Brownian Motions - Revision history

Bradley Monk at 04:42, 23 June 2024

2024-06-23T04:42:18Z

Bradley Monk: /* Stochastic Diffusion */

2024-06-23T04:27:50Z

Stochastic Diffusion

← Older revision		Revision as of 21:27, 22 June 2024
Line 467:		Line 467:
	=Stochastic Diffusion=		=Stochastic Diffusion=

	~~==Introduction==~~
	This exercise shows how to simulate the motion of single and multiple particles in one and two dimensions using Matlab. You will discover some useful ways to visualize and analyze particle motion data, as well as learn the Matlab code to accomplish these tasks. Once you understand the simulations, you can tweak the code to simulate the actual experimental conditions you choose for your study of Brownian motion of synthetic beads. These simulations will generate the predictions you can test in your experiment.		This exercise shows how to simulate the motion of single and multiple particles in one and two dimensions using Matlab. You will discover some useful ways to visualize and analyze particle motion data, as well as learn the Matlab code to accomplish these tasks. Once you understand the simulations, you can tweak the code to simulate the actual experimental conditions you choose for your study of Brownian motion of synthetic beads. These simulations will generate the predictions you can test in your experiment.

Bradley Monk: /* Stochastic Diffusion */

2024-06-23T04:27:14Z

Stochastic Diffusion

← Older revision		Revision as of 21:27, 22 June 2024
Line 465:		Line 465:


	==Stochastic Diffusion==		=Stochastic Diffusion=

	==Introduction==		==Introduction==

Bradley Monk: /* Introduction */

2024-06-23T04:26:57Z

Introduction

← Older revision		Revision as of 21:26, 22 June 2024
Line 467:		Line 467:
	==Stochastic Diffusion==		==Stochastic Diffusion==

	=Introduction=		==Introduction==
	This exercise shows how to simulate the motion of single and multiple particles in one and two dimensions using Matlab. You will discover some useful ways to visualize and analyze particle motion data, as well as learn the Matlab code to accomplish these tasks. Once you understand the simulations, you can tweak the code to simulate the actual experimental conditions you choose for your study of Brownian motion of synthetic beads. These simulations will generate the predictions you can test in your experiment.		This exercise shows how to simulate the motion of single and multiple particles in one and two dimensions using Matlab. You will discover some useful ways to visualize and analyze particle motion data, as well as learn the Matlab code to accomplish these tasks. Once you understand the simulations, you can tweak the code to simulate the actual experimental conditions you choose for your study of Brownian motion of synthetic beads. These simulations will generate the predictions you can test in your experiment.

Bradley Monk: /* Stochastic Diffusion */

2024-06-23T04:26:17Z

Stochastic Diffusion

← Older revision		Revision as of 21:26, 22 June 2024
Line 465:		Line 465:


	==[[Stochastic Diffusion]]==		==Stochastic Diffusion==

	=Introduction=		=Introduction=

Bradley Monk at 04:25, 23 June 2024

2024-06-23T04:25:21Z

Show changes

Bradley Monk at 04:11, 23 June 2024

2024-06-23T04:11:47Z

← Older revision		Revision as of 21:11, 22 June 2024
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	==[[Brownian Motion]] Matlab Code==		==[[Brownian Motion]] Matlab Code==

	<syntaxhighlight lang="matlab" line start="1~~" highlight="1" enclose="div~~">		<syntaxhighlight lang="matlab" line start="1">
	function [out] = BradsBrownianMotionScript()		function [out] = BradsBrownianMotionScript()
	format compact;		format compact;

Bradley Monk at 04:10, 23 June 2024

2024-06-23T04:10:29Z

Show changes

Bradley Monk at 04:09, 23 June 2024

2024-06-23T04:09:04Z

← Older revision		Revision as of 21:09, 22 June 2024
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	: <big><code>D (in units: '''µm²/s''')</code></big> is the mean diffusion rate per unit time (velocity), often in µm²/s for biological motion on a molecular level (n.b. µm²/s are the units for particle diffusion on a surface/membrane; the surface can actually be curved or ruffled such that the surface fills a 3D space, however the diffusion equations for those instances are slightly more complex. Units for 3D space are naturally: µm³/s). This refers to how fast, on average, the particle moves along its surface trajectories. This value is often of ultimate interest, that is, the goal of single-particle tracking studies is often to define the average diffusion rate of a molecule and report it as a Diffusion Coefficient; once '''D''' is known, this Diffusion Coefficient can then be easily implemented in [[computer]] simulations of 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 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:		<big><code>D (in units: '''µm²/s''')</code></big> is the mean diffusion rate per unit time (velocity), often in µm²/s for biological motion on a molecular level (n.b. µm²/s are the units for particle diffusion on a surface/membrane; the surface can actually be curved or ruffled such that the surface fills a 3D space, however the diffusion equations for those instances are slightly more complex. Units for 3D space are naturally: µm³/s). This refers to how fast, on average, the particle moves along its surface trajectories. This value is often of ultimate interest, that is, the goal of single-particle tracking studies is often to define the average diffusion rate of a molecule and report it as a Diffusion Coefficient; once '''D''' is known, this Diffusion Coefficient can then be easily implemented in [[computer]] simulations of 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 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:

	Instead, '''D''' is often calculated from the ''mean squared diffusion'' (MSD) path of the particle, defined below.		Instead, '''D''' is often calculated from the ''mean squared diffusion'' (MSD) path of the particle, defined below.


			==== <big>MSD</big>  : mean squared displacement ====
			<br>

			::<math>
			MSD = { 2d \cdot D}
			</math>


			:: <big><code>d</code></big> : dimensions
			:: <big><code>D</code></big> : diffusion coefficient


			: <big><code>MSD (in units: '''µm²''')</code></big> is the mean square displacement of a particle over some time period. If '''D''' (the diffusion coefficient) is unknown, MSD can be computed using the following equation:


			[[File:Mean Squared Displacement Equation.png\|left\|150px]]
			{{Clear}}


			...where the quantity '<big><code>''x''</code></big> ' refers to the displacement of a particle over '<big><code>''n''</code></big> ' total steps, with '<big><code>''k''</code></big> ' indicating the current step.

			==== <big>K</big>  : standard deviation of the normal distribution for '''D''' ====
			<br>

			::<math>
			k = \sqrt{d \cdot D}
			</math>

			:: <big><code>d</code></big> : dimensions
			:: <big><code>D</code></big> : diffusion coefficient


			: <big><code>k (in units: '''µm''')</code></big> is the standard deviation (σ) of the normal distribution of step lengths that, when randomly sampled, will give rise to a diffusion rate '''D'''. This value is useful for simulating Brownian motion for a particular diffusion rate.


			==== <big>L</big>  : mean step length ====
			<br>

			::<math>
			L = \sqrt{2d \cdot D}
			</math>

			:: <big><code>d</code></big> : dimensions
			:: <big><code>D</code></big> : diffusion coefficient


			<big><code>L or Λ (in units: '''µm''')</code></big> is the average step length per interval of observation. In diffusion simulations, this is the step size per iteration (equivalent form: <big><code>L = (2d * D)<sup>.5</sup></code></big>).



			==== <big>λ</big>  : mean 1D step length component of L ====
			<br>

			::<math>
			\lambda = {L \over{\sqrt{2}}}
			</math>


			<big><code>λ (in units: '''µm''')</code></big> is the average 1-dimensional step length for each component (X,Y,Z) dimension of '''''L'''''. For example, simulating 2D particle diffusion will require the generation of individual step lengths for both the X and Y dimension. The total step distance from the origin will be the length of the hypotenuse created by the individual X and Y component step lengths. In fact, the equation: λ {{=}} L / √2 is derived from the Pythagorean theorem for right triangles, such that 2λ² = λ² where 2λ² represents a² + b² and λ² represents c².



			==== <big>ε</big>  : step length scalar coefficient ====
			<br>

			::<math>
			\epsilon = {\sqrt{\delta \over{D}}}
			</math>


			:: <big><code>δ</code></big> : new desired diffusion rate

			: <big><code>ε (in units: au)</code></big> is a coefficient value that, when multiplied by each λ component step length, will scale those lengths to achieve a new diffusion rate '''δ'''.


			{{Clear}}

Bradley Monk at 04:07, 23 June 2024

2024-06-23T04:07:08Z

← Older revision		Revision as of 21:07, 22 June 2024
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	[[File:Brownian5.gif\|right]]		[[File:Brownian5.gif\|right]]


			==Common Diffusion Terms & Parameters==

			I've found it most helpful to learn about the terms and parameters used to characterize stochastic diffusion as they apply to diffusion phenomena along a flat 2D surface. These parameters include:


			==== <big>D</big>  : diffusion rate coefficient ====
			<br>

			::<math>
			D = {L^{2} \over 2d \cdot t}
			</math>

			:: <big><code>L</code></big> : step length
			:: <big><code>d</code></big> : dimensions
			:: <big><code>t</code></big> : time


			: <big><code>D (in units: '''µm²/s''')</code></big> is the mean diffusion rate per unit time (velocity), often in µm²/s for biological motion on a molecular level (n.b. µm²/s are the units for particle diffusion on a surface/membrane; the surface can actually be curved or ruffled such that the surface fills a 3D space, however the diffusion equations for those instances are slightly more complex. Units for 3D space are naturally: µm³/s). This refers to how fast, on average, the particle moves along its surface trajectories. This value is often of ultimate interest, that is, the goal of single-particle tracking studies is often to define the average diffusion rate of a molecule and report it as a Diffusion Coefficient; once '''D''' is known, this Diffusion Coefficient can then be easily implemented in [[computer]] simulations of 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 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:

			Instead, '''D''' is often calculated from the ''mean squared diffusion'' (MSD) path of the particle, defined below.