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Page 1: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Spontaneous movement of particles due to thermal agitation

Lecture 4: Diffusion: The Macroscopic and Microscopic Theories

R. Ranganathan Green Center for Systems Biology, ND11.120E

Robert Brown 1827

Adolf Fick 1855

Albert Einstein 1905

Page 2: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Linear systems at the thermodynamic limit….

Linear

Nonlinear

n = 1

n = 2 or 3

n >> 1

continuum

exponential growth and decay

single step conformational change

fluorescence emission

pseudo first order kinetics

fixed points

bifurcations, multi stability

irreversible hysteresis

overdamped oscillators

second order reaction kinetics

linear harmonic oscillators

simple feedback control

sequences of conformational change

anharmomic oscillators

relaxation oscillations

predator-prey models

van der Pol systems

Chaotic systems

electrical circuits

molecular dynamics

systems of coupled harmonic oscillators

equilibrium thermodynamics

diffraction, Fourier transforms

systems of non-linear oscillators

non-equilibrium thermodynamics

protein structure/function

neural networks

the cell

ecosystems

Diffusion

Wave propagation

quantum mechanics

viscoelastic systems

Nonlinear wave propagation

Reaction-diffusion in dissipative systems

Turbulent/chaotic flows

adapted from S. Strogatz

Page 3: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

First, the macroscopic view...

The observations:

Page 4: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

First, the macroscopic view...

The observations:

Page 5: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

First, the macroscopic view...

The physical model:

Page 6: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

First, the macroscopic view...

The physical model:

Page 7: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

But the, along came Einstein in 1905....

The physical model:

Page 8: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

But the, along came Einstein in 1905....

The physical model:

How does this explain the phenomenological properties of diffusion?

Page 9: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Does the (unbiased) random walk account for all these properties? Let’s look in 1-D....

What are the consequences?

Page 10: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

First what is the position of each particle after steps of the walk? Well....

i n

1. The average displacement of particles....

Each step takes seconds, distance

moved is

τδ

A “stochastic iterative map”....we will come back to this.

Page 11: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

First what is the position of each particle after steps of the walk? Well....

i n

1. The average displacement of particles....

Each step takes seconds, distance

moved is

τδ

Thus, the particles go nowhere on average

Page 12: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

First what is the squared position of each particle after steps? Well....

i n

2. How much do the particles spread out over time?

〈xi2 (n)〉

Each step takes seconds, distance moved is

τδ

Page 13: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

First what is the squared position of each particle after steps? Well....

Now, let’s take the average...

i n

2. How much do the particles spread out over time?

〈xi2 (n)〉

Each step takes seconds, distance moved is

τδ

Page 14: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

How much do particles spread out over time?

〈xi2 (n)〉

Each step takes seconds, distance moved is

τδ

Page 15: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

How much do particles spread out over time?

〈xi2 (n)〉

We can simplify....

Each step takes seconds, distance moved is

τδ

Page 16: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

How much do particles spread out over time?

〈xi2 (n)〉

We need to change n into time....

Each step takes seconds, distance moved is

τδ

But...we want the RMS displacement, so....

Page 17: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

We want the RMS displacement:

How much do particles spread out over time?

〈xi2 (n)〉

We need to change n into time....

Each step takes seconds, distance moved is

τδ

Thus, the particles spread out as the square root of time...

Page 18: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

3. What about the shape of the distribution of particles?

Think about coin tossing....

Page 19: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

What about the shape of the distribution of particles?

This is the binomial density function again,….

Page 20: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

What about the shape of the distribution of particles?

Page 21: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

What about the shape of the distribution of particles?

But if the number of trials is very large and p is not too small.....

Page 22: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

H.C. Berg. “Random Walks in Biology”, (1993) Princeton Press

What about the shape of the distribution of particles?

But if the number of trials is very large and p is not too small.....the binomial distribution approaches the Gaussian distribution. The bell shaped curve!

Page 23: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

So the random walk does indeed account for the motion of particles...

A seminal example of how simple physical theory (the random walk) can explain the rather complex behavior of particles moving under thermal agitation...

Page 24: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

So the random walk does indeed account for the motion of particles...

But, what happened to good old Fick’s Law, which does indeed also account for the properties of diffusion? Well, it works and it still works with this new understanding....

Page 25: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

The relationship of the random walk (the microscopic view) to Fick’s first law (the macroscopic view).

Now, how do we write the flux of particles going from to ? x +δ x

x

Page 26: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

The relationship of the random walk (the microscopic view) to Fick’s first law (the macroscopic view).

Page 27: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

The relationship of the random walk (the microscopic view) to Fick’s first law (the macroscopic view).

Page 28: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

The relationship of the random walk (the microscopic view) to Fick’s first law (the macroscopic view).

Page 29: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

The physical model:

So, Fick’s mapping of diffusion to Fourier’s or Ampere’s Laws of heat conduction and current flow is correct.

But what kind of force is a concentration gradient?

Page 30: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....the thermodynamic basis for diffusion.

To understand this, we begin with some definitions and some review of thermodynamics....

Page 31: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....the thermodynamic basis for diffusion.

Page 32: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....the thermodynamic basis for diffusion.

where the gradient operator is defined as....

Page 33: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....the thermodynamic basis for diffusion.

Page 34: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....some basic laws of thermodynamics.

free energy is a function of a number of so-called “natural variables”...

Page 35: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....some basic laws of thermodynamics.

and the derivative of free energy involves taking partial derivatives of the function G with respect to these natural variables...

so what are there partial derivatives? They have key physical interpretations...

Page 36: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Now....some basic laws of thermodynamics.

And so we get the basic definition of infinitesimal changes in Gibbs free energy....the basic equation of equilibrium thermodynamics.

Page 37: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 38: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 39: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 40: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 41: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 42: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

....and one more step get’s us to back to Fick’s law....

Page 43: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

One important point here....

Page 44: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 45: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Ok, with this, let’s go back to our problem of diffusion....

Page 46: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 47: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

To understand this, we return to our 1D diffusion problem....

Fick’s Second Law....The Diffusion Equation

Page 48: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 49: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 50: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Taking the limits as both tau and delta approach zero...

Fick’s Second Law....The Diffusion Equation

Page 51: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 52: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 53: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 54: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 55: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 56: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 57: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 58: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Fick’s Second Law....The Diffusion Equation

Page 59: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

One can solve higher dimensional versions of the diffusion equation...in general many complex phenomena can be explained by solutions to this equation.

Fick’s Second Law....The Diffusion Equation

Page 60: Lecture 4: Diffusion: The Macroscopic and Microscopic Theoriesranganathanlab.org/.../2018/05/QB2017_lecture4_Diffusion.pdfdiffraction, Fourier transforms systems of non-linear oscillators

Next time…the theory of diffraction

Linear

Nonlinear

n = 1

n = 2 or 3

n >> 1

continuum

exponential growth and decay

single step conformational change

fluorescence emission

pseudo first order kinetics

fixed points

bifurcations, multi stability

irreversible hysteresis

overdamped oscillators

second order reaction kinetics

linear harmonic oscillators

simple feedback control

sequences of conformational change

anharmomic oscillators

relaxation oscillations

predator-prey models

van der Pol systems

Chaotic systems

electrical circuits

molecular dynamics

systems of coupled harmonic oscillators

equilibrium thermodynamics

diffraction, Fourier transforms

systems of non-linear oscillators

non-equilibrium thermodynamics

protein structure/function

neural networks

the cell

ecosystems

Diffusion

Wave propagation

quantum mechanics

viscoelastic systems

Nonlinear wave propagation

Reaction-diffusion in dissipative systems

Turbulent/chaotic flows

adapted from S. Strogatz


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