Modelling cytoskeletal transport by clusters of non-processive molecular motors with limited binding sites

Molecular motors are responsible for intracellular transport of a variety of biological cargo. We consider the collective behaviour of a finite number of motors attached on a cargo. We extend previous analytical work on processive motors to the case of non-processive motors, which stochastically bind on and off cytoskeletal filaments with a limited number of binding sites available. Physically, motors attached to a cargo cannot bind anywhere along the filaments, so the number of accessible binding sites on the filament should be limited. Thus, we analytically study the distribution and the velocity of a cluster of non-processive motors with limited number of binding sites. To validate our analytical results and to go beyond the level of detail possible analytically, we perform Monte Carlo latticed based stochastic simulations. In particular, in our simulations, we include sequence preservation of motors performing stepping and binding obeying a simple exclusion process. We find that limiting the number of binding sites reduces the probability of non-processive motors binding but has a relatively small effect on force–velocity relations. Our analytical and stochastic simulation results compare well to published data from in vitro and in vivo experiments.

The Gillespie algorithm is the method where the time until the next event dt is drawn from an exponential distribution with rate parameter given by the sum of the rates of all events possible from the current state. The main concept is that one, and only one, event happens each time step and that the duration of each time step changes. The leading motor can move forward and backward with rates p 1 and q 1 and the following motors with the rates p and q, respectively. Therefore, there are four possible events, however, due to the simple exclusion process, not all these events will be available in each time step. The time until the next event is calculated each time step as τ = 1 α ln 1 r 1 where α is the total probability of all events possible in that time step and r 1 is a random number from a uniform distribution (0, 1). Which event happens in that time step is determined by a second random number, r 2 , drawn from the uniform distribution.
The leading motor's velocity for different number, N , of bound motors from our Gillespie simulations is shown in figure S16 (a) and (b). Our simulation results using the Gillespie algorithm match those using our fixed time step method within the error bars. The error bars for the Gillespie algorithm are so small as to be barely visible in figure S16 since they lie inside those of our fixed time step method. Given the equivalence of our results using each method we are satisfied to use the faster fixed time step method to present our results in the main text.

B. NON-PROCESSIVE MOTORS GILLESPIE ALGORITHM
We also adapt the Gillespie algorithm from processive motors to non-processive model drawn in the main text figure 5. Including the binding on and off the filament means we now have six possible events to include in the total probability α with the addition of nk off and (N − n)k on for the probabilities of binding off or on respectively. The rest of the algorithm is the same as that described in section A. We compare this algorithm with our results from our fixed time step that the Gillespie algorithm is more accurate. However, in our case the Gillespie algorithm takes ten times longer to run than our fixed time step Monte Carlo simulations. Therefore in the main text we present results using our fixed time step Monte Carlo simulations since they are more time efficient and sufficiently accurate for our purposes.

C. SEQUENCE PRESERVATION
In figure S18 we present probability distributions for the case of variable number of binding sites (scenario B) but without sequence preservation for both extremes of force f = 0 and f = f s .
In the simulations presented in this section the simple exclusion process ensures sequence preservation for bound motors stepping but motors can swap positions by unbinding and rebinding. This is in contrast to the scenario B case in the main text section 3.3 in which motors are constrained to rebind in a way that preserves their sequence. The agreement between the simulations results shown in figure S18 and the analytical distributions with limited number of binding sites (equation (3.6)) provides evidence that the mismatch between the analytical probability distribution and the simulation results in the main text section 3.3 is due to including sequence preservation on motor rebinding in those simulations.

D. STALL FORCE FOR LARGE NUMBERS OF MOTORS
In figure S19 we show an extended version of figure 12 of the stall force against number of motors N up to N = 100. This additional data shows that the numerical result for non-processive motors with unlimited number of binding sites increases linearly for large number of motors (N > 10) with the same gradient as that for processive motors. This is also the case for binding sites limited to the average found in simulations.
The black circles in figure S19 show that for small number of motors the stall forces for non-processive Ncds extracted from our simulations correspond well with those those numerically calculated from the analytical solutions with the number of binding sites unlimited (green crosses) and limited (light blue crosses). However for N > 3 the results from our simulations are are lower than those obtained from the analytical expressions and remain lower for N > 10.

E. SINGLE MOTOR APPROXIMATION
For q pe −f /2 , we can approximate equation (2.2) by V N ≈ pe −f /2 − qe f /2 (q/p) N −1 which is the single motor case with the backwards stepping rate modified by the factor (q/p) N −1 . This can reproduce an approximation to the curves seen in figure 3 (see figure S20) but has the disadvantage of only being valid for q pe −f /2 which is not well satisfied for our parameter values.  Figure S20: Force-velocity curves of N = 1 (blue), 2 (red) and 10 (green) processive motors using the full analytical expression, equation (2.2) (solid lines) and an approximate analytical expression V N ≈ pe −f /2 − qe f /2 (q/p) N −1 which corresponds to a single motor with backwards stepping rate modified by the factor (q/p) N −1 (dashed lines). The equivalent simulation points are shown on the main text figure 3.