Microscopy Biomedical Imaging

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Detection of Non-Brownian Diffusion in the Cell Membrane inSingle Molecule Tracking

Behind the hop diffusionBased on: Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking

Ken Ritchie, Xiao-Yuan Shan, Junko Kondo, Kokoro Iwasawa, Takahiro Fujiwara, and Akihiro Kusumi

Biophysical Journal Volume 88 March 2005

Rodrigo Rojas Moraleda

January 2011

Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 1/53


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Outline

1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary



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Outline

1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary



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Objectives

This work presents the use of simulation to create a baseline from which to study andanalyze:

Confined diffusion phenomena in plasma membrane.

The relationship between the data acquisition rate and the interpretation of thediffusion.

Using a Monte Carlo algorithm, molecules undergoing simple brownian, totallyconfined, and hop diffusion, where simulated. Next, the characteristics determined areexperimentally examined.



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Hypothesis

The implicit assumption hypothesis of this work is:

Is possible to establish a relationship between the data acquisition rate

(observation frame rate) and the interpretation of the diffusioncharacteristics of individual particles/molecules



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Outline

1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary



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Diffusion in plasma membrane Why?

Many cellular processes, such as signaling processes, involve the interaction of severalindividual molecules that must come together to transmit information across the

plasma membrane to the cell interior. Hence, it is of great importance to understandthe mechanism by which the motion of transmembrane and membrane-associatedmolecules is regulated in the cell membrane.



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Diffusion in plasma membrane Difficulty

However, in the cells, molecular behavior is very inhomogeneous, even molecules ofsingle species interact stochastically with distinct molecules or cellular structures in avariety of local environments. Furthermore, molecular interactions are by naturestochastic.



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Outline

1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary



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Background

Main features about this simulation:

Particles were represented by a Gaussian intensity of 250nm width.

Particle motion were simulated in time steps of 1s.

Each captured frame contains the sum of individual 1s particle motions.

The pixel resolution in the image acquisition was simulated to 40nm/pixel.



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Background







B k d


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Background







B k d


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Background







B k d


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Background







B k d


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Background Process

To characterize the particle motion mode:

Obtain the trajectory of a single molecules.

Calculate the mean square displacement (MSD) for every lag time ( )

Plotting it as a function of corresponding


Background


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Background Process






Background P


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Background Process






Background P


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Background Process






Background


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Background

Figure: a kernel image of the diffusion probe was taken from the first frame of the video.


Background Process


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Background Process





MSD is determined byThe position information (r1(x1, y1), r2(x2, y2), r3(x3, y3), . . .) of single moleculesin the recorded trajectory at a fixed acquisition time, t .

Lag time is given as n = nt where n is the number of lags between two steps.


Background Process


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Background Process








Background Process


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Background Process








Background Process


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g








Background Process


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g








Background Process


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g








Background Mean Square Displacement


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For instance, for 1 = 1t , the displacements are calculated as follows;

Then the square displacements are calculated as




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The MSD is obtained as an average of all steps corresponding to a single lag time bydividing the sum of square displacements to the number of sample periods between

start and end points in the trajectory.

The quantitative analysis of molecular movement was carried out based on the MSD

methods

MSD(nt) = (N 1 n)1N1n

j=1

{[x(jt + nt) x(jt)]2

+[y(jt + nt) y(jt)]2]}

t is the time resolution.x(jt + nt), y(jt + nt) describes the particle position following an interval nt.




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The MSD is obtained as an average of all steps corresponding to a single lag time bydividing the sum of square displacements to the number of sample periods between

start and end points in the trajectory.

The quantitative analysis of molecular movement was carried out based on the MSD

methods

MSD(nt) = (N 1 n)1N1n

j=1

{[x(jt + nt) x(jt)]2

+[y(jt + nt) y(jt)]2]}

t is the time resolution.x(jt + nt), y(jt + nt) describes the particle position following an interval nt.




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Background Confinement


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Convention: D24 mean estimate D fitting MSD at 2t, 3t and 4t using a straightline, commonly because behavior of the MSD plot between 0 and time 2t is complex

MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}

whereL : compartment size

D : Fitting parameter that estimate the microscopic diffusion coefficient.n : is the frame number.t : time for each frame.

Convention: Dmacro The macroscopic diffusion coefficient describing the HopDiffussion over the compartments.

Hop Diffussion is characterized by the compartment size L the shor -term diffusionD24 and the average residence time

=L2

4DMacro

mas...




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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...




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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...




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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...




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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...




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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...



C D fi MSD d h


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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...



C i D i D fi i MSD 2 3 d 4 i i h


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MSD = x(nt)2conf

=L2

6

16L2

4

inf

k=1(odd)

1

k4exp{

1

2(

k

L)22Dnt}





=L2

4DMacro

mas...


Outline


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1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary


Simulation Experimental Setup


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Using a Monte Carlo algorithm, molecules undergoing simple brownian, totallyconfined, and hop diffusion, where simulated.



Video acquisition rate was setup to 25s( 140500

sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1

30 sec).Each captured frame contains the sum of individual 1s particle motions.


The image contrast reduction (that involves and uncertainty increase to locate aparticle) caused by the reduction of frame exposure time, was not simulated.

All simulations were performed for 1000 frames per run and 100 runs per case.




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sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1








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sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1








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sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1








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sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1








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sec), 0.11ms( 19090

sec),

0.2ms(1

500 sec), 33ms(1








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sec), 0.11ms( 19090

sec),

0.2ms(

1

500 sec), 33ms(

1








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sec), 0.11ms( 19090

sec),

0.2ms(

1

500 sec), 33ms(

1








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sec), 0.11ms( 19090

sec),

0.2ms(

1

500 sec), 33ms(

1






Simulation About Brownian diffusion

Brownian Motion .


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is the assumably random movement of particles suspended in a fluid. Einsteinpredicted that Brownian motion of a particle in a fluid at a thermodynamictemperature T is characterized by a diffusion coefficient

D = kbT/b [nm2/s]

kb is the Boltzman constant, T temperature, b is the resistance coefficient on theparticle.

In free diffusion with a large number of molecules, it was predicted that after

time, , a molecule will end up somewhere within a sphere of radius R.Mean square displacement of particles during diffusion time, , and diffusioncoefficient,D in any direction.

< R2 >= 2D, < R2 >= 4D, < R2 >= 6D


Simulation Brownian diffusion


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Brownian diffusion was simulated by allowing a point particle to walk randomly on asquare lattice. Each timestep consisted of a choice of moving to one of the fournearest-neighbor sites. The scale of the simulation was set such that the spacingbetween lattice sites was 6nm and the timestep was 1ms. As such, the base diffusion

coefficient was 9m2/s = (6nm)2/4(1s).


sec), 0.11ms( 19090

sec),

0.2ms( 1500

sec), 33ms( 130

sec).


Simulation Brownian motion


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Figure: Brownian Motion: Monte Carlo trajectories (1000 frames each) observed at frame rates of33msecframe and

25frame

Important similarity between the 33msec trajectory and the expanded 25sectrajectory. As expected simple Brownian motion is unaffected by the frame rate of

acquisition.


Simulation Mean Square Displacement MSD


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Figure: MSDs calculated from the trajectories above (single-molecule MSDs) plotted as a functionof the time interval. Both MSDs grow linearly with time, indicating that, in both of these verydifferent time-windows, the motion is simple Brownian characterized by similar single diffusioncoefficients within the error of the measurement. Note that both x and y axes are expanded by afactor of 1000 in the figure on the right.


Simulation Brownian motion


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Figure: Average microscopic diffusion coefficient (error bars represent the standard error of themean) expected to be observed at different frame times (at least 100 simulations for each frametime). The set diffusion coefficient in the simulation was 9mm2/s (shown by a lateral broken line).


Simulation Confined diffusion


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Figure: In confined diffusion, the particle is free to randomly diffuse inside an area surrounded byimpermeable walls.




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Early diffusion measurements immediately showed that different environments and/orpatterns of membrane organization (intramembrane barriers, skeletal interactions,rafts,and other phenomena) would have differential effects on the lateral diffusion behaviorof membrane components.




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(1) membrane diffusion; (2) two forms of cytosolic diffusion: restricted diffusion(arrowheads) and unrestricted diffusion (arrows); (3) active transport; and (4)confined diffusion.


Simulation Confined motion


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Confined motion simulations were performed in a square of size 42, 120, and 240 nmbounded by impenetrable barriers.




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Figure: Tiypical 1000 frames trajectories of a molecule trapped in a squarer compartment.




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Figure: Tiypical 1000 frames trajectories of a molecule trapped in a squarer compartment.

Centralized average position over a circular area is related to the frame times.




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Figure: Particle trapped within a 120 nm lenght square compartment.




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Figure: Particle trapped within a 120 nm lenght square compartment.

The asymptotic values of L2/6 in the equation for particle trapped are to be grater in25sec/frame than 33ms/frame




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Figure: Average microscopic diffusion coeficient in D24 as function of the camera frame time in a120nm square compartment.




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Figure: Aparent compartment size as function of frame time, determined fiting with MSD-t curve




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Figure: assess 100,000 consecutive positions simulated undergoing a free free but confineddiffusion, frame times were, 25sec, 2ms, 33ms

Effective potential:U(x) = kBTlog(P(x))

U(x): Effective potential.

P(x): Probability of finding a particle at position x.kBT: Thermal energy.


Simulation Hop diffusion


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Evidence, from single particle tracking and optical tweezers, studies implies thatthe cytoplasmic portion of transmembrane proteins collides nonspecifically with

the membrane skeleton, causing a temporary confinement of the diffusing protein.Hop diffusion has also been observed for lipid motion in the outer-leaflet of themembrane. Implying that obstacles are immobilized on the underlying membraneskeleton meshwork and hence reflect its structure in the hindrance of lipiddiffusion.




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Evidence, from single particle tracking and optical tweezers, studies implies thatthe cytoplasmic portion of transmembrane proteins collides nonspecifically with

the membrane skeleton, causing a temporary confinement of the diffusing protein.Hop diffusion has also been observed for lipid motion in the outer-leaflet of themembrane. Implying that obstacles are immobilized on the underlying membraneskeleton meshwork and hence reflect its structure in the hindrance of lipiddiffusion.




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Hop diffusion was simulated using a two-dimensional square array of partiallypermeable barriers (probability of transmission per attempt 0.0008) separated by 42,

120, or 240 nm.




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Figure: Typical 1000-frame trajectories simulated for a particle undergoing hop diffusion over120-nm length square compartments




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Figure: Typical 1000-frame trajectories simulated for a particle undergoing hop diffusion over120-nm length square compartments




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At a frame time of 33 ms (right), the plot can be fitted with a linear line,showing (apparent) simple Brownian character.

At a frame time of 25 ms, typical hop diffusion characteristics are apparent: fastrise in the short-time regime and slower linear growth of MSD with time in thelong-time regime, with a slope comparable to that found in the 33-ms MSD-t

plot.




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plot.




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plot.




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At shorter frame times, the diffusion coefficient within a compartment dominates.

At much longer frame the diffusion coefficient within a compartment becomesnegligible (2ms/fr, 0.14m2/s), and the apparent diffusion coefficient isdetermined by the hop diffusion between the compartments.




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At much longer frame the diffusion coefficient within a compartment becomesnegligible (2ms/fr, 0.14m2/s), and the apparent diffusion coefficient isdetermined by the hop diffusion between the compartments.




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At much longer frame the diffusion coefficient within a compartment becomesnegligible (2ms/fr, 0.14m2/s), and the apparent diffusion coefficient is

determined by the hop diffusion between the compartments.




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The plot of log(MSD/time) against log(time), covering six orders of magnitude intime (2s 2s).

The individual solid curves are those obtained for each frame time.

The vertical broken lines show the time taken to reach the barriers (at 0.1 ms)and the median residency time within a compartment (at 23 ms).

Anomalous diffusion is observed in between these events.




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Outline

1 Obj ti


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1 Objectives

2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary


Discussion The system

PtK2 kangaroo rat kidney cells


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PtK2 kangaroo rat kidney cells.

Diffusion of a transmembrane protein, transferrin receptor was measured.

Colloidal gold particles of 40-nm in diameter conjugated with bovine holotransferrin

Single fluorescent-molecule video imaging was performed using a 1.45 NA TIRFobjective

The precision of the position determination was estimated from the standarddeviation of the coordinates of 40-nm diameter gold particles attached to apoly-L-lysine-coated coverslip, it were 17 nm and 6.9 nm at time-resolutions of 25ms and 2 ms, respectively.

The positional resolution begets a limit on the smallest diffusion coefficient thatmay be measured. At a time-resolution of 25 ms, the smallest measurablediffusion coefficient was found to be 0.021 mm2/s.

The frame time has been systematically varied from the standard videos 33 ms

to 220 s and 25 s.





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The precision of the position determination was estimated from the standarddeviation of the coordinates of 40-nm diameter gold particles attached to a

poly-L-lysine-coated coverslip, it were 17 nm and 6.9 nm at time-resolutions of 25ms and 2 ms, respectively.



to 220 s and 25 s.





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to 220 s and 25 s.





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to 220 s and 25 s.





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to 220 s and 25 s.





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g y








to 220 s and 25 s.





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g y








to 220 s and 25 s.





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to 220 s and 25 s.


Model evaluation results


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Experimental trajectories obtained at frame times of 25s, 220s, and 33 ms. In thecase of 25s trajectory, various plausible compartments detected.

Decreasing of the frame time, the number of trajectories classified as undergoingsimple Brownian motion decreases from 77% at standard video rate to 7% at a frametime of 25 ms.




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Apparent microscopic diffusion coefficient, D13, for transferrin receptor plottedagainst the frame time of the camera (red triangles). and the results of Monte Carlosimulation (blue squares) for 54-nm compartments with underlying diffusion coefficientof 9 mm2/s and probability of passing a barrier of 0.0045.

The average compartment size for transferrin receptor in PtK2 cells was determined tobe 47 0.03 nm, (n=30) by a fit to the MSD-t

The average residency time for gold-tagged transferrin ( = L2

4Dmacro), was 2.8 ms.




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These results can be simulated very well, assuming the hop diffusion for 54-nmcompartments with the microscopic diffusion coefficient of 9 mm2/s, a probability of

passing the boundaries of 0.0045.


Outline

1 Objectives


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2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary


Discussion

Th l l l i di h h h diff i b il i k l


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These results clearly indicate that the hop diffusion can be easily mistaken as slow

simple Brownian diffusion, if observation is made only at relative slower ratesrespect particle motion.

Although pure simple Brownian motion is unaffected by this time-averaging, theapparent motion of particles undergoing confined or hop diffusion motion isstrongly affected.

The distinction of the frame rate, frame time and the total time of observations,

may not be clear. However these three represent different concepts, and thisdifference becomes important in understanding their effects on hop diffusion.

A frame time of 2-ms suggests that 50 determinations have to be made before acompartment is detected as such, which is in general agreement with ourexperience


Discussion

Th l l l i di h h h diff i b il i k l


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Discussion

Th lt l l i di t th t th h diff i b il i t k l


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Discussion



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Discussion



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Questions ?

Rodrigo Rojas [email protected]


Outline

1 Objectives


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2 Introduction

3 Background

4 Simulation

5 Model Evaluation

6 Discussion

7 Glosary


Glosary


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A stochastic process is one whose behavior is non-deterministic. volver


Glosary


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Monte Carlo methods (or Monte Carlo experiments) are a class of computationalalgorithms that rely on repeated random sampling to compute their results.

Define a domain of possible inputs.

Generate inputs randomly from the domain using a certain specified probabilitydistribution.

Perform a deterministic computation using the inputs.

volver


Glosary

A simple Monte Carlo simulation to approximate the value of could involverandomly selecting points (xi, yi)

ni=1 in a unit square and determining the ratio =

mn

, where m is the number of points that satisfy x2i + y2i 1. In a typical simulation of

l h f 2 2


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sample size n = 1000 there were 787 points satisfying x2i

+ y2i 1.

Using this data, we obtain

pointsinsidethecircle

pointsinsidethesquare=

14r2

r2

=1

4

=m

n=

787

1000= 0.787

4 = 3.148volver



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