Diffusion Part 1

Diffusion equationrandom walk derivation

Random walk reconsidered (PBoC 13.2.1)

Imagine you have calculated the probability p(x,t) that a random walker finds himself at x

after a time t.

p(x,t+!t) depends on p(x,t) in the immediate vicinity of x (±!x, to be exact):

Our previous random walk notation had k!t=l=r and !x=h

k!t=1/2 is the classic random walk; k!t < 1/2 is a random walk with drunken pauses

Diffusion equation

k!t

1-2k!t

k!t

k!t

x x+!xx-!x

x x+!xx-!x

t

t+!t

k!t

k!t

1-2k!t

p(x, t + !t) = k!t · p(x!!x) + (1! 2k!t) · p(x, t) + k!t · p(x + !x)

Diffusion equationTo first order in !t this becomes

which becomes (to second order in !x)

This is the diffusion equation with diffusion coefficient

To connect this to our previous expressions for the unbiased 1D random walk, use k!t=1/2

to identify k ! (2!t)-1, x ! R, !x ! h, and t/!t ! N giving Dt=Nh2/2.

We will see below that the free-space solution to the diffusion equation iswhich is identical to our 1D random walk expressions with the substitutions above.

Not surprisingly, the “random walk with drunken pauses” corresponds to Dt < Nh2/2: slower diffusion.

p(x, t + !t) = p(x, t) + k!t · {p(x + !x) +!2p(x, t) + p(x!!x)}

!p

!t= k!x2 !2p

!x2

!p

!t= D

!2p

!x2 D ! k!x2

p(x, t) = 1!4!Dt

e"x2/4Dt

Diffusion equationkinetic theory derivation

Diffusion equationKinetic theory of gases

Fick’s law

The net particle flux is:

Jx = !Ddc

dx

!J = !D!"c

J ! #area time

D ! length2

time

1D

3D

J–(x)

J+(x)

J–(x+dx)

J+(x+dx)

area Adx

Diffusion equationEinstein showed that:

where ! is the drag coefficient defined by F = ! v.

For a sphere of radius a in a fluid with viscosity ",

For a typical small ion,

Since , we expect

Since the drag of a nonspherical object at low Reynolds number is dominated by its longest dimension, thin proteins diffuse less than their mass suggests.

D =kBT

!

! = 6"#a

D ! 1 µm2/ms

a ! 3"

m D ! m!1/3

globular proteins

needlelike proteins

Diffusion equationConservation of mass (or number)

leads to the continuity equation:

!c

!t= !!Jx

!x

!c

!t= !"" · "J

1D

3D

J(x!dx/2) J(x+dx/2)

area Adx

n(x) = c(x) A dx

Diffusion equationTogether, Fick’s law + conservation law lead to the diffusion equation:

This is identical to (and often called) heat conduction

This is a second order PDE: it requires an initial condition and two boundary conditions to be solved.

This form assumes that D is constant in space.

We could add source or sink term to right hand side if appropriate.

This is identical to the equation for p(x,t) from the random walk, except

here we talk about the c(x,t).

Since c(x,t) = N p(x,t) (where N is the total number of particles in the system), I will use p and c interchangeably.

!c

!t= !"" ·

!!D""c

"= D"2c

!c

!t= D

!2c

!x23D: 1D:

Free-space solutions

Solutions to diffusion equation in free space (with c=0 at infinity) look like spreading Gaussians

or

You can derive this using Fourier transforms, but it’s easier to guess it and plug back into PDE to check its validity

Mass is conserved:

Diffusion profiles always smear out: they never sharpen.

Free-space diffusion

c(x, t) =N!4!Dt

e!x2/4Dt

! !

"!c(x, t)dx = N

c(x, t) =N!

2!"2(t)e!x2/2!2(t)

!2(t) ! 2Dt

The 1D solution looks like

Free-space diffusion

!10 !5 5 10x

0.2

0.4

0.6

0.8

1.0

c

snapshots in time animation

Free-space diffusionThe “edge” (position of the half-height, for instance) propagates like

details can change, but any diffusive process ends up with

So the speed of propagation is constantly decreasing!

Diffusion is a reasonable transport process only over short times and distances.

This is not the typical speed of a diffusing molecule, which is similar to the speed of sound (100 m/s or so)

Typical values for D range from 1 "m2/s to 1 "m2/ms:

d"/dt will slow to a few "m/s within a millisecond or second

!(t) !"

2Dtx !

"t

Free-space diffusionEven if your initial concentration profile is weird, if it has a finite width you always end up with a spreading Gaussian:

= +

Free-space diffusionWe can solve diffusion equation in spherical coordinates:

by noting that it’s a superposition of 3 individual 1D diffusions:

Similarly for cylindrical coordinates, where

c(x)c(y)c(z) dx dy dz = N4!r2

(4!Dt)3/2e!r2/2Dt dr ! c(r) dr

!c

!t= D

1r

!

!r

!r2 !c

!r

"

c(x)c(y) dx dy = N2!"

(4!Dt)2/2e!!2/2Dt d" ! c(") d"

Free-space diffusionIn summary, we have

!r2

"=

# !

0r2c(r) dr = 6Dt

!x2

"=

# +!

"!x2c(x) dx = 2Dt

!!2

"=

# !

0!2c(!) d! = 4Dt

1D

2D

3D

c(x, t) = N1

(4!Dt)1/2e!x2/4Dt

c(!, t) = N2"!

(4"Dt)1e!!2/2Dt

c(r, t) = N4!r2

(4!Dt)3/2e!r2/4Dt

Diffusion = random walkThe end-to-end distance of a random coil is the same as the net displacement in a random walk is the same as the diffusion of concentrated burst.

Always find <x2> ! t if correlation between links is “short”:

for instance, finite length m (including m=0) or exponentially decaying

Long-distance correlations (for instance, power law) lead to unusual superdiffusive behavior:

defined as <x2> ! tn with n>1

often called Lévy flights

not to be confused with small t ballistic regime

has been demonstrated in

membranes (active transport?)

cell interiors (active transport)

ecology (foraging / search algorithms of albatrosses, bumblebees and deer.)

(Nature 449:1044 or Ecology 88:1962)

great stock market crash of 2008

(The Economist January 24th 2009)

Advection-diffusion

Advection-diffusionWhat if your random walk is not unbiased?

Go back to the earlier derivation, but make rightward and leftward step rates (k+ and k-) unequal:

The expression for p(x,t+!t) is now

which eventually becomes the Smoluchowski equation (v1) PBoC eqn 13.54:

where we identify D and v with the microscopic rates: D = (k++k–) !x2/2 and v = (k+–k–) !x.

k -!t

1-(k-+k+)!t

k -!t

k+ !t

x x+!xx-!x

x x+!xx-!x

t

t+!t

k+ !t

k -!t

1-(k-+k+)!t

p(x, t+!t) = k+!t ·p(x!!x)+(1!(k++k!)!t) ·p(x, t)+k!!t ·p(x+!x)

!c

!t= D

!2c

!x2! v

!c

!x

Advection-diffusionWhat if you have bulk flow as well as diffusion?

Moving fluid will transport its local concentration c(x). This is called advection (or sometimes, incorrectly, convection).

This contributes an advective flux

We add this to our original diffusive flux and put the total flux into the continuity equation to give

In 1D with constant D and v, this is the same as Smoluchowski v1 from the previous slide.

In 1D with constant D but not assuming v is constant, this becomes Smoluchowski v2:

This PDE describes the evolution of a concentration profile due to diffusion and advection simultaneously.

!Jadv = !vc!J = !D!"c

!c

!t= !"" ·

!!D""c + "vc

"

!c

!t= D

!2c

!x2! !(vc)

!x

!c

!t= ! !

!x

!!D

!

!c+ vc

"1D:3D:

Advection-diffusionFor constant v, the solution is simple: a diffusing Gaussian profile that drifts at speed v.

Advection-diffusionMicroscopic Smoluchowski equation

The movement v could be caused by an applied force F rather than bulk fluid flow.

Using the definition of !, (F= !v), the Einstein relation (D=kBT/!) and the definition of force (F = - #G/#x), and writing p for c, we get Smoluchowski v3

Now we apply this to a generalized energy landscape G(x). It describes how the population

(or probability) p(x,t) flows in time and space.

The only unknown is the microscopic diffusion parameter D. We have to estimate it, but since everything is

proportional to D as long as it’s constant it will change the time scale but not the shape of p. To see this,

Nondimensionalize the Smoluchowski equation:

x*=x/l, G* = G/G0, t* = t D/l2, T*=kBT/G0, giving

!p

!t= D

!!2p

!x2+

1kBT

!

!x

"!G

!xp

#$

!p

!t!=

!!2p

!x!2+

1T !

!

!x!

"!G!

!x!p

#$

We drop the *’s to give Smoluchowki v4 (the last one):

Everything in this equation is of order 1. This is easier to visualize and easer to solve numerically.

There is only one parameter that matters, the dimensionless temperature kBT/G0.

When you see a complicated free energy surface G(x), imagine the protein diffusing around (generally toward the minimum of G) according to the Smoluchowski equation.

news and views

Extensive research in protein folding andunfolding have uncovered rate constantsfor some proteins far below one second,implying the existence of definedroadmaps for the folding process12,16. Theexistence of fast folding events has impor-tant consequences for the understandingof the initiation of protein folding. Itseems that the sequence and the topologyof the native state determine how a pro-tein folds and that model simulations and

structure predictions may therefore suc-ceed in finding the correct fold of anunknown sequence18. This view assumesthat local amino acid sequences initiatefolding in subdomain structures, whichthen lead to the final fold in a stepwisemanner.

The major driving force in proteinfolding is considered to be the hydropho-bic effect, which causes the formation ofconformations stabilized by packing the

8 nature structural biology • volume 7 number 1 • january 2000

side chains of hydrophobic amino acidsinto the interior of the protein. The fold-ing process of a protein generally dependson the strength of this hydrophobic effectas well as on the stability of the proteinand can be described by the energy land-scape theory16,19. The topology of thefolding energy landscape can best bedescribed as a funnel with a rough surface(Fig. 1). The intermediate states populatethe local minima within the rough surfaceand the inner funnel wall, sometimesentrapping misfolded conformations.The roughness of the funnel surface, andthus the formation of the native state canvary significantly between different typesof proteins. Pure !-helical proteinsalmost simultaneously undergo an enor-mous reduction in conformational spaceand hydrodynamic radius, form local seg-ments of structure, and immediately gainsome tertiary interactions9. !/" proteinsinitially collapse their hydrophobiccores20 before assembling into orderedstructures where the formation of "-sheets seems to be the rate-limitingstep. Some folding pathways thereforeinclude the formation of transient inter-mediates or more stabilized states such asthe molten globule21. !-lactalbumin, an!/" protein, is able to form such a stable

Fig. 2. Stereo views of the structure of !-lactalbumin (1HFZ). a, The basic structure of the protein with the !-helices in red, the "-sheet in dark yellow,the four disulfide bridges in light yellow and the calcium in light blue. The "-domain is located in the upper rear section of the protein, whereas thecalcium binding site is positioned at the front. The !-domain is in the lower half of the protein structure. b, The same view as in (a) but the sidechains of the hydrophobic amino acids Val, Leu, and Ile are indicated in green spheres. The densely packed hydrophobic core of !-lactalbumin isclearly visible. c, The same view as in (a) but with Arg, Lys, and His indicated in blue spheres, and d, with Asp and Glu indicated in red spheres. Thelocations of the different polar amino acids on the surface of the protein are clearly visible.

a b

c d

Fig. 1. Schematic of thefolding energy landscape ofa protein molecule wherethe energy of the protein isdisplayed as a function ofthe topological arrange-ments of the atoms. Themultiple states of theunfolded protein located atthe top fall into a foldingfunnel consisting of analmost infinite number oflocal minima, each of whichdescribes possible foldingarrangements in the pro-tein. Most of these statesrepresent transient foldingintermediates in the processof attaining the correctnative fold. Some of theseintermediates retain a morestable structure such as the molten globule, whereas other local minima act as folding traps irre-versibly capturing the protein in a misfolded state (see ref. 16 for details).

© 2000 Nature America Inc. • http://structbio.nature.com

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Free

Advection-diffusion

!p

!t=

!!2p

!x2+

1T

!

!x

"!G

!xp

#$

The Smoluchowski equation

The Smoluchowski equationThe steady-state solution to the Smoluchowski equation is the Boltzmann distribution.

Smoluchowski describes the process of reaching thermal equilibrium.

In general, the equation is hard to solve.

I’ve only seen analytical solutions for

constant G(x) (the diffusion equation)

linear G(x) (the advection-diffusion equation)

quadratic G(x) (no name: thermalization in a springlike potential)

The Smoluchowski equationDiffusion over a barrier shows thermalization within the barrier (for T*<1) followed by barrier crossing. At lower T it takes longer to cross the barrier.

T*=1/2 T*=1/8

The Smoluchowski equationWhen the landscape is asymmetrical, the system ends up predominantly in the lower-energy state. At lower T, this is more pronounced and takes longer.

T*=1/4

T*=1/16

The Smoluchowski equationMore complicated landscapes lead to more complicated behaviors, including intermediate states. These appear as kinetic intermediates even if they aren’t present in the long-time, thermalized, Boltzmann distribution limit.

T*=1/8; different biases to the landscape

The Smoluchowski equationWhen T* is small (meaning either large energies G0 or small temperatures kBT), the distribution tends to “linger” in local minima. There is little probability of being outside a local minimum, so the local minima can plausibly be considered “the states” of this landscape, with the deepest minimum (the global minimum) the “ground state”.

T*=2 T*=1/8

The Smoluchowski equationA “quench” simulation shows the idea behind the conformational substate model: a single state at high T resolves into three distinct states at low T.

The Smoluchowski equationThe rate of transition from state A to B depends on the barrier between them and on the temperature. This can be calculated analytically for the Smoluchowski equation if T* is small. Kramer’s theory ultimately gives an expression for the transition from A to B:

# depends on details of the model, but in its simplest form is

where kA is the curvature of the free energy surface at the bottom of the well and k‡ is the curvature at the top.

For our purposes the details don’t matter: just that transition rates are exponential in !G‡ = G‡-GA, referenced to temperature.

This is simply the Arrhenius relation. Different treatments give different prefactors #, but all have the activation energy dependence.

kA!B = !e"!G‡/kBT

! =!

kAk‡/2"#

Exponential kinetics

Exponential kineticsSequential transitions

If there is a mandatory path from A to B to ... to the native state and the barriers between states are large, then the folding process can be represented by

If the rates are similar, you will see intermediate states accumulate. If not, the rate to N will be dominated (at long times) by the slowest of all the individual rates.

A ! B ! C ! ... ! NkA!B kB!C

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kA I

<< kI B

time

num

ber

A

intermediate

B

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1equal rates

time

num

ber

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kA I

>> kI B

time

num

ber

Exponential kineticsFor a finite (small) number of states, the kinetics of the final state will reflect all the individual rates.

If there are only a few states and rate constants, you may be able to identify different rates and infer how many intermediates there are.

If there are many intermediates, and especially if there are many intermediates with closely spaced rates (i.e. energy barriers), you may not be able to identify individual rates and the long-time behavior may not look exponential.

This type of non-exponential kinetics is typical of an energy landscape that is of intermediate roughness:

not so rough that the protein is trapped in free minima

rough enough that many minima contribute and no single barrier definitively controls the folding process.

news and views

Extensive research in protein folding andunfolding have uncovered rate constantsfor some proteins far below one second,implying the existence of definedroadmaps for the folding process12,16. Theexistence of fast folding events has impor-tant consequences for the understandingof the initiation of protein folding. Itseems that the sequence and the topologyof the native state determine how a pro-tein folds and that model simulations and

structure predictions may therefore suc-ceed in finding the correct fold of anunknown sequence18. This view assumesthat local amino acid sequences initiatefolding in subdomain structures, whichthen lead to the final fold in a stepwisemanner.

The major driving force in proteinfolding is considered to be the hydropho-bic effect, which causes the formation ofconformations stabilized by packing the

8 nature structural biology • volume 7 number 1 • january 2000

side chains of hydrophobic amino acidsinto the interior of the protein. The fold-ing process of a protein generally dependson the strength of this hydrophobic effectas well as on the stability of the proteinand can be described by the energy land-scape theory16,19. The topology of thefolding energy landscape can best bedescribed as a funnel with a rough surface(Fig. 1). The intermediate states populatethe local minima within the rough surfaceand the inner funnel wall, sometimesentrapping misfolded conformations.The roughness of the funnel surface, andthus the formation of the native state canvary significantly between different typesof proteins. Pure !-helical proteinsalmost simultaneously undergo an enor-mous reduction in conformational spaceand hydrodynamic radius, form local seg-ments of structure, and immediately gainsome tertiary interactions9. !/" proteinsinitially collapse their hydrophobiccores20 before assembling into orderedstructures where the formation of "-sheets seems to be the rate-limitingstep. Some folding pathways thereforeinclude the formation of transient inter-mediates or more stabilized states such asthe molten globule21. !-lactalbumin, an!/" protein, is able to form such a stable

Fig. 2. Stereo views of the structure of !-lactalbumin (1HFZ). a, The basic structure of the protein with the !-helices in red, the "-sheet in dark yellow,the four disulfide bridges in light yellow and the calcium in light blue. The "-domain is located in the upper rear section of the protein, whereas thecalcium binding site is positioned at the front. The !-domain is in the lower half of the protein structure. b, The same view as in (a) but the sidechains of the hydrophobic amino acids Val, Leu, and Ile are indicated in green spheres. The densely packed hydrophobic core of !-lactalbumin isclearly visible. c, The same view as in (a) but with Arg, Lys, and His indicated in blue spheres, and d, with Asp and Glu indicated in red spheres. Thelocations of the different polar amino acids on the surface of the protein are clearly visible.

a b

c d

Fig. 1. Schematic of thefolding energy landscape ofa protein molecule wherethe energy of the protein isdisplayed as a function ofthe topological arrange-ments of the atoms. Themultiple states of theunfolded protein located atthe top fall into a foldingfunnel consisting of analmost infinite number oflocal minima, each of whichdescribes possible foldingarrangements in the pro-tein. Most of these statesrepresent transient foldingintermediates in the processof attaining the correctnative fold. Some of theseintermediates retain a morestable structure such as the molten globule, whereas other local minima act as folding traps irre-versibly capturing the protein in a misfolded state (see ref. 16 for details).

© 2000 Nature America Inc. • http://structbio.nature.com©

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mStructure / folding summary

Schultz, “Illuminating folding intermediates”, Nature Structural

Biology 7:7-10 (2000)

secondary structuremolten globule

tertiary structurenative conformation

Free

primary structurerandom coil

hydrophobic collapse

kineticintermediate

rate-limitingbarrier kinetically trapped

misfold

Structure / folding summary

Folding on a free energy landscapefrom the Dill group (UCSF)