Mean Squared Displacement

MSD Explained

In statistical mechanics, the mean squared displacement (MSD or average squared displacement) is the most common measure of the spatial extent of random motion; one can think of MSD as the amount of the system "explored" by a random walker.

More Background

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The Probability Density Function (PDF) for a particle in one dimension is found by solving the one-dimensional Diffusion equation. (This equation states that the position probability density diffuses out over time - this is the method used by Einstein to describe a Brownian particle.

As a particle travels, the molecule is jostled by collisions with other molecules which prevent it from following a straight line. If the path is examined in close detail, it will be seen to be a good approximation to a random walk. Mathematically, a random walk is a series of steps, one after another, where each step is taken in a completely random direction from the one before. This kind of path was famously analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. In two dimensions this relationship can be written as:

r²=4•D•t

In 1D, since both forward and backward steps are equally probable, we come to the surprising conclusion that the probable distance travelled sums up to zero! This is clearly a useless property to calculate. If however, instead of adding the distance of each step we added the square of the distance, we realise that we will always be adding positive quantities to the total. In this case the sum will be some positive number, which grows larger with every step. This obviously gives a better idea about the distance (squared in this case) that a particle moves. If we assume each step happens at regular time intervals, we can easily see how the square distance grows with time, and Einstein showed that it grows linearly with time.

In a molecular system a molecule moves in three dimensions, but the same principle applies. Also, since we have many molecules to consider we can calculate a square displacement for all of them. The average square distance, taken over all molecules, gives us the mean square displacement. This is what makes the mean square displacement (or MSD for short) significant in science: through its relation to diffusion it is a measurable quantity, one which relates directly to the underlying motion of the molecules.

In molecular dynamics the MSD is easily calculated by adding the squares of the distance. The linear (i.e. straight line) dependence of the MSD plot is apparent. If the slope of this plot is taken, the diffusion coefficient D may be readily obtained.

At very short times however, the plot is not linear. This is because the the path a molecule takes will be an approximate straight line until it collides with its neighbour. Only when it starts the collision process will its path start to resemble a random walk. Until it makes that first collision, we may say it moves with approximately constant velocity, which means the distance it travels is proportional to time, and its MSD is therefore proportional to the time squared. Thus at very short time, the MSD resembles a parabola. This is of course a simplification - the collision between molecules is not like the collision between two pebbles, it is not instantaneous in space or time, but is `spread out' a little in both. This means that the behaviour of the MSD at short time is sometimes more complicated than this MSD plot shows.

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Albert Einstein in a study of Brownian motion and he showed that the mean square of the distance traveled by particle following a random walk is proportional to the time elapsed. This relationship can be written as:

r²=2d•D•t

• r² MSD
• d dimensions
• D diffusion coefficient (diffusion rate)
• t time step

The next important equation is the standard deviation of the diffusion rate distribution. It would be more convenient to call this the step size distribution but the two values are not exactly proportional. That equation is given as:

k = sqrt(d * D * t)

• k σ or standard deviation of the diffusion rate distribution
• d dimensions
• D diffusion coefficent
• t time step

Knowing k we can then generate random steps from a normal distribution with a mean µ = 0 and sd σ = k pulling a value from this distribution for both the X and Y dimensions for each step. In Matlab this would look like:

xyds = k * randn(2,Ndots);

This will generate a random walk with a diffusion rate = D and MSD = 2d•D. We might also be interested in the average step size that is being generated from this distribution to create such a diffusion rate. This can be done by taking the mean absolute value of the generated steps:

mStep ≈ mean(mean(abs(xyds),2))

The analytical solution to this problem however is the mean µ of the half normal distribution, with a standard deviation σ = k :

λ = k * √2 / √π

• mS = k*sqrt(2)/sqrt(pi)

Image Gallery

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File:Brownian4.ogv

Notes

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) ^FIG: {{#info: {{{2}}} CLICK AWAY FROM IMAGE TO CLOSE }}

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 several good resources for explaining and calculating MSD:

• wolfram alpha
• Democritus
• Matlab toolbox

VIDEO

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MEAN SQUARED DISPLACEMENT

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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

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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).

Articles

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

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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.

Czondor Choquet • 2012 • PNAS - PDF

Unified quantitative model of AMPA receptor trafficking at synapses

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^FIG: {{#info: Error creating thumbnail: File missing(A) Schematic diagramof themodel. Kinetic parameters include Dout (extrasynaptic diffusion), Din (synaptic diffusion), DPSD (diffusion coefficient of the PSD), kon (AMPAR/ scaffold binding rate), koff (AMPAR/scaffold dissociation rate), and kendo (endocytosis rate). (B) Simulated trajectory (50 s). The synapse is in green, the PSD in red, and dendrite borders in gray. Geometric parameters are: a (synapse spacing), b (synapse width), c (PSD width), w (dendrite width). (C) High-magnification 30-s trajectory of an AMPAR-bound Qdot on the surface of a DIV 9 neuron. and simulated (plain curves) AMPAR median diffusion coefficients obtained at different neuronal ages (DIV 4–15) or varying synaptic spacing (0.75–30 μm), respectively, were plotted against synapse density. The simulated curves were computed for different kon values, keeping koff: 0.1 s−1. (F) Interquartile distributions of AMPAR diffusion coefficients from experiments (black) and simulations (green). Neuroligin-1 expression, which doubled the number of Homer1c-positive puncta, was mimicked by a decrease in synapse spacing (a: 1 μm). Overexpressing PSD-95 was modeled by enhancing kon (2.5 s−1) to mimic an increase in PSD binding sites, plus an increase in PSD size (c: 0.4 μm). CLICK AWAY FROM IMAGE TO CLOSE }}

Dendrite

• length: 10 - 60 µm
• width: 2 µm
• D.out: 0.05 - 0.3 µm²/s

PSD

• length: 0.3 µm
• width: 0.3 µm
• distance: 1 - 30 µm apart

PSA (perisynaptic area PSD pad)

• length: 0.3 µm
• width: 0.3 µm
• D.in: 0.03 µm²/s

Total synaptic area: .6 x .6 µm In some cases, we varied the size of the synapse and the PSD, keeping a constant ratio of 2 between them (b: 0.1–1.6 μm; c: 0.05–0.8 μm).

AMPARs bound to the PSD were allowed to diffuse at a very low diffusion coefficient DPSD: 0.0002 μm2/s, reflecting the intrinsic movement or morphing of the PSD. We neglected the turnover of individual scaffold molecules, typically much longer than the AMPAR diffusion time scale