Mean Squared Displacement

Finding the MSD of the simulation is (a trivial task) using a Matlab toolbox (see video below). The target extrasynaptic MSD was 0.1 ^µm²⁄_s (from Choquet ^FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}). Scaling the model PSD size and distance between synapses to real-world values based on the target MSD is not a trivial task, and may actually be related only indirectly. Just because a particle moves quickly, does not mean it moves from A to B quickly - it depends on how much stuff there is to collide with. Linear estimates have shown that glycine receptor movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s).

First, the randomly generated step-size was scaled to produce an MSD of 0.1 ^units²⁄_step using the Matlab Brownian motion toolbox. Second, the dimensions of the model need to be scaled. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 ^units²⁄_step. Next, the arbitrary 0.4 units need to be given meaning...

Given movements along dendrites at a speed of 1–2 µm/min (.016-.008 µm/s) observation, we can use a derivation of the MSD equation to help scale the model. The root-mean-square distance after N unit steps, with a step length of (L) is:

• d = L*sqrt(N)

In order to travel a distance d, N steps are required by this equation

• N = (d/L)²
• N = (1000 nm /400 nm)² = 6.25
• N = (2000 nm /400 nm)² = 25
• N = (4000 nm /400 nm)² = 100

• Lets say a particle is traveling linearly at a rate of 1 µm / 100 s
• And takes between 6-25 steps to cover 1-2 µm.

here are two good resources explaining MSD:

• wolfram alpha
• Democritus

VIDEO

---

{{{2}}}

MEAN SQUARED DISPLACEMENT

---

Brownian Motion Mean Squared Displacement

• The goal of this calculation is to relate the simulated particle diffusion to real world values, namely velocity.
• Particle velocity will be a function of MSD x units ²⁄s which scales on space (units) and time (s) parameters.
• Space and time in the model are defined arbitrarily as Step_Size and Step where each Step a particle moves a distance randomly chosen from a normal distribution (µ=1,σ=.2)
• a step size of 1 unit/step will produce a brownian motion MSD of ~0.52 ±0.2 units ²/s
• empirical observations show that reasonable values for MSD are:
  • PSD 0.01 µm ²/s
  • synaptic 0.05 µm ²/s
  • extrasynaptic 0.1 µm ²/s

• given an MSD of 0.52 ±0.2 units ²/s at the current parameters: 1 step = 1 unit (at µ=1,σ=.2), the model will need to be scaled such that particles move at an extrasynaptic rate of 0.1 µm ²/s.
• spines are on average 1 to 10 µm apart, if the model is comparing two spines 1 µm apart, they should be separated by 5 units of model space. This is because the current particle diffusion rate of the model is .5 µm ²/s and the empirical MSD is .1 µm ²/s

---

to make 0.1 ^units²⁄_step ≈ 0.1 ^µm²⁄_s. It was found that an XY random step-size of µ=0.4 (σ=.2) units produced an MSE ≈ 0.1 ^units²⁄_step. Then, the arbitrary 0.5 units were given meaning (converted to 0.5 µm) by scaling the model according to real-world values (see below) by making 1 unit = 1 µm; as a convention, a subunit will be 1/10th of a unit, thus 1 subunit = 0.1 µm). The PSD areas were set to 3-subunits (.3 µm) square, 20 subunits (2 µm) apart, within a rectangular field 20 subunits (2 µm) wide and 60 subunits (6 µm) long. Given these scaled dimensions where 10 subunits ≈ 1 µm, a particle with an XY step-size of 0.5 units moving in a straight line, could theoretically go from PSD1 to PSD2 in 4 steps (obviously given the simulated particles are moving with Brownian motion, this lower-bound would be extremely rare).

Michalet • 2010 • Phys Rev E Stat Nonlin Soft Matter Phys - PDF

Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium

Expand to view experiment summary

---

We examine the capability of mean square displacement analysis to extract reliable values of the diffusion coefficient D of single particle undergoing Brownian motion in an isotropic medium in the presence of localization uncertainty. The theoretical results, supported by simulations, show that a simple unweighted least square fit of the MSD curve can provide the best estimate of D provided an optimal number of MSD points is used for the fit. We discuss the practical implications of these results for data analysis in single-particle tracking experiments.