biological andsoft systems

bss

Department of PhysicsCavendish Laboratory

Biological Physics

4. DiffusionDiffusion in external potentials. First passage times. Kramer’s theory. Enzyme kinetics.

1

!

j = "D#c

#x

!

"c

"t= #

"j

"x

!

"c

"t= D

" 2c

"x2

j

Diffusion

Brownian motion leads to diffusion

Fick’s law

continuity equation

diffusion equation

!

D" = kT

Einstein relation

By including additional fluxes we derived:

!

j x( ) = "Ddc

dx+F

#c

2

Puzzle: Why do cells not burst ?

The cytoplasm has a very different composition from the extra-cellular environment

Macromolecules (many of which are negatively charged) attract counter-ions

But if ionic concentration

inside cell is high, water

will flow in by osmosis,

swelling the cell

3

Ion flow & Nernst relation

If concentration of e.g. KCl is higher inside than outside, there will be

a potential difference across a membrane only permeable for K+

IN OUT

!

j x( ) =qE x( )c x( )

"#D

dc

dx

Modified Fick’s law

!

j x( ) = D "dc

dx+qEc

kBT

#

$ %

&

' (

Nernst-Planck formula

!

1

c

dc

dx=

q

kBTE

In equilibrium

!

lncin

cout

"

# $

%

& ' =

(q)Vequ

kBT

Nernst relation

Membrane potential

!

"V =Vin#V

out

4

Donnan potential

Cell contains 3 types of ions in significant concentrations: Na+, K+, Cl-

If cell is at equilibrium:

Donnan potential

but

cell would burst

5

Ion pumps

Solution: actively pump cations out of cell

• reduces and

• makes more negative

eg. neuron

Ion pumps can transduce chemical energy to electrostatic energy

10 0.1 20 10

clearly not at equilibrium

“Sodium anomaly”: Na+ concentration is higher outside than inside

6

!

t0

=1

j x0( )

Diffusion in a force field: Fokker-Planck equation

Fokker-Planck

equation

flux of probability density

F-P-equation can be used to estimate the mean first passage time

• time for particle to diffuse a certain distance

• important for all diffusion limited processes

7

First-passage time w/o external force

No external force:

!

p x( ) = "2x

x0

2+2

x0

!

"p x, t( )"t

= D" 2p x, t( )"x 2

Solution:

!

j x0( ) = "D

dp x( )dx

=2D

x0

2

!

t0

=x0

2

2D

8

First-passage time w/ external force

Transport processes in the cell often take place by biased Brownian motion

Ion channel Protein translocation

Kinetics may be determined by considering diffusion in a potential landscape

9

!

t1

=1

j x1( )

=1

j1

First-passage time in external potential

!

j1

= "Ddp x( )dx

+F x( )#

p x( )

Integrate from x to x1...

Can be rewritten as...

Absorption at boundary:

10

multiply by and integrate from 0 to x1....

First-passage time in external potential

11

First passage time:

Example: diffusion with constant force

!

t1 =1

D

kT

F

"

# $

%

& ' 2

exp (Fx1

kT

"

# $

%

& ' (1+

Fx1

kT

)

* +

,

- .

Time for a 100 kDa protein to diffuse 8 nm in !s:

!

t1"kT

D

x1

F

!

t1 "1

D

kT

F

#

$ %

&

' (

2

exp )Fx1

kT

#

$ %

&

' (

12

Kramers’ problem: escape over a potential barrier

Time that it takes to escape is

called Kramer’s time tK

(Kramers, 1940)

13

!

U x( ) "U x1( )#

1

2x # x

1( )2

$ $ U x1( )

!

t1 "#

D

1

$ $ U x1( ) $ $ U 0( )exp

U1

kT

%

& '

(

) * "

LoL1

Dexp

U1

kT

%

& '

(

) *

Landscapes and intermediate states

!

U x( ) "1

2# # U 0( )x

2

Maximum

Minimum

L0

L1Not only the barrier height but also the energy landscape influences escape rates

Intermediate states are important

14

Kramers’ time in molecular biology and biological physics

• linear molecular motors (kinesin, myosin) can be modelled as Brownian ratchets, diffusing on an energy landscape modified by ATP-hydrolysis

• protein folding is often modelled by diffusion on a 1D energy landscape

• single-molecule force spectroscopy on biomolecules using AFM, optical tweezers

• chemical reactions and enzyme activity (kinetic-proofreading...)

15

!

"G!

k"1#exp "$G*

+ $G

kT

%

& '

(

) *

!

N2,eq

N1,eq=k"1

k1= exp "

#G

kT

$

% &

'

( )

Brownian motion and chemical reactions

A chemical reaction is a random walk on a free energy landscape

Reaction rate:

!

k1 =1

t1

"exp #$G*

kT

%

& '

(

) *

!

"G*

k1

k1 k

–1

Equilibrium situation:k1

k-1

S1S2

S1

S2

!

"G*

(detailed balance)

16

!

˙ N 1"

dN1

dt= k

1N

2t( ) # k#1

N1

t( )

˙ N 2"

dN2

dt= #k

1N

2t( ) + k#1

N1

t( )

!

N1

+ N2

= Ntot

!

˙ N 1

= k1

Ntot" N

1( ) " k"1N

1

!

N1 t( ) " N1,eq = N1 0( ) " N1,eq( )exp " k1 + k"1( )t[ ]!

˙ N 1

= 0

!

1 " = k1 + k#1 = Cexp #$G*

kT

%

& '

(

) * 1+ exp #

$G

kT

%

& '

(

) *

%

& '

(

) *

!

N1,eq = k

1Ntot k

1+ k"1( )

!

"

Reaching equilibrium

Suppose we begin with nonequilibrium populations N1(t) and N2(t)

!

"G

k1 k

–1

S1

S2

!

"G*

k1

k-1

S1S2

In equilibrium:

Relaxationto equilibrium:

Time to equilibrium depends on energy barrier!

17

Enzymes speed up reactions

Biochemical reactions in the cell often never reach equilibrium

eg. H2O2 H2O + 1/2 O2

but only 1% degraded in 3 days

S P

k1

k-1

*

S

P

S P

E EES

EP

Enzyme catalase speed reaction by 1012

Enzymes reduce the energy barrier between substrate and product …

… thus they increase the rate of both forward and back reactions

Enzymes are not consumed in the reaction!

ESEP

E+P

E+S

*

!

"G = #41kBT

18

Enzyme kinetics

Assume E.S E + P is • fast

• irreversible

E + S E.S E + P

quasi-steady state

reaction velocity

k1

k-1

k2

!

d

dtPE

= "k1cSPE

+ k"1 + k2( )PES

!

PE

=1" PES

19

Michaelis-Menten relation

saturating velocity

Michaelis constant

20