Spatial diffusion
F7
Diffusion
Inhomogeneous concentration
some examples
role in biology
calcium diffusion
Communication, e.g. Ca-signaling
Macroscopic organization
general
compartmentalized
stochastic
interaction with buffers
Simplified neuron models
1D propagation in axons, growth of nails
2D diffusion in membranes
3D diffusion between cells or within cells
1
2
How can neurons find their
place?
Examples
calcium waves in fertilized egg cells
cAMP waves in amoeba movement
Chemoattractant towards target cells
Chemoattractant from growth cones causes bundling
Chemorepellant from growth cones, dependent on target
cell chemoattractant, causes dispersion near target
Cone growth towards chemical gradient.
Drosofila (fruit fly) egg
red=ftz gene
blue=eve gene
3
Diffusion speed
4
Diffusion time
glucose (192 u) 660 [D/107 cm2/s]
hemoglobin (64500 u) 6.9 [D/107 cm2/s]
tobacco mosaic virus (40590000 u) 0.46 [D/107
cm2/s]
5
10nm (cell membrane) 0.1s
10m (small cell) 100ms
1mm ???
1m (length of longest axon)
6
Diffusion time
Diffusion time
10nm (cell membrane) 0.1s
10m (small cell) 100ms
1mm 16.7 min
1m (length of longest axon) 32 years
10nm (cell membrane) 0.1s
10m (small cell) 100ms
1mm 16.7 min
1m (length of longest axon) ???
Need for active transport!
7
Reaction-Diffusion Systems
Self Organisation
An important class of pattern forming systems consists
of chemicals diffusing and reacting with each other.
If the interactions are nonlinear and the diffusion
coefficients different, interesting instabilities result.
Can be modelled with partial differential equations or
cellular automata.
dx i
dt
=f x1 ,x 2 , x n +D i 2 x i
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Complex patterns can be produced by application of
simple rules to simple initial states.
In biological systems chemical signals combined
with non-linear local interaction are able to produce
very complex patterns.
Symmetry breaking:the initial undifferentiated state
is unstable. Feedback amplifies deviations, turning
noise into pattern.
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Fish Patterns
10
Two interacting species
A basic example with two morphogens A and
B (lab 3a). Let a and b be their deviations
from equilibrium in each cell:
da
=c 1 ac 2 br a a3 D a 2 a
dt
db
=c3 ac 4 br b b3 D b 2 b
dt
+A
11
-
+B
12
Belousov Zhabotinsky reaction
Calcium diffusion
0.2 M Malonic Acid
0.3 M Sodium Bromate
0.3 M Sulfuric Acid
.005M Ferroin
Combine to form a solution. Add approximately 5 mL of the
solution to a Petri dish, 6 cm in diameter, so that the thickness of the
layer is 0.5 -1 mm. Watch until colorful spatio-temporal patterns
emerge. In thicker layers there is an interference of hydrodynamic
flows with the reaction.
13
14
Calcium metabolism in neurons
Calcium is a vital second messenger
Ca2+ plays an important role in practically every cell type.
Ca2+ controls secretion, cell movement, muscular contraction, cell differentiation, ciliary beating, and so on.
Important in both excitable and non-excitable cells.
Well regulated in cells. Toxic at high concentrations.
Can activate Ca-sensitive ion channels
15
Summary of calcium homeostasis
Ca
2+
PM pumps
RyR
ER
Typical Calcium
Oscillations
A: Hepatocytes (liver)
B: Rat parotid gland (gland)
ICa
Ca2+
C: Gonadotropes (gonad)
leak
IPR
D: Hamster eggs (post-fertilization)
Ca -B
(buffering)
2+
serca
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E, F: Insulinoma cells (pancreas)
Mitochondria
17
18
Calcium dynamics: exponentially decaying pool
I
d [ Ca2+ ]
= Ca [ Ca2+ ] [ Ca 2+]min dt
2 Fv
Generic modeling
Total buffer
Set up a typical reaction diffusion equation for calcium:
C
=D 2 C JIPR J RyR J Serca J leak JPM JI Jm,out Jm,ink 1 C
bt bk 2 b
t
mitochondria
buffering
l
fluxes
This reaction-diffusion equation is coupled to a system of ODEs (or PDEs), describing the various sources and sinks of a species.
ER fluxes
PM fluxes
Advantage: only three parameters
[Ca2+]min
baseline calcium concentration ~50nM.
decay rate constant, summarises diffusion, buffering,
pumps and exchangers.
v
volume of calcium pool, usually submembrane shell
(relevant for activation of KCa channels).
The specifics of the coupled ODEs depend on which particular model is being used.
Limitations:
Sometimes the PM fluxes appear only as boundary conditions, sometimes not, depending on the exact assumptions made about the spatial
properties of the cell.
In general the buffering flux is a sum of terms, describing buffering by multiple diffusing buffers.
Not possible to study calcium dependent processes in the cytoplasm
(e.g. calcium induced calcium release CICR).
Different KCa channels sense different [Ca2+]?
Possible extension:
19
Calcium diffusion: spatial discretization
20
Use several calcium pools with different i.
Calcium diffusion
Spatial discretisation depending on structure of compartment and
location of calcium influx. Divide cylindrical compartments into shells,
spines into stacked cylinders.
Assume uniform calcium flux across compartment membrane only need to consider radial (1D) diffusion.
Fick´s first law: flux JCa (in mol/s) through area a is proportional to
negative concentration gradient:
J Ca =- aD Ca
J
[ Ca 2+]
x
Discretised concentration change in volume vi:
J
[ Ca 2+] i,t+1[ Ca 2+]i,t
t
ai,i+1 [ Ca 2+ ]i+1,t [ Ca2+ ]i,t
vi
x
Only parameter: diffusion constant DCa. Depends on ion size and
medium, e.g. DCa (water) ~ 600 µm2/s, DCa (cytoplasm) ~ 200 µm2/s.
Limitations:
Localised calcium fluxes local calcium domains.
Calcium induced calcium release calcium waves, calcium sparks.
Model 3D diffusion (computationally expensive).
= D Ca
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22
Stochasticity in intracellular Ca dynamics
Computationally Efficient
Stochastic Diffusion Algorithm
Intracellular pathways can involve very small numbers of ions
or molecules.
100 nM [Ca2+] = 5 calcium ions in a spine head with 0.5 m diameter.
Use stochastic simulation methods instead of mass action kinetics.
AB AB
k
Subdivide dendrites and spines
into sub-volumes
Probability that one molecule of A
changes to AB during the time t:
k
1
2
k 1 t
p A AB=1e
Pre-calculate the probability that one molecule leaves the
compartment
Look-up tables store the probability that k out of N molecules
leave a compartment
The number, k, molecules leaving out of N is determined with
single random number
Stochastic methods are computationally expensive
Use adaptive stochastic methods that switch to deterministic
calculations for large numbers of molecules (Vasudeva & Bhalla, 2003).
Representation of spatial microstructure: use Monte-Carlo simulation
where all molecules are represented individually and perform random
walks. Simulation software: MCell
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24
Calcium influx and efflux
Generic modeling
Total buffer
Set up a typical reaction diffusion equation for calcium:
C
=D 2 C JIPR J RyR J Serca J leak JPM JI Jm,out Jm,ink 1 C
bt bk 2 b
t
ER fluxes
mitochondrial
fluxes
PM fluxes
buffering
This reaction-diffusion equation is coupled to a system of ODEs (or PDEs), describing the various sources and sinks of a species.
The specifics of the coupled ODEs depend on which particular model is being used.
Plasma membrane
Ca2+ ATPase
3 Na+ / Ca2+ exchanger
Electrogenic affect resting membrane potential and excitability.
Intracellular calcium stores
Ca2+ ATPase
Calcium induced calcium release CICR
IP3 induced calcium release IICR
Calcium dependent waves, sparks, coincidence detection…
Ca2+ ATPase model
Sometimes the PM fluxes appear only as boundary conditions, sometimes not, depending on the exact assumptions made about the spatial
properties
of the cell.
Time
dependent
or
In general the buffering flux is a sum of terms, describing buffering by multiple diffusing buffers.
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Calcium buffers
1
2
d [Ca 2+]
=k 1 [Ca2+ ][B]+k 2 [CaB]+J
dt
Four parameters for each buffer:
Forward and backward rate constants k1 and k2.
Total concentration [B]T = [B] + [CaB].
Diffusion constant DB.
Derived parameters (assume rapid
buffer approximation, i.e. steady-state):
Dissociation constant
Buffer capacity
K n[Ca 2+ ]n
dJ J J
=
dt J
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EBA appropriate when the saturability of mobile buffer is negligible.
For example, this is the case for millimolar concentrations of
Calbindin-D28K in the saccular hair cell.
RBA appropriate when there is significant saturability of mobile
buffer and when buffer kinetics are fast relative to Ca2+ diffusion.
This is often the case near Ca2+ channels in synapses, and near
IP3 or ryanodine receptors in the ER/SR.
Smith et al. (2001) did an asymptotic analysis of buffered Ca2+
diffusion near a point source, and determined mathematical
conditions for when RBA or EBA are appropriate.
[ Ca2+ ][ B ] k 2
=
[ CaB]
k1
K D [ B ]T
[B]
=
=
T
d [ Ca2+ ] K [ Ca 2+] 2 K D
D
K D=
d [ CaB ]
(between 50 and 4000 in neurons)
Buffering factor
steady state
V max [Ca 2+ ] n
EBA vs. RBA
k
J Ca2+ B CaB
k
For each diffusion shell
and each buffer:
J =
=
d [ Ca2+ ]
d [ Ca2+ ]T
=
1
1 +
lim r 0 BBbk (EBA)
buffer unperturbed
lim r 0 B0 (RBA)
buffer saturates
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28
HH dimension reduction
Simplified neuron models
Each compartment gives one ODE
Each ion channel gives one (e.g. K) or two
(e.g. Na) ODEs.
Simple Ca-diffusion gives one ODE
very fast variable (Na act) at s-s
fast variable u (membrane potential)
slow variable w (Na inact + K act)
du
1
= u u³ w I
dt
3
dw
= eb0 b1 u w
dt
Expensive to simulate
Needs many parameters
29
time in units of , ratio of time scales e=/w
30
Integrate and fire IF
Focuses on the capacitive property of the
membrane
Assumes the action potential acts like a “reset”
VT
dV
=V t R I t
dt
=R C
if V V T set V =V reset
Vreset
t
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32
Calcium excitability
Example: Diffusion of Calcium or Second
Messengers in Dendrite with Spines
Both IPR and RyR release calcium in an excitable manner. They both respond to a calcium challenge by the release of even more calcium
The precise mechanisms are not known for sure (although detailed models can be constructed).
Spines Concentrate Molecules
An IPR behaves like a Na+ channel (in some ways). In response to an increase in [Ca
Large fluctuations in small compartments
A lot of attention has been focused on IPR and RyR. Less on pumping. But the dynamics of pumping is equally important.
Spine
1.75
7.75
Dendrite
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