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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

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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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 10/53

    B k d

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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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 10/53

    B k d

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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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 10/53

    B k d

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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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 10/53

    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

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    Background

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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

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 11/53

    Background P

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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

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    Background P

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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

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 11/53

    Background

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    Background

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

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    Background Process

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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

    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.

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    Background Process

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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

    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.

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    Background Process

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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

    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.

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    Background Process

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    g

    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

    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.

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    Background Process

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    g

    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

    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.

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    Background Process

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    g

    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

    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.

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    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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    Background Mean Square Displacement

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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 Mean Square Displacement

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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 Mean Square Displacement

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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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    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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    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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    Background Confinement

    C D fi MSD d h

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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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    Background Confinement

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

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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...

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 18/53

    Outline

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    1 Objectives

    2 Introduction

    3 Background

    4 Simulation

    5 Model Evaluation

    6 Discussion

    7 Glosary

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    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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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    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 20/53

    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 20/53

    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 20/53

    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 20/53

    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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.

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 20/53

    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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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    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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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    Simulation Experimental Setup

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

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

    Particle motion were simulated in time steps of 1s.

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

    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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    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

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    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).

    Video acquisition rate was setup to 25s( 140500

    sec), 0.11ms( 19090

    sec),

    0.2ms( 1500

    sec), 33ms( 130

    sec).

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    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.

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    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.

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    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).

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    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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    Simulation Confined diffusion

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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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    Simulation Confined diffusion

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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.

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    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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    Simulation Confined motion

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

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    Simulation Confined motion

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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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    Simulation Confined motion

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

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    Simulation Confined motion

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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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    Simulation Confined motion

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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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    Simulation Confined motion

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

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    Simulation Confined motion

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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.

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    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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    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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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 is

    determined by the hop diffusion between the compartments.

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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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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    Simulation Hop diffusion

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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

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    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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    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 a

    poly-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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    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 a

    poly-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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    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 a

    poly-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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    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 a

    poly-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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    Discussion The system

    PtK2 kangaroo rat kidney cells.

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    g y

    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 a

    poly-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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    Discussion The system

    PtK2 kangaroo rat kidney cells.

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    g y

    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 a

    poly-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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    Discussion The system

    PtK2 kangaroo rat kidney cells.

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    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 a

    poly-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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    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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    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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    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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    Model evaluation results

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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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    Model evaluation results

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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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    Model evaluation results

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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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    Model evaluation results

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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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    Model evaluation results

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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.

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    Outline

    1 Objectives

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    2 Introduction

    3 Background

    4 Simulation

    5 Model Evaluation

    6 Discussion

    7 Glosary

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    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

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    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

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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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    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

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    Discussion

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

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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

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 48/53

    Discussion

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

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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

    Rodrigo Rojas Moraleda Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking 48/53

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    Questions ?

    Rodrigo Rojas [email protected]

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    Outline

    1 Objectives

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    2 Introduction

    3 Background

    4 Simulation

    5 Model Evaluation

    6 Discussion

    7 Glosary

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    Glosary

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

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    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

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