The effect of diffusional space restriction with a rigid cage

Simulations of diffusional movement were performed as described in section 2.3. In this sub-section, a reflective wall was introduced at different heights H parallel to the plane of the coverslip (see Fig. 2.12a). A diffusing fluorescent object hence was enclosed inside the spatial cage 0<Z <H , because it bounced back after every collision from the coverslip or the wall during random walk diffusion. As expected, this constraint resulted in deflection of the MSD plots from the straight line (‘free’ diffusion) and formation of the saturation plateau, indicative of ‘caged’ diffusion (Fig. 4.9a). The saturation level of the MSD plots corresponds to the effective cage radius (approximately H/2). A simulated ‘immobile control’ (an object fixed at the coverslip plane Z=0 and generating Poisson noise with the average intensity 100 a.u.) revealed a flat MSD with a small (50 nm²) vertical offset due to noise. When the cage size was kept constant (H=150 nm) but the diffusion coefficient was variable (Fig. 4.9b), it became apparent that the speed of

NPA, nm² M18 KO

WT

KO + Munc18-1

***

0 2 4 6 8 10 12 14

NPA, nm² 0 1 2 3 4 5 6 7 8 9 10 Fixed cells

Beads

Live cells _**^*

a) b)

NPA, nm² M18 KO

WT

KO + Munc18-1

***

0 2 4 6 8 10 12 14

NPA, nm² 0 1 2 3 4 5 6 7 8 9 10 Fixed cells

Beads

Live cells _**^*

a) b)

MSD plot saturation strongly depends on the diffusion coefficient, i.e. on the average speed of the diffusing object reaching the opposite walls of the cage. These results also confirmed that the algorithm employed for simulations (section 2.3) functioned correctly.

Fig. 4.9. MSD plots calculated from the Z-trajectories, obtained using random-walk diffusion simulations in restricted space. (a) A restrictive wall enclosing the object in the space 0<Z <H led to MSD saturation at the level corresponding to the cage size H. ‘Immobile control’ (Z=0, only Poisson noise) revealed a flat MSD. Z-diffusion coefficient assumed was 0.005 μm²/s. (b) The Z-diffusion coefficient was varied while the cage size kept constant: H=150 nm. MSD plots plateau faster (larger D_Z) or slower (smaller D_Z) to the same level.

Next, the velocity autocorrelation functions (ΔZ-ACF) were examined, that result from the simulated trajectories with diffusional space restrictions. The data shown in Fig. 4.10 demonstrate the effect of diffusional space constriction on the ΔZ-ACF. The ‘immobile control’ (Poisson noise only) resulted in an autocorrelation function not differing significantly from zero (Fig. 4.10a), as in the TIRFM experiments with immobile controls described in the previous section. When the object was allowed to diffuse by increasing the height H of reflective cage from zero, the negative amplitude of ΔZ-ACF also increased. This increase was due to the more prominent signal component arising from repetitive collisions with the walls compared to the random noise from random walk and fluorescence intensity. Additionally, the negative component’s decay time to zero became longer with increasing H. The latter is explained by longer time required for the object to reach the opposite wall of the cage after collision. However, further increase in H, along with slowing ΔZ-ACF decay time course, leads in turn to a decrease of the ΔZ-ACF negative amplitude and elevation of the variance (error bars are ±SEM). This decrease is caused by the fact that at large H the object can travel far from the coverslip plane, and changes in the direction of its movement direction are determined more by the random walk probability than by reflections from the restricting wall. Interestingly, when the object is set free in the simulated semi-space (H =∞), the negative component can still be detected only if no additional intensity noise is added in the simulations (or if the number of averaged traces is increased). This small negative component results from the

0

restriction of the space only by the coverslip, since random walk diffusion loses its symmetry and randomness due to possible reflections of the object from the coverslip plane. The described behavior of the ΔZ-ACF is illustrated by the NPA value (Fig.

4.10b): it is essentially zero for Poisson noise (fixed object) and increases with the object freedom until random changes in movement direction become prevalent over the reflections from restrictive walls at a given diffusion coefficient. An object in a semi-free space is found to have an NPA value different from zero, but the significance of this difference may be compromised by random noise.

Fig. 4.10. Summary of the ΔZ-ACF analysis of the simulated diffusion trajectories in restricted space. (a) Average ΔZ-ACF functions obtained for diffusion coefficient D_Z=0.005 in a space with the restrictive wall at Z =H. Fluorescence noise was added in all cases except Z <∞

(which otherwise resulted in a very noisy function). (b) Quantification of ΔZ-ACF negative amplitude by the NPA value reveals maximum at intermediate H values (100-200 nm) and small amplitudes for either strongly restricted or free objects. (c) Average ΔZ-ACF functions obtained for different diffusion coefficients in the space restricted at H=150 nm. The negative amplitude component, quantified in (d) with the NPA, increases within the range of diffusion coefficients assumed in simulations.

Similar effects are produced by varying the diffusion coefficient for the object in the cage with fixed size. Fig. 4.10c illustrates the increase in ΔZ-ACF negative amplitude and acceleration of the autocorrelation decay when D_Z increases, due to shortening time intervals between collisions of the object with the walls. The NPA (Fig. 4.10d) reports the rise in the negative ΔZ-ACF amplitude in the range of tested diffusion coefficients.

-50

0 0.005 0.01 0.015 0.02 0.025 0.03 D_Z, μm²/s -Cage height H, nm

NPA, nm2

0 0.005 0.01 0.015 0.02 0.025 0.03 D_Z, μm²/s

0 0.005 0.01 0.015 0.02 0.025 0.03 D_Z, μm²/s -Cage height H, nm

NPA, nm2 -Cage height H, nm

NPA, nm2

noise D_Z=0.005 μm²/s

Faster diffusion coefficients could not be probed for the spatial grid size used, but the tendency of decay in NPA could be noticed already at D_Z=0.03 μm²/s. One can predict further amplitude decay with increasing D_Z, since it is equivalent to reducing the cage size while keeping the diffusion coefficient unchanged (shoulder at small H<150 nm in Fig. 4.10b).