Diffusion Confusion
Mark Anderson
Fran Butek
Andy Dettinger
Dan Hecker
Mark Osegard
Overview
•Diffusion
•A Biological Ohm’s Law
•Facilitated Diffusion
•Muscle Respiration
Mechanisms of
Cellular Homeostasis
ho·me·o·sta·sis:
The ability or tendency of an organism or
cell to maintain internal equilibrium by
adjusting its physiological processes.
Cellular Homeostasis
Cell Membrane
The Cell Membrane
Acts as a boundary that separates the internal
working of the cell from its external environment.
It allows free passage of some material and
restricts the passage other. In this way the cell
membrane regulates the passage of materials
into and out of the cell.
The Cell Membrane (cont.)
It consists of a double layer of
phospholipid molecules.
Examples of phospholipids include
lecithin, cephalins, phosphoinositides
(in brain), and cardiolipin (in heart).
The membrane contains water-filled pores and
protein-lined pores, called channels. These channels
allow passage of specific molecules.
The Cell Membrane (cont.)
The membrane’s intracellular and extracellular
environment are made up primarily of NaCl and KCl
The cell membrane works as a barrier to the free flow of
these ions. The membrane also maintains concentration
differences between ions, and acts as a barrier to water flow.
Cellular Transportation
Two Types of Transportation
Molecules can be transported across the cell membrane
by passive or active processes.
Active process requires the expenditure of energy.
Cellular Transportation
(cont.)
The passive process results from the, random movement of the
molecules.
There are three passive transport mechanisms
1) Osmosis - is the process by which water is
transported through the cell membrane
2) Simple diffusion - accounts for the passage of
small molecules through pores and of lipid-soluble
molecules through the bilipid layer.
3) Carrier-mediated diffusion - occurs when a
molecule is bound to a carrier molecule that
moves readily through the membrane.
Making the Connection
The cell membrane is a
medium where diffusion
takes place.
Our study this semester of
diffusion, and what will be
shown in the following
slides, is the diffusion
where the membrane is a
group of cells, such as
muscle tissue.
Hammering out
the Diffusion Equation
Compartmental Analysis
Diffusion through a Region
Divergence Theorem
The Diffusion Equation
Fick’s Law
Refined Diffusion Equation
Compartmental Analysis
X in
X
X out
X is some variable of interest
A change in X = X in – X out
in derivative terms:
dx
= xin (t ) − xout (t )
dt
What is Diffusion?
Definition – the process by which a
substance disperses within an ambient
medium over time
Modeled using compartmental analysis
Net change of substance at a point =
(inflow rate of substance) – (outflow rate of the substance)
Diffusion through a Region
The region may contain
sources and sinks.
( x, y , z ) ∈ Ω
Modeling Diffusion
(the nuts and bolts)
u = u(x, y, z, t) = density at a point in
the region
f = f(x, y, z, t) = production at a point
(a density) in the region
due to sources and
sinks
J = J(x, y, z, t) = flux density at a point
Some Calculus
(u )dV = total mass in region
Ω
d
dt
( u ) dV
= rate of mass change in region
Ω
( f ) dV = total production of mass in
Ω
region due to sources and sinks
The Confusing Quantity
of Flux
J = J(x, y, z, t) = flux density at a point
(density x velocity)
The units of flux are:
mass
area * time
small flux in the
negative x direction
large flux in the
positive x direction
What can we do with Flux?
Mass leaving the region through the boundary will
be the component of the flux that is in the direction
of the surface normal
J ⋅ n = total flux out of the region in
the direction of at a point on the n
boundary
( J ⋅ n )dA
∂Ω
= total flux (accumulation
of mass) out of the region
through the boundary
J ( x, y , z , t )
n
Putting Together the Pieces
Using compartmental analysis we can write an
equation relating all the components we have
developed thus far.
rate of substance change = amount produced in
the region – amount that escapes the region
d
(u )dV = ( f )dV − ( J ⋅ n )dA
dt Ω
Ω
∂Ω
Divergence Theorem
(Gauss’s Theorem)
a Method of modifying the flux integral
gradient
∂ ∂ ∂
∇=( , , )
∂x ∂y ∂z
The amount of mass that overflows the
boundary must be equal to the total change in
mass within the boundary.
The Bathtub Overflow
Theorem
( J ⋅ n)dA = (∇ ⋅ J )dV
∂Ω
Ω
Johann Carl Friedrich Gauss
Born: 30 April 1777
Brunswick, Germany
Died: 23 Feb 1855
Göttingen, Germany
Simplification
d
(u ) dV = ( f ) dV − ( J ⋅ n ) dA
dt Ω
Ω
∂Ω
Divergence
Theorem
∂u
( ) dV = ( f ) dV − (∇ ⋅ J ) dV
∂t
Ω
Ω
Ω
∂u
( ) dV = ( f − ∇ ⋅ J ) dV
∂t
Ω
Ω
Almost there…
∂u
( ) dV − ( f − ∇ ⋅ J ) dV = 0
∂t
Ω
Ω
∂u
( − f + ∇ ⋅ J ) dV = 0
∂t
Ω
If you integrate over all
subregions of “OMEGA”
and
get
zero,
the
expression integrated must
also be equal to zero
∂u
− f +∇⋅J = 0
∂t
The Unrefined
Diffusion Equation
∂u
= f − ∇⋅ J
∂t
density change at a point over time =
density production at that point per
unit time – the divergence (the rate of
flow from that point)
Gradient
Fick’s Law
∂ ∂ ∂
∇=( , , )
∂x ∂y ∂z
During diffusion we assume particles move in the direction of
least density. They move down the concentration gradient
In mathematical terms we will assume
J = − D∇u
Where D is a constant of proportionality
called the Diffusion Coefficient
Adolph Fick
Born: 1829
Cassel, Germany
Died: 1879
Ockham’s Razor
∂u
− f +∇⋅J = 0
∂t
When faced with a choice between two things,
choose the simpler.
If we assume D is constant and substitute the new
expression we found for flux using Fick’s Law we can
remove the flux density component of our diffusion equation
as follows:
∂u
= f − ∇⋅ (−D∇u)
∂t
∂u
= f + D∇⋅ ∇u
∂t
∂u
= f + D∇2u
∂t
William of Ockham
Born: 1288
Ockham, England
Died: 9 April 1348
Munich, Bavaria
The Laplacian
∂
∂
∂
∇ =( 2 + 2 + 2)
∂x ∂y ∂z
2
2
2
2
Pierre-Simon Laplace
Born: 23 March 1749
Normandy, France
Died: 5 March 1827
Paris, France
The Refined
Diffusion Equation
∂u
2
= f + D∇ u
∂t
A Biological Ohm’s Law
(a formulation of the relationship of voltage, current, and resistance)
Ohm’s law is defined most simply using three terms V, I, and R.
V=IxR
Where:
V = voltage measured in volts
I = current measure in amperes
R = resistance measure in Ohms
So what does Ohm’s law have
to do with Diffusion?
Using Fick’s Law we can derive a chemical counterpart of
Ohm’s Law that can be used to describe diffusion
through a membrane of thickness L.
large chemical
reservoir
cl
membrane
L
large chemical
reservoir
cr
As you can see from the model,
this membrane has a concentration of cl on the left
and a concentration of cr on the right
Applying the diffusion Equation
Let c(x,t) denote the concentration at x at time t in previous one
dimensional model.
If we assume no local production we can model the
concentration using the diffusion equation and write:
2
∂c
∂c
=D 2
∂t
∂x
Boundary conditions
c(0,t) = cl
c(L,t) = cr
Simplification by Assumption
In order for us to simplify our lives and the following equation,
we ignore time and assume it to be at steady-state, therefore the change in
concentration will fall off to zero:
∂c
=0
∂t
Using the steady state equation and our equation from the
previous slide we can write:
D
2
∂ c
∂x
2
=0
One
dimensional
Laplacian
A Linear Model
Dividing by D and integrating with respect to x we get
∂c
=a
∂x
where “a” is a constant.
Then we integrate with respect to x one more time
and see that c(x) is linear in steady state:
c ( x ) = ax + b
Boundary Conditions
c(0) = cl
and
c( L) = cr
b = cl
cr = aL + b
we define a and b to be:
cr − cl
a=
L
So recalling the linear formula we found previously
c ( x ) = ax + b
we plug in the a and b values defined above.
The resulting equation then becomes:
x
c( x) = cl + (cr − cl )
L
Differentiating
∂c cr − cl
=
∂x
L
differentiating
with respect to x
Now recalling Fick’s Law, with a one dimensional
gradient (aka a derivative), which Mark Anderson used earlier we write:
Note that in one
dimension J is
scalar, rather
than a vector as
before
Plugging in:
∂c
∂x
J = − D∇c
∂c
= −D
∂x
cl − cr
J =D
L
Made It!
We can rewrite as
D
J = (cl − cr )
L
Using the equation from my first slide, solved for I
We can see a correspondence between our
1
I= V
R
model for flux and Ohm’s Law.
The ratio L / D is similar to “resistance” of the membrane and
subsequently the ratio D / L is the conductance, or the permeability of
the membrane.
J = I(current)
D/L = 1/R(conductivity)
(cl-cr) = V(voltage drop)
A Biological Ohm’s Law
D
J = (cl − cr )
L
Ohm’s Law is the simplest diffusion scenario.
It is conducted in one dimension
when there are no reactions occurring.
Reaction-Diffusion Systems
Reactants in an enzymatic reaction are
free to diffuse, so that one must keep track
of the effects of both diffusion and
reaction.
An example in which both diffusion and
reaction play a role is known as facilitated
diffusion.
Facilitated Diffusion
Facilitated diffusion occurs when the flux
of a chemical is increased by a reaction
that takes place in the diffusing medium.
An example of facilitated diffusion occurs
with the flux of oxygen in muscle fibers.
Oxygen in Muscle Fibers
In muscle fibers, oxygen is bound to
myoglobin, a much larger molecule, and is
transported as oxymyoglobin. This
transport is greater than that of free
oxygen, and hence the flux of oxygen is
enhanced.
Whoa!
I know what you are thinking: “ain’t that
kind of weird since myoglobin has a much
larger molecular weight (16,890amu) than
oxygen (32amu), and therefore a much
smaller diffusion coefficient!?”
Doxygen=4.4*10-7cm2/s
Dmyglobin=1.2*10-5cm2/s
A Simple Model
To
help
us
understand
this
phenomenon, consider a slab reactor
(or reservoir) in which oxygen and
myoglobin react to form oxymyoglobin.
Myoglobin is not allowed to leave the
slab in a free or bound form, but oxygen
may flow in and out freely.
Slab Reactor
k+
→
←
k
O2 + Mb
large chemical
reservoir of pure O2
x=0
slab
0<x<L
−
MbO2
large chemical
reservoir of pure O2
x=L
Definitions
Let s = s(x,t) denote the concentration of
oxygen at a point x and time t
Let e=e(x,t) denote the concentration of
myoglobin at a point x and time t
Let c=c(x,t) denote the concentration of
oxymyoglobin at a point x and time t
Applying the Diffusion Equation
If we let f denote the rate of uptake of oxygen into
oxymyoglobin, we can use the diffusion equation to
model the concentrations
s = [O2], e = [Mb], c = [MbO2] as shown below:
Note that f will play the role of
either a source or a sink
depending its sign.
∂ s
∂s
− f
= Ds
2
∂t
∂x
2
∂ e
∂e
− f
= De
2
∂x
∂t
2
∂
c
∂
c
3 instances of
+ f
= Dc
the diffusion
2
∂t
∂x
equation
2
Assumptions
Myoglobin and oxymyoglobin are nearly
identical in molecular weight and structure,
therefore we assume: De ≈ Dc
We will assume that as we approach the boundaries of the
slab the changes in myglobin and oxymyoglobin taper off,
giving us the boundary conditions at x=0 and x=L to be :
∂e ∂c
=
=0
∂x ∂x
Compartmental Analysis
O2
Mb
k+ se
MbO2
k-c
k+
→
←
k
O2 + Mb
−
MbO2
Diagram
Equation
From the compartmental diagram we can write
the following equation
dc
= k + se − k − c
dt
This equation describes the uptake of oxygen
into oxymyoglobin, hence we can write:
f = k + se − k − c
Elimination
Because total enzyme in the slab reactor is
conserved we know that in the total free enzyme +
the total bound enzyme must = the total enzyme
before reactions began at time t=0
e + c = e0
e = e0 − c
Since e actually depends on c we can
eliminate the following equation from our
system
∂e
∂ e
= De 2 − f
∂t
∂x
2
Quasi Steady State
Recall our current system of equations, now that e has been
eliminated as a variable
∂s
∂2s
= Ds
− f
2
∂t
∂x
∂c
∂ 2c
+ f
= Dc
2
∂t
∂x
f = k + se − k − c
At Quasi Steady state we
assume rates of formation
and rates of breakdown are
the same, and that at steady
state the rates of change are
0.
∂s ∂c
+
=0
∂t ∂t
Substituting
Taking our previous equation and substituting
from our current system of equations we get
0 = Ds
∂2s
∂x 2
+ Dc
∂ 2c
∂x 2
∂c
∂
∂s
+ Dc
Ds
0=
∂x
∂x
∂x
∂s
∂c
+ Dc
=c
Ds
∂x
∂x
Fick’s Law Revisited
Recall that by Fick’s law
Flux = -(diffusion coefficient)*(spatial derivative of density)
Using this we can model the total oxygen flux J=J(x) at a
point as
∂c
∂s
+ Dc
J = − Ds
∂x
∂x
Integrating
ℑ=
L
0
ds
dc
− Ds
+ Dc
dx
dx
dx
ℑ = − Ds (s0 − s L ) − Dc (c0 − cL )
The quantities s0 and sl are known , but cl and
c0 are not!
We can find cl and c0 these by using
nondimensionalization .
Nondimensionalization
To further simplify and understand this system of
equations we need to define some new quantities.
k+
σ =
s
k−
c
u=
e0
x = Ly
Scaled
Substrate
Scaled Complex
Rescaled spatial
variable
More Nondimensionalization
If we use these with our current system of
equations we arrive at the following equation:
ε1σ yy = σ (1 − u ) − u = −ε 2u yy
where
ε1 =
Ds
2
e0 k + L
ε2 =
Dc
2
k− L
Using Experimental Data
Experimental data suggests the following epsilon values
ε1 = 1.5 *10
−7
ε 2 = 8.2 *10
These number are small enough that we
can approximate them with zero, leaving
us with the equation to solve for u
σ (1 − u ) − u = 0
σ
u=
σ +1
−5
Back Substituting
By back substituting and going through a LOT of
algebra we can get the following
k+
s
k−
σ
u=
=
σ +1
k+
k−
s +1
c
=
e0
c
s
=
e0 s + k −
k+
cL and c0 Revealed!
k−
Let K :=
k+
e0 s
c=
s+K
Now, if we use our boundary
conditions of c(0)=c0 and c(L)= cl
we get
e0 s 0
c0 =
s0 + K
e0 s L
cL =
sL + K
Total Oxygen Flux
Dc
Ds
(s0 − sL ) + (c0 − cL )
ℑ=
L
L
s0
Dc
Ds
sL
)
ℑ = (s0 − sL ) + e0 (
−
L K + s0 K + sL
L
=
Back substituting are
new values for cL and c0
into our equation for the
total flux of oxygen in
the slab leads to the
following
D
e0 K
Ds
(s0 − sL )(1+ c
)
Ds (s0 + K)(sL + K)
L
Ds
=
(1 + µρ )( s 0 − s L )
L
D c e0
ρ =
Ds K
K2
µ=
( s0 + K )(sL + K )
What does it all mean!?
Dan H. introduced us to the concept of cellular
homeostasis.
Mark A. developed the Diffusion Equation.
Mark O. developed a Biological version Ohm’s
Law.
Andy D. developed a general model of
Facilitated Diffusion, and walked us through a
case study of a Reaction Diffusion system.
Muscle Respiration
Definition – Muscle respiration refers to the
ability of muscle to extract oxygen from the
blood being delivered to it by the heart and
circulatory system.
This can be modeled using the Reaction
Diffusion system.
A Cylindrical Model
z
Cylindrical Coordinates:
(r, , z)
r
y
x
Muscle Fiber:
a cylindrical model
Assumption and Analogy
Physical Assumptions:
i) The Oxygen concentration is
fixed on the outside of the “fiber”.
ii) At steady state the oxygen (s)
and oxymyoglobin (c) are
distributed radially and
symmetrically about the z axis.
In effect, the substrate only
depends on r, rather than (r, , z).
Recall the Diffusion Equation
∂u
2
= D∇ u + f
∂t
Where u is the concentration at a point and time.
u = u ( x, y , z , t )
and the Laplacian of u is:
∂ u ∂ u ∂ u
∇ u= 2 + 2 + 2
∂x
∂y
∂z
2
2
2
2
Laplacian in Cylindrical
Coordinates
∂
1 ∂
1 ∂
∂
+ 2
∇ =
+
r
2
2
r ∂r ∂r
r ∂θ
∂z
2
2
2
Using our assumption of radial symmetry the Laplacian can be
rewritten as:
∂
1 ∂
∇ =
r
r ∂r ∂r
2
In the Absence of Oxygen
Consumption…
∂s
2
= Ds ∇ s − f
∂t
∂c
2
= Dc ∇ c + f
∂t
The rate of change of oxygen concentration with respect to
time = diffusion coefficient of the oxygen * Laplacian of the
oxygen concentration – uptake of oxygen into oxymyoglobin.
Note: f denotes the rate of uptake of oxygen into oxymyoglobin.
Oxygen Consumption
in Steady - State
∂s
2
2
= Ds ∇ s − f
0 = Ds ∇ s − f
∂t
1 ∂
∂s
Expanding the
0 = Ds
r
−f
Laplacian
r ∂r ∂r
Accounting for
oxygen
depletion
1 ∂
∂s
r
− f −g
0 = Ds
r ∂r ∂r
Oxymyoglobin
in Steady – State
∂c
=0
∂t
Expanding the
Laplacian
0 = Dc ∇ c + f
2
1 ∂
∂c
+f
r
0 = Dc
r ∂r ∂r
Modeling Muscle Respiration
Describing the rate of change of oxygen concentration during
diffusion:
1 ∂
∂s
r
− f −g
0 = Ds
r ∂r ∂r
Describing the rate of change of oxymyoglobin concentration during
diffusion:
1 ∂
∂c
0 = Dc
r
+f
r ∂r ∂r
Boundary Conditions
s = sa
∂c
=0
∂r
At r = a where a is the maximum radius
at r = a
∂s ∂c
=
=0
∂r ∂r
at r = 0
-a
0
a
What’s our Conclusion?
We’ve used mathematical models, in particular
the diffusion equation, to explain the metabolic
process of oxygen consumption in muscle
respiration. The final model we developed can
be used in combination with asymptoptic
methods to discern information about the steady
state behavior such processes.
We will be continuing our study of this topic next
semester.
Special Thanks
Dr. Steve Deckelman - (patience and pizza)
Dr. Ann Parsons - (expertise)
Mr. Adolph Fick - (great flux model)
Sir William of Ockham - (razor)
Mr. Johann Carl Friedrich Gauss - (divergence)
Mr. Pierre-Simon Laplace - (Laplacian)
References
Mathematical Physiology
Keener and Sneyd
Springer-Verlag, 1998
Thomas’ CALCULUS
Revised by Finney, Weir and Giordano
Addison Wesley Longman, 2001