velocities of cargoes transported by single and two motor proteins decreases. Coincidentally at decreased ATP concentration there was an an appreciable increase in the run-length of cargoes transported by two motors, while no such effect was seen in the case of transport by one motor. The authors proposed that the increased run-length observed in the presence of two motors results from the lowered dissociation of each motor from the microtubule at decreased ATP concentrations and the increased probability that the cargo stayed bound to the microtubule. Studies such as [21,29] using probabilistic models of single motors have also predicted in that an ensemble of kinesin motors is a robust system and the robustness increases under high loads [37]. The study of cargo transport by a heterogeneous ensemble of motor proteins composed of both wild type and mutant motors is important to inform our understanding of how mutant motors impact intracellular transport and lead to an onset of diseases. Recent studies have implied that alterations in the kinesin-1 motor domain may have a role in impaired axonal transport. Phosphorylation of a mammal kinesin motor domain by kinase c-Jun N-terminal kinase-3 (JNK3) at a conserved serine residue (Ser-176 in A and C isoforms and Ser-175 in B isoform) is implicated in Huntington’s disease [24]. However the mechanisms affected by Ser-175 phosphorylation are not well understood. An experimental study by Selvin and coworkers in [25] reported that a negative charge at Ser-175, acquired through mutation or phosphorylation, leads to a lower stall force and decreased velocity under external loads of 1 pN or more, while leaving the ATPase, microtubule-binding affnity and processivity unchanged. Using a semi-analytical method, we reveal surprising emergent transport behaviors arising when a cargo is transported by multiple motor proteins, some of which are mutated and some are not. In particular we analyze the impact of Ser-175 kinesin mutation such as those reported in [25] on cargo transport in the presence of wild type motors. The detailed investigation made possible by our method leads us to hypothesize that under certain conditions, a ‘mooring mechanism’ is activated where proteins moor the cargo to the microtubule and prevent it from being lost. While these mooring proteins do not contribute to the motion of the cargo, they enhance the probability of attachment of other cargo-bound motor proteins to the microtubule that subsequently contributes to an increase in average cargo displacement. The activation of mooring mechanism depends on a number of external factors such as load force and ATP concentration. However, it is also determined by intrinsic properties of the motor protein such as the stall force of the individual motors. Remarkably, mutant motors that have stalling forces matched to the external load force can act dominantly and determine emergent transport properties such as longer run-length, even when they are outnumbered in the ensemble by wild-type motors. Such mechanisms could potentially point to how diseased states emerge and progress coincident with the accumulation of the mutant motor species. A separate contribution of the article is a semi-analytical method for determining the probability distribution of various configurations of a cargo carried by multiple number and types of motor-proteins. A detailed experimental study of the various modalities of transport by multiple motor proteins (homologous or otherwise) requires significant instrumental resolution than what is needed to investigate single motor behavior. As a consequence, observing the transport dynamics of multiple motors is experimentally challenging [26,27]. It is further compounded by the combinatorial complexity introduced by the multitude of scenarios possible when many motors and motor types participate in transport. Such challenges motivate the use of analytical and computational tools. The mean-field approach in [19,21] makes use of simplifying assumptions, such as equal load sharing among all motors, to achieve analytical results, thereby sacrificing significant detail for computational benefits. The Monte Carlo approach in [22], provides better fidelity where complex models can be employed; however, the accuracy of results depend on the number of iterations and on the rarity of the events that occur. Unlike the Monte-Carlo simulations or any currently implemented simulation methods that study multi-motor ensembles, our Master Equation based method [28] analytically solves for the probability distributions of all possible scenarios at any time point. The methodology is uniquely powerful and enables the calculation of various biologically relevant quantities such as average velocity and run-length, for reasonably sized ensembles and with high accuracy while using lesser computational resources and time as compared to Monte Carlo simulations (the methodology proposed in [6,7] also obtains results without resorting to Monte Carlo simulations with a single motor attached to the cargo). The underlying concepts behind our methodology are motivated by earlier work reported in [29]. The key enabling concept is that of ‘relative configuration’ of motor proteins, determining the transition probabilities between relative configurations from the transition probabilities of the absolute configuration space and subsequently determining the biologically relevant quantities from the relative configuration space. The computation engine is implemented using MATLAB and can be used to simulate cargo transport by any two unidirectional species. Our method provides a general platform to study the transport of cargo by multiple motors of two different types where each type of motor protein can be individually characterized by a probabilistic model describing its stepping, detachment and attachment rates. For this article, the technique has been utilized to introduce alterations of various parameters from the nominal ones to model a mutation of the serine residue and compute the effect of such a mutation on cargo carried by a mixture of wild-type and mutant motor-proteins. In summary, we developed a simulation engine to study the transport of cargo by multiple motor proteins with distinct properties that in concert can exhibit emergent transport behaviors.
  The knowledge of the steady state conditional probability distributions allows one to compute biologically relevant quantities such as average velocity and run-length. The results obtained for ensembles containing two and three motors for varying load forces are reported here.
  Consider a cargo that is carried by both wild-type and mutant motor proteins on a microtubule. The microtubule is modeled as directed linear lattice formed by equally sized dimers with dimension d. Here the kth dimer is located at location a ¯ k = k d and indexed by the set of integers I = {…, −2, −1,0, 1,2, …}. Each motor protein bound to the cargo can attach, take a forward step or detach from the microtubule. The absolute configuration Ω≔{ Ωh, kΩd, k }k∈I of motor-protein arrangement on the microtubule specifies the number, Ωh, k, of wild-type motor proteins and the number, Ωd, k, of mutant motor proteins at the kth location on the microtubule. For example, the absolute configuration of the ensemble of motors illustrated in Fig 16 is given by Ω = [ ⋯ Ω - 1 Ω 0 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 Ω 6 ⋯ ] = ⋯ 0 1 0 1 0 0 0 1 ⋯ ⋯ 0 0 0 1 0 1 0 0 ⋯. The relative configuration of an ensemble of motors is represented using a string of three symbols. Given an absolute configuration we first identify the rearguard motor which is the motor that is attached to the microtubule and lags behind all the other motors on the microtubule. Using the location of the reargurad motor as a reference, the relative configuration ϑ is obtained as a string of three symbols ‘Mh’, ‘Md’ and ‘|’, where ‘Mh’ and ‘Md’ denote wild-type and mutant motors respectively, with ‘|’ denoting a separator that distinguishes different microtubule locations. The motor located the farthest from the rearguard motor on the microtubule is identified as the vanguard motor. For example, the relative configuration of the ensemble in Fig 17 (a) is the string ‘|Mh Md||Md||Mh|’. The configuration that results after the furthermost mutant motor in Fig 17 (a) takes a step is shown in Fig 17 (b) which has a relative configuration given by ‘|Mh Md|||Md|Mh|’. Both the mutant as well as the wild-type motor proteins are characterized via their own set of stepping, attachment, and detachment probabilities (for wild-type (PS, PD, PA) and for mutant (P ¯ S, P ¯ D, P ¯ A) ). The individual motors for both species are modeled as hookean springs when stretched that offer no resistance when compressed. A single motor is assumed to have a linkage rest length L0 and spring stiffness constant Ke. Motors of both the species are assumed to not step backward and are bound to the cargo particle irreversibly. More complex models of motor-proteins can be easily accommodated. It is further assumed that the there exists a force Fstall called the stalling force, where if the force on the motor protein F ≥ Fstall then the motor does not take a forward step and stepping probability is zero [31]. The stalling force can be that is also used to estimate how many motors are carrying the cargo. For an ensemble of wild-type and mutant motors carrying a cargo, the following result holds: Result 1: Given an ensemble of M molecular motors attached to a common cargo that is subjected to a load force Fload, the distance between the rearguard and the vanguard motor is bound by n = m a x (M + 1) m a x (F s, F ¯ s) - F l o a d K e + d, F l o a d K e + 2 L 0 (1) where Fs is the minimum load force for which the stepping probability of the wild-type motor protein becomes zero (i. e. the stalling force for the wild-type motor protein), F ¯ s is the minimum load force for which the stepping probability of the mutant motor protein becomes zero (i. e. the stalling force for the mutated motor protein), L0 is the rest length of the motor linkage, Ke is the linkage stiffness and d is the step-size of the motor. A detailed derivation is provided in the S1 Text. It is to be noted that the absolute configuration space admits infinitely many representations as there is always a small probability of finding the cargo at any location on the microtubule. However, the above result concludes, that given a stall force for both wild-type and mutant motors the relative configuration space is finite, since there are no motors beyond n units away from the rearguard motor in any relative configuration. In this section we present a general scheme for determining transition rates between absolute configurations. We begin with a structural model for single motor protein that consists of motor head, stalk and cargo binding tail domain. The linkage between the motor-heads and tail for single motor is modeled as a hookean spring when stretched, that has a rest length L0. It offers no resistance when compressed [22]. The motor heads move along the microtubules exerting a force F on a cargo that is expressed as a function of its length L by, F (L) = K e (L - L 0) if L ≥ L 0,0 if | L | < L 0, K e (L + L 0) if L ≤ - L 0. Zeq is the mean position of the cargo that is its equilibrium position determined by the forces exerted by the motors on the cargo through their linkages and the load force Fload on the cargo. The mean cargo position for a fixed Fload > 0 is a function of the absolute configuration i. e. Zeq = Zeq (Ω). If the cargo position is assumed to follow a truncated Gaussian distribution Θ (z) with variance σ, its probability density Θ (z) for |z| < 3σ is given by, Θ (z) = (e - z 2 2 σ 2) / (2 ∫ 0 3 σ e - z 2 2 σ 2 d z). The effect of thermal noise can be incorporated by determining the steady state variance σ of the cargo position. We assume that when a motor in the ensemble takes a forward step or detaches, the probability distribution of the cargo position reaches a new distribution with negligible transient. Here, the time scale of the cargo dynamics is assumed to be faster than the rate of change of motor configurations. A transition to another configuration Ω′ occurs if either the wild type or mutant motor at a location a ¯ k steps forward to a ¯ k + 1, detaches from the location a ¯ k or reattaches to the location a ¯ k on the microtubule. By representing Ω′ as Ω + S, S is a sequence that corresponds to the type of transition (step, detach or attach) and the type of motor (wild-type or mutant) that has transitioned. The transition rate from Ω to Ω′ is determined by averaging the associated probability rate over the position of the cargo. The model of a single motor-protein is specified via the probability PS (F) of the motor taking a step, the detachment probability, PD (F), of the motor detaching from the microtubule, and the probability of attachment PA of an unattached motor-protien to the microtubule, per second. Here F is the force acting on the motor which is considered positive if it is directed opposite to the motor stepping direction (e. g. kinesin forward stepping is towards the mictorubule + end). Here in order to calculate the transition rates between absolute configurations it is assumed that the probability rates of step, detachment and attachment are known; later we illustrate a way to compute these probabilities for kinesin motors. The results put forth in this article correspond to an instantiation of our methodology for kinesin motor protein. The probability rates of stepping, detachment and attachment of a single kinesin motor are determined using several available studies [16,21,30,31].
summary: Molecular motors such as kinesin and dynein facilitate directed transport of intracellular cargo over tracks called microtubules. Inside cells, multiple motor proteins are known to bind and move cargoes. These teams of motors enable the transport of cargoes over longer distances, extending beyond the processive runlengths of a single motor. Impaired transport, possibly due to mutations that affect single motor parameters, is known to cause neurodegenerative diseases. A recent study reported that phosphorylation of a kinesin motor implicated in Huntington’s disease, leads to a reduction in the single motor stalling force. In this work, we investigate how heterogeneity in motor stall forces can affect the coordinated transport properties of multi-motor ensembles. Our model predicts that motors with reduced stall force, even when in the minority, can determine emergent transport properties of average velocity and run-length. Under appropriate external loads, our analysis predicts that motor ensembles containing mutant motors travel longer distances, potentially contributing to the dysregulation of coordinated cargo transport, impairment of neuronal function and the onset of neurodegeneration. These results are enabled by development of a novel semi-analytic methodology to study cargo transport by multiple motors with distinct transport properties. This method is computationally less extensive than existing Monte-Carlo based approaches, easy to implement, and holds potential for understanding how individual motor proteins and properties contribute to the coordination of transport by motor ensembles.
keywords: phosphorylation
  kinesins
  microtubules
  molecular motors
  mathematics
  probability distribution
  statistics (mathematics)
  cellular structures and organelles
  cytoskeleton
  motor proteins
  research and analysis methods
  proteins
  mathematical and statistical techniques
  monte carlo method
  probability theory
  biochemistry
  cytoskeletal proteins
  cell biology
  post-translational modification
  biology and life sciences
  physical sciences
  microtubule motors
  statistical methods
title: Interrogating Emergent Transport Properties for Molecular Motor Ensembles: A Semi-analytical Approach