Diffusion - Revision history

Bradley Monk at 17:24, 21 May 2017

2017-05-21T17:24:04Z

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	:: <big><code>d</code></big> : dimensions		:: <big><code>d</code></big> : dimensions
	:: <big><code>t</code></big> : time		:: <big><code>t</code></big> : time
	: <big><code>D (in units: '''µm²/s''')</code></big> is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this: {{Fig\|[[File:Brownian5sn.gif]]}}. Instead, '''D''' is often calculated from the ''mean squared diffusion'' (MSD) path of the particle, defined below.		: <big><code>D (in units: '''µm²/s''')</code></big> is the mean diffusion rate per unit time, often in µm²/s for biological motion on a molecular level. This refers to how fast, on average, the particle moves along its trajectories. This value is often of ultimate interest, particularly for simulating Brownian motion; however under most circumstances, the diffusion rate cannot be observed directly in empirical experiments - this would require the ability to visualize all the microscopic particles and collisions that dictate the particle's movement, and would have to be done on a nanosecond timescale. In the animation on the right, the diffusion rate can actually be quantified directly; but what is often seen when observing particle diffusion through a microscope would more closely resemble this: <!-- {{Fig\|[[File:Brownian5sn.gif]]}} --!>. Instead, '''D''' is often calculated from the ''mean squared diffusion'' (MSD) path of the particle, defined below.

Bradley Monk: /* Brownian Motion */

2014-11-24T17:44:32Z

Brownian Motion

← Older revision		Revision as of 10:44, 24 November 2014
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	==Brownian Motion==		==Brownian Motion==

	Brownian ~~motion~~ describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a ''random walk'', which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space. [[File:Brownian5.gif\|right]] The mathematical description of this process often includes these terms:		[[Brownian Motion]] describes the stochastic movement of particles as they travel through space. This type of random movement is often referred to as a ''random walk'', which is typical of particles that diffuse about a 1D 2D or 3D space filled with other particles or barriers. To understand Brownian motion, lets start by characterizing this phenomenon in 2D space. [[File:Brownian5.gif\|right]] The mathematical description of this process often includes these terms:

Bradley Monk at 05:22, 10 August 2014

2014-08-10T05:22:47Z

← Older revision		Revision as of 22:22, 9 August 2014
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	[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]		[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]][[Category:Neurobiology]]

Bradley Monk at 06:16, 23 November 2013

2013-11-23T06:16:48Z

Show changes

Bradley Monk: /* Articles */

2013-11-23T06:13:31Z

Articles

← Older revision		Revision as of 23:13, 22 November 2013
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	}}<!-- END ARTICLE -->		}}<!-- END ARTICLE -->

	~~{{Article\|Czondor Choquet\|2012\|PNAS - [http://www.ncbi.nlm.nih.gov/pubmed/22331885 PDF]\|22331885\|Unified quantitative model of AMPA receptor trafficking at synapses}}~~
	~~{{ExpandBox\|Expand to view experiment summary\|~~
	{{Fig\|[[File:UnifiedFig1.png\|500px]]\|(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).}}


	~~;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

	~~}}<!-- END ARTICLE -->~~









	[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]		[[Category:ReDiClus]] [[Category:Diffusion]] [[Category:Statistics]]

Bradley Monk at 22:45, 3 November 2013

2013-11-03T22:45:05Z

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	==Types of Diffusion==		==Types of Diffusion==
	[[File:Surface diffusion hopping.gif\|thumb\|300px\|[https://en.wikipedia.org/wiki/Surface_diffusion SURFACE DIFFUSION] -- Model of a single adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.]]		[[File:Surface diffusion hopping.gif\|thumb\|300px\|[https://en.wikipedia.org/wiki/Surface_diffusion SURFACE DIFFUSION] -- Model of a single adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.]]



	* Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients		* Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
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	~~{{Clear}}~~
	==Slot Diffusion==		==Slot Diffusion==
	[[File:Slots Diffusion.gif\|thumb\|300px\|Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.]]		[[File:Slots Diffusion.gif\|thumb\|300px\|Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.]]

Bradley Monk at 22:43, 3 November 2013

2013-11-03T22:43:06Z

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			{{TOC right}}

	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 the approximation of 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:		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 the approximation of 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:

Bradley Monk at 22:38, 3 November 2013

2013-11-03T22:38:57Z

← Older revision		Revision as of 15:38, 3 November 2013
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	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 the approximation of 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:		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 the approximation of 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:



	For a list of common mathematical concepts used to model diffusion see: [[Diffusion Mathematics]]		For a list of common mathematical concepts used to model diffusion see: [[Diffusion Mathematics]]

Bradley Monk at 22:38, 3 November 2013

2013-11-03T22:38:03Z

← Older revision		Revision as of 15:38, 3 November 2013
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			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 the approximation of 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:



			For a list of common mathematical concepts used to model diffusion see: [[Diffusion Mathematics]]


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	~~==Common Mathematical Concepts in Diffusion==~~

	* See [[Diffusion Mathematics]]

Bradley Monk: /* Common Mathematical Concepts in Diffusion */

2013-11-03T21:02:07Z

Common Mathematical Concepts in Diffusion

← Older revision		Revision as of 14:02, 3 November 2013
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	==Common Mathematical Concepts in Diffusion==		==Common Mathematical Concepts in Diffusion==

	~~{{ExpandBox\|Laplace operator\|~~		* See [[Diffusion Mathematics]]
	In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows.
	~~}}<!-- END ARTICLE -->~~