Run length distribution of dimerized kinesin-3 molecular motors: comparison with dimeric kinesin-1

Kinesin-3 and kinesin-1 molecular motors are two families of the kinesin superfamily. It has been experimentally revealed that in monomeric state kinesin-3 is inactive in motility and cargo-mediated dimerization results in superprocessive motion, with an average run length being more than 10-fold longer than that of kinesin-1. In contrast to kinesin-1 showing normally single-exponential distribution of run lengths, dimerized kinesin-3 shows puzzlingly Gaussian distribution of run lengths. Here, based on our proposed model, we studied computationally the dynamics of kinesin-3 and compared with that of kinesin-1, explaining quantitatively the available experimental data and revealing the origin of superprocessivity and Gaussian run length distribution of kinesin-3. Moreover, predicted results are provided on ATP-concentration dependence of run length distribution and force dependence of mean run length and dissociation rate of kinesin-3.


SI text S1. Potential of interaction between one kinesin head and MT in a mechanochemical coupling cycle
Based on argument (i) (see main text), we take the interaction potential of one kinesin head with MT as described as follows. In nucleotide-free state the kinesin head binds strongly to MT, with the interaction potential being written as S S ( , , , , , ) ( ) ( ) ( ) ( ) ( ) ( ) where coordinate oxyz is defined in Term V Sx (x) < 0 (with the maxima equal to zero) represents the interaction potential between the kinesin head and MT along a MT protofilament and is approximately shown in Fig. S1a. The period of V Sx (x), d = 8.2 nm, is equal to the distance between two successive binding sites on MT filaments. Here we take V Sx (x) in one periodicity having an asymmetric form here, with asymmetric ratio d 1 /d 2 = 3/5, as done before [S1]. It should be mentioned that taking other forms for V Sx (x) (including the symmetric form) has little effect on the results for the dimer. Terms denote the potential changes arising from the head rotations, with A  , A  and A  characterizing the interaction distances. Here, we define in the state of the head bound to MT. These potential changes V y (y), V z (z), the Morse potential form that describes the van der Waals interaction. To be consistent with the Debye length that is in the order of 1 nm in solution, we take A y = A z = 1 nm and head r A  = head r A  = head r A  = 1 nm, where the kinesin head is approximately a sphere of radius r head = 2.5 nm. After ATP binding and then hydrolysis to ADP.Pi the kinesin head remains bound strongly to MT, with the interaction potential still being approximately described by S ( , , , , , ) V x y z    .
Immediately after Pi release, the interaction potential becomes one that can be the same as those defined above. Note that immediately after Pi release the binding affinity (E w1 ) of the kinesin head for the local binding on MT site, where the kinesin head in ADP.Pi state has just bound, is weaker than that (E w2 ) at other binding sites.
After a period of time t r , the affinity of the local binding site on MT for ADP-head relaxes to the normal value and the interaction potential becomes where the MT-bound head is located at (x, y, z) = (0, 0, 0).

S3. Potential of interaction between two kinesin heads
We take the potential of interaction between the MT-bound head and detached ADP-head with the NL of the MT-bound head being undocked having following form where (x, y, z) is the center-of-mass coordinate of the detached ADP-head relative to that of the MT-bound head (which is taken as the origin of the coordinate) during one stepping period, (x 1 , y 1 , z 1 ) is the position of the detached ADP-head in the  , 1  and 1  has nearly no effect on our results. When the NL of the MT-bound head is docked, the potential is still described by Eq. (S4), but with E I1 being replaced with E I2 (< E I1 ).

S4. Interaction between the NL and kinesin head
To determine the interaction between the NL and kinesin head, we should determine the elasticity of the linker by using all-atom MD simulations (see Section S8), as used elsewhere [S3-S5]. Here, we take the NL of wild-type Drosophila kinesin-1 as an example to describe the simulation procedure. We take residues 324 through 338 from the structural data (PDB 3KIN), where residues 325 -338 constitute the NL. We adjust the line connecting the alpha carbon (CA) atom of residue 324 and that of residue 338 along a given direction. We fix the CA atom of the residue 324 and impose a series of constant forces on the CA atom of the residue 338 along the given direction, as done before [S6]. We then calculate the distance, r NL , between the two terminal CAs of the NL after reaching the equilibrium for 20 ns. In the literature, the calculated data of the force-extension relation of a flexible peptide were usually fitted by using the worm-like-chain (WLC) model. However, it is noted that the simulated data of the force-extension relation of the kinesin's NL are better fitted by using the exponential function than WLC model, especially at small values of pulling force (see Fig. S2). Thus, here we use the exponential function to fit the simulated data where a and b are constants. Then the force on the detached head, which results from the stretched NL, can be calculated by using Eq. (S5).
Similarly, we obtain the force-extension relation of the NL of kinesin-3 head. The simulated data for kinesin-3's NL can also be fitted well using Eq. (S5) (Fig. S3).    , ,

S6. Equations to describe the movement of kinesin dimer in the intermediate state with one ADP-head bound to MT and the other ADP-head bound strongly to the MT-bound head
When the two ADP-heads are bound together strongly, the movement of the MT-bound ADP-head relative to MT can be described by following equations The initial conditions for Eqs. (S12) -(S17) are: ( 0 , , x y z    ) = (0,0,0,0,0,0).

S8. All-atom MD simulations
The all-atom MD simulations are carried out by using GROMACS5.1 [S7] with OPLS-AA/L all-atom force field [S8]. To avoid the edge effect, the distance between the peptide of the NL and the boundary of the box is at least 1.5 nm and much longer along the direction of the pulling force to stretch the NL. We add solvent and necessary ions with favorable concentration. Counter-ions are also added to neutralize the system. All MD simulations are run at 300K and 1 bar. The time step is set as 2 fs, and the output data is updated every ps. All chemical bonds are constrained using LINCS algorithm [S9]. The short-range electrostatics interaction and the cutoff for van der Waals interaction is set as 1 nm. Velocity-rescaling temperature coupling [S10] and Berendsen pressure coupling [S11] are used. The energy minimization is performed using the steepest descent method. Before the dynamic simulations, the systems are equilibrated successfully for 2 ns at 300 K and 1 bar pressure in the NVT ensemble and NPT ensemble, respectively. For calculations of the force-extension relation of the NL, after a 20-ns constant force-extension simulation, the distance between the two terminal CAs of the linker is extracted from the output trace files using the VMD1.9.2 [12].